Post on 02-Nov-2020
Hindawi Publishing CorporationISRN Signal ProcessingVolume 2013 Article ID 605035 18 pageshttpdxdoiorg1011552013605035
Research ArticleAbout a Partial Differential Equation-Based Interpolator forSignal Envelope Computing Existence Results and Applications
Oumar Niang123 Abdoulaye Thioune24 Eacuteric Deleacutechelle2
Mary Teuw Niane3 and Jacques Lemoine2
1 Departement Informatique et Telecommunications Ecole Polytechnique de Thies (EPT) Thies BP A10 Senegal2 Laboratoire Images Signaux et Systemes Intelligents (LISSI-EA3956) Universite Paris-Est Creteil Val-de-Marne Creteil France3 Laboratoire drsquoAnalyse Numerique et drsquoInformatique (LANI) Universite Gaston Berger (UGB) Saint-Louis BP 234 Senegal4Departement de Mathematique et Informatique Faculte des Sciences et Technique Universite Cheikh Anta Diop de DakarDakar BP 5005 Senegal
Correspondence should be addressed to Oumar Niang oniangucadsn
Received 6 November 2012 Accepted 3 December 2012
Academic Editors H Hu and S Li
Copyright copy 2013 Oumar Niang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper models and solves the mathematical problem of interpolating characteristic points of signals by a partial differentialEquation-(PDE-) based approach The existence and uniqueness results are established in an appropriate space whose regularity issimilar to cubic spline one We show how this space is suitable for the empirical mode decomposition (EMD) sifting processNumerical schemes and computing applications are also presented for signal envelopes calculation The test results show theusefulness of the new PDE interpolator in some pathological cases like input class functions that are not so regular as in the cubicsplines case Some image filtering tests strengthen the demonstration of PDE interpolator performance
1 Introduction
Interpolators are widely used in signal and processing or dataanalysis In particular for the empirical mode decomposition(EMD) algorithm [1 2] the iterative estimation of the signaltrend is based on the computing of the envelopes obtainedby the cubic spline interpolation of local extrema The splineinterpolation has been recognized as being very effective forEMD But for signals that have no local extremum the cubicspline interpolation fails We proposed a PDE-based modelwhich overcomes this limit of classical EMD implementationin the computing of the envelopes for signals that haveno local extremum [2ndash4] Recently the Spectral IntrinsicDecomposition (SID) method [5] based on the spectraldecomposition of the PDE interpolator provides a newapplication of our model This PDE interpolator contributeto the mathematical modeling of the EMD and has providedvarious applications in signal and image processing [4 6]In this paper we describe the mathematical modeling of thenew PDE interpolator by variational methodsThe resolutionof the variational problem leads to existence and uniqueness
results in appropriate spaces The paper is organized asfollows In Sections 2 and 21 recalls some PDE models insignal and image processing and in Section 22 some math-ematical preliminary notions are set out In Section 3 themathematical modeling is exposed In Sections 4 and 5 theresolution of the variational problem is dealt Subsequentlynumerical implementation and applications are presented inSection 6 At last we finish by conclusions
2 Some Preliminaries
21 PDE Models in Signal and Image Processingand EMD Principle
211 Some Diffusion Equations This part consists of a briefand nonexhaustive presentation of classical nonlinear diffu-sion filters for 1D and 2D signal processing with more focuson the 2D case In this purpose we can recall the modelfor nonlinear diffusion in image filtering proposed by Catteet al [7] This filter is a modified version of the well-know
2 ISRN Signal Processing
Perona andMalik model [8]The basic equation that governsnonlinear diffusion filtering is
119906119905(x 119905) = div (119892 (|nabla119906 (x 119905)|2) nabla119906 (x 119905)) (1)
with x = (1199091 119909
2) and where 119906(x 119905) is a filtered version of the
original image 119906(x 119905) = 1199060(x) as the initial condition with
reflecting boundary conditions In (1) 119892(sdot) is the conductivity(or diffusivity) function which is dependent (in space andtime) on the image gradient magnitude Several forms ofdiffusivity were introduced in the original paper of Peronaand Malik [8] All forms of diffusivity are chosen to be amonotonically decreasing function of the signal gradientPossible expressions for conductivity are
1198921(x 119905) = 1
1 + (|nabla119906 (x 119905)| 120573)2
1198922(x 119905) = exp (minus(|nabla119906 (x 119905)| 120573)2)
(2)
Parameter 120573 is a threshold one which influences the aniso-tropic smoothing process The nonlinear equation (1) actsas a forward parabolic equation smoothing regions whilepreserving edges Other methods based on high-order PDEare provided for image restoration like in [9ndash11] Efficientnumerical schemes were introduced in [12] based on additiveoperator splitting (AOS) scheme or based on alternatingdirection implicit (ADI) scheme See [12ndash15] for a review andextensions of these methods Unlike these methods of high-order PDE that are specially developed for denoising ourmodelwas constructed to interpolate the characteristic pointsof a signal
The major problem of nonlinear diffusion-based processis that it is generally difficult to correctly separate the highfrequency components from the low frequency ones In caseof denoising applications the objective of this process is touse the diffusivity function as a guide to retain useful dataand suppress noise
Numerous authors have proposed fourth-order PDEsfor image smoothing and denoising with the hope thatthese methods would perform better than their second-orderanalogues [16ndash18] Indeed there are good reasons to considerfourth-order equationsThefirst reason that can be retained isthe fact that fourth-order linear diffusion damps oscillationsat high frequencies (eg noise) much faster than second-order diffusion On the other hand the theory of fourth ordernonlinear PDEs is far less developed than the second orderone Also such equations often do not satisfy a maximumprinciple or comparison principle and implementation ofthe equations could thus introduce artificial singularities orother undesirable behavior In recent studies Tumblin [19]Tumblin and Turk [16] and Wei [17] proposed the followingform
119906119905(x 119905) = minus div (119892 (119898 (119906)) nablaΔ119906 (x 119905)) (3)
where 119892(sdot) = 1198921(sdot) as in (2) and 119898 is some measurement of
119906(x 119905) In [16] (3) is called a ldquolow curvature image simplifierrdquo(LCIS) and a good choice for 119898 is defined as 119898 = Δ119906 to
enforce isotropic diffusion [19] These PDE tools for digitalsignal and image processing make more reachable the 2Dextension of the 1D PDE-based method for characteristicpoints interpolation presented in this paper
212TheEmpiricalModeDecomposition Principle TheEMD[1] method decomposes iteratively a signal into amplitudemodulation-frequency modulation (AM-FM) type compo-nents called intrinsic mode functions (IMF) The underlyingprinciple of this decomposition is to locally identify in thesignal the most rapid oscillations defined as the waveforminterpolating interwoven local maxima and minima To dothis local extrema points are interpolated with a cubicspline to yield the upper and lower envelopes The meanenvelope (half sum of upper and lower envelopes) is thensubtracted from the initial signal and the same interpolationscheme is reiterated on the remainder The so-called siftingprocess stops when the mean envelope is reasonably zeroeverywhere and the resulting signal is called the first IMFThe higher-order IMFs are iteratively extracted applying thesame procedure to the initial signal after the previous IMFshave been removed
22 Some Useful Mathematical Concepts Throughout thepaperΩ denotes an open-bounded subset ofR119899 119899 = 1 or 2In the sequel we will need the following definitions andresults
Definition 1 we define one the spaces 119881 andV as follows
119881 = 119907 isin 1198673(Ω) |
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
V = 119907 isin 1198673(Ω) |
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω
open and verifying int
ΓΩ
119907119889119909 = 0 and 120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(4)
Definition 2 For a given function 119907 isin 1198671(Ω) if119863
119889(119907)(119909) and
119863119892(119907)(119909) denote respectively the right and left derivative of
119907 at 119909 the minmod derivatives of 119907 is define by
120597119907
120597119909
(119909) = minmod (119863119892(119907) (119909) 119863
119889(119907) (119909)) where
minmod (119901 119902) =
sign (119901)min (1003816100381610038161003816119901100381610038161003816100381610038161003816100381610038161199021003816100381610038161003816) if 119901119902 gt 0
0 if 119901119902 le 0
(5)
A 1198671(Ω) function being absolutely continuous admits right
and left derivatives then 1199060isin 119867
1(Ω) has obviously left
and right derivatives so that we can validate numericallycomputing of the diffusivity function 119892 defined in Section 3
ISRN Signal Processing 3
Definition 3 Let (119909119899)119899isinN be a sequence of elements in a vec-
torial normed space (119864 sdot 119864) it is said that (119909
119899)119899isinN converge
weakly [20] in 119864 and noted by 119909119899 119909 if exists an element
119909 isin 119864 such that forall119891 isin 1198641015840 lim
119899rarrinfin119891(119909
119899) = 119891(119909) where 1198641015840
denotes the set of continuous linear forms on 119864
Definition 4 (the Green formula) Let 119906 119907 isin 1198622(Ω) then
int
Ω
Δ119906119889119909 = int
120597Ω
120597119906
120597119899
119889119904
int
Ω
nabla119906nabla119907119889119909 = minusint
Ω
119906Δ119907119889119909 + int
120597Ω
120597119907
120597119899
119889119904
int
Ω
(119906Δ119907 minus 119907Δ119906) 119889119909 = int
120597Ω
119906
120597119907
120597119899
minus 119907
120597119906
120597119899
119889119904 Green formula
(6)
where 120597119906120597119899 denotes the normal derivative of 119906 on theboundary 120597Ω ofΩ
Definition 5 (the Poincare-Wirtinger inequality) LetΩ be anopen-bounded set and let 119906 isin 119867
1(Ω) then there exists a
constant 119862 gt 0 such that the norm of 119906 in 1198671(Ω) and the
norm of 119906 in 1198712(Ω) are linked by the following inequality
10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119904
100381710038171003817100381710038171003817100381710038171198671(Ω)
le 119862nabla1199061198712(Ω) (7)
where |Ω| denotes the length ofΩ The best constant 119862 in thePoincare-Wirtinger inequality is 1120582 for example the inverseof the first positive eigenvalue 120582 of the Laplace operator withhomogenous Neumann boundaries conditions In our caseΩ sub R and |Ω| is the diameter ofΩ
Theorem 6 (regularity theorem) Let
119906 isin 119906 isin 1198673(Ω) 119904119906119888ℎ 119905ℎ119886119905
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119906119889119909 = 0 (8)
then there exist two strictly positive reals 119862 and 119870 such thatfirstly
nabla1199062
1198712(Ω)
le 119862Δ1199062
1198712(Ω) (9)
and secondly
1199062
1198673(Ω)
le 119870Δ1199062
1198671(Ω) (10)
The main existence and uniqueness result of the solutionof our henceforth problem (29) is due to the application ofthe following Lions theorem [21]
Theorem 7 (the Lions theorem) Let119867 and 119881 be two Hilbertspaces with 119881 sub 119867 One considers a bilinear application 119886 on119881times119881 and 119861 an operator on119881 Under the following conditions
(1) exist120583 and 120588 gt 0 such that |119886(119906 119907)| ge 1205881199072119881minus 120583|119907|
2
119867
(2) exist 1198882gt 0 such that |(119907 119861119907)
119881| le 119888
2|119907|
2
119881
For 1199060= 119906(0) isin 119867 the problem
119906 isin 1198712(0 119879 119881) cap 119862 (0 119879119867) (11)
such that
forall119907 isin 119881 (
119889119906
119889119905
119861119907)
119867
+ 119886 (119906 119907) = 0 (12)
and 1199060= 119906(0) has a unique solution
Furthermore 119889119906119889119905 isin 1198712(0 119879119867
1015840)
3 Mathematical Modeling of the NewPDE-Interpolator
Recall that the PDE model aimed to contribute to themathematical modeling of the EMD as it is in [2] In a firststep to be in line with the classical EMD our goal is tomodel the upper or lower envelope as the asymptotic solutionof a PDE system whose initial condition is the input signalthat we want to interpolate local extrema (or more generallycharacteristic points) Initially this envelope is obtained bycubic spline interpolation Let Ω be the open domain of angiven finite energy signal 119906
0isin 119871
2(Ω) We construct an
operator 119860 of appropriate domain119863(119860) as follows
120597119906
120597119905
+ 119860119906 = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(13)
The need of asymptotic solution existence denoted by 119906infin
implies (120597119906120597119905)(infin 119909) = 0 for example according to (13) wehave 119860119906(infin 119909) = 0 Moreover the requirement of the sameregularity between the solution and a cubic spline leads to1205974119906120597119909
4(infin 119909) = 0 Thus in the first analysis we can choose
119860 = 1205974119906120597119909
4 To leave invariant features points during thediffusion process simply multiply in the expression 119860119906 theterm 120597
4119906120597119909
4 by function 119892 depending on spatial variable 119909and vanishing at characteristic points That gives
(119860119906) (119909) = 119892 (119909)
1205974119906
1205971199094 (14)
Another reminiscent form of long-range diffusion [22] is thefollowing
(119860119906) (119909) =
120597
120597119909
(119892 (119909)
1205973119906
1205971199093) (15)
Function 119892 can be interpreted as a diffusivity function whoserole is to control the diffusion process We take it necessarilypositive to have a direct diffusion and the existence ofsolution Other forms of operators are possible [3] we presentsome ones as follows for (14)
(119860119906) (119909) = 119892 (119909) [minus120579
1205972119906
1205971199092+ (1 minus 120579)
1205974119906
1205971199094] (16)
4 ISRN Signal Processing
or for (15)
(119860119906) (119909) =
120597
120597119909
[119892 (119909) (minus120579
120597119906
120597119909
+ (1 minus 120579)
1205973119906
1205971199093)] (17)
or accomplete diffusion involving a flow
(119860119906) (119909) =
120597
120597119909
[minus120579119892 (119909)
120597119906
120597119909
+ (1 minus 120579)
120597
120597119909
(119892 (119909)
1205972119906
1205971199092)]
(18)where 0 le 120579 le 1 is the tension factor which controls theregularity of the solution
Both forms allow more freedom on the regularity ofthe solution 119906 To access the mathematical properties of thesolution we chose the second form in (17) with 120579 = 0 and thecorresponding equation (15)
At the local scale between two consecutive character-istic pointsmdashtwo local maxima for examplemdashthe diffusioninduces a smoothing phenomenon that deletes the local min-imum A simple form for 119892 to calculate the envelope is givenby a positive piecewise function lower than 1 that is constantbetween two characteristic points of 119906
0and zeroed only at
these points Characteristic points are often being definedby their values and the signs of first second and third localderivatives of119906
0We characterize the function119892 as depending
on sign(1205971199060120597119909) sign(1205972119906
0120597119909
2) and sign(1205973119906
0120597119909
3)
For the purpose of existence and regularity of the solu-tion we are led to work with the regularized version of sign
sign120599(119911) =
2
120587
arctan(120587119911120599
) (19)
where 120599 is a regularization coefficientIn the following we define 119892plusmn
119890for extrema detection
119892119891for turning points detection and 119892plusmn
mc for maximum andminimum curvature points detection
119892plusmn
119890(119909) =
1
9
[
10038161003816100381610038161003816100381610038161003816
sign(120597119906
0
120597119909
)
10038161003816100381610038161003816100381610038161003816
plusmn sign(12059721199060
1205971199092) + 1]
2
(+) for maxima and (minus) for minima
119892119891(119909) = [sign
120599(
12059721199060
1205971199092)]
2
for turning points
119892plusmn
mc (119909) =1
9
[
100381610038161003816100381610038161003816100381610038161003816
sign(12059731199060
1205971199093)
100381610038161003816100381610038161003816100381610038161003816
plusmn sign(12059721199060
1205971199092) + 1]
2
(20)
(+) for maximum curvature points and (minus) for minimumcurvature points All these functions are of the form 119892(119909) =
[ℎ(119909)]2 with ℎ(119909) = 0 at characteristic points
So we have 1198921015840(119909) = 0 if 119892(119909) = 0 This property formally
allows us to cancel119860119906 and 120597119906120597119905 of the form (15) at the pointswhere 119892 is null The factor 19 is a normalization term andverifies 0 le 119892(119909) le 1 A more general diffusivity function 119892could be envisaged
Let us consider the following differential operators119860119890and
119860119891(119890 for upper and 119891 for inflexion or curvature)
forall119906 isin 119863 (119860119890) 119860
119890119906 =
120597
120597119909
(119892plusmn
119890(119909)
1205973119906
1205971199093) (21)
or
forall119906 isin 119863 (119860119891) 119860
119891119906 =
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) (22)
If 1199060isin 119862
2(Ω) as it is the case in the classical imple-
mentation EMD there is no problem for interpolation bycubic spline method But it is indeed a restriction on theregularity of 119906
0which is due to the choice of the interpolation
technique that requires a good detection of local extremafor the envelopes calculation The basic EMD principle mustapply for input functions that are not regular as functions in119862
2(Ω) for example for signal 119906
0isin 119867
1(Ω) as it is the case
in reality [23] But numerically without even a regularizationfunction sign if the derivatives are taken in the sense ofminmod called flux limiter or slope the function 119892(119909) stillhas a meaning
31 Interpretation of the Diffusivity Action in the DiffusionProcess Between two consecutivemaxima119883max119894 and119883max119894+1 the function 119892+
119890is piecewise constant and can be written as
follows
119892 (119909) =
0 at 119883max119894 119883max119894+1
4
9
at 119883inf 119894 119883inf 119894+1 119883min119894
1 on ]119883inf 119894 119883min119894[ cup ]119883min119894 119883inf 119894+1[
1
9
sur ]119883max119894 119883inf 119894[ cup ]119883inf 119894+1 119883max119894+1[
(23)
We have a diffusion in the sense that 119906119905= 120597119906120597119905 unhook
to the maximum of the curve 1199060 The reason is that the
diffusion is more pronounced in local minimum becausethe smoothing effect tends to regularize curves Thus theupper envelope 119906(119905 cong infin sdot) of 119906
0is less oscillating than
1199060is The same behavior occurs for the mean envelope We
would then try to interpret locally the relationship 119906 ge 1199060
by evoking the maximum principle [24] But this principleis not immediately checked for equations of the type (15)The justification lies in the fact that smoothing impliesnabla119906(119905 sdot)
1198712(Ω)
le nabla11990601198712(Ω)
In summary we have just built a PDE system
(1) having the input signal 1199060to decompose as an initial
value condition
(2) such that the result of a diffusion process preservescharacteristic points of 119906
0
(3) with a regularity similar to a cubic spline
32 Formulation of the Mathematical Problem Let 119860 be oneof the above constructed interpolation operators and 119879 gt 0
be a diffusion time sufficiently long To fix ideas we solvethe problem of upper envelope calculus Indeed for the otherkinds of characteristic points the same problem arises withanalogous mathematical resolution methods
ISRN Signal Processing 5
The problem posed by our mathematical modeling is tofind a solution 119906 of the system
120597119906
120597119905
+ 119860119906 = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(24)
Then the mean envelope (interpolating turning points) cal-culus is stated as follows
For given 1199060 find 119906 such that
120597119906
120597119905
+
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(25)
33 Comments on theDegeneracy in Problem (24) To preventthe zero degeneracy case in the system (25) whose solution isnot directly accessible by variational methods we are led tosolve an intermediate problem with
0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)
That is the nondegenerate problem defined hereafter
34 Formulation of the Zero Degeneracy Induced byDiffusivityFunction Thediffusivity function in (26) is very close to zeroat the characteristic points while refraining to cancel and tosolve the system (25) now just to move on to the limit when120572 tends to zero to retrieve (25) and its solution
Thus in particular we can define the function
119892119891119899(119909) = 119892
119891(119909) +
1
119899
119899 isin Nlowast (27)
which verifies the conditions of (26)The new system that we call nondegenerate problem is
fomulated as followsFind 119906
119899such that
120597119906119899
120597119905
+
120597
120597119909
(119892119899(119909)
1205973119906119899
1205971199093) = 0 in [0 119879] times Ω
119906119899(0 119909) = 119906
0in Ω
plus boundaries conditions
(28)
35 Formulation of the Nondegenerate Problem For 119899 suffi-ciently large 119906
119899is a close approximation of the envelope 119906
interpolating the characteristic pointsBy a density technique we can demonstrate that this
sequence (119906119899)119899isinN converges to the solution of the system (25)
By posing for the convenience of notation 0 lt 120572 le 119892119891119899
= ℎ le
120573 le 1 the non-degenerate problem is reworded as
Find 119906 such that
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(29)
Equation (29) is an operational differential equationstype The techniques for the resolution of this kind ofproblem are numerous we mainly refer to the variationalformulation to find weak solutions [25] and the method ofthe approximation of evolutionary operators The last allowsto work on a new elliptic problemThe solution is obtained bypassing to the limit of the approximate solution nstead of theequation
In the next section we solve (29) by a variationalapproach according to the resolution methods for parabolicproblems described in [7 20]
4 Existence and Uniqueness of Solutions forPDE Interpolator
41 The Guideline for the Resolution of the MathematicalProblem Now we summarize the general procedure to solvethe problem
(1) First step resolution of the problem (29) This stepincludes
(i) the variational formulation in Section 43(ii) the resolution of the variational problem in
appropriate functional space(iii) the converse showing that the solution of the
variational problem solves the departure prob-lem in Section 434
(2) Second step in Section 5 construction of the sequ-ence (119906
119899)119899isinN of solutions of problem (29) where 119906
119899
is the solution obtained with the diffusivity ℎ119899(119909) =
ℎ(119909) + (1119899) 119899 isin Nlowast see Section 5 Finally demon-strate that the sequence (119906
119899)119899isinN converges to a limit 119906
where 119906 is the solution of problem (25) in Section 5
42 The Main Results Formulation The main result which isgoing to be solves is the following
Theorem 8 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909+int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V
(30)
such that 119906(0) = 1199060and 119889119906119889119905 isin 1198712
(0 119879119867) Reversely let 119906be a solution of the variational problem (30) then (120597119906120597119905) +(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3)) = 0
And the complete follows result
6 ISRN Signal Processing
Theorem 9 For 1199060isin 119867
1(Ω) exists a unique solution
119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867
1(Ω)) (31)
verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(32)
43 Variational Formulation Let 119907 isin 1198673(Ω) and (to make
sense to integrals) multiply the equation
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (33)
of the system (29) by the test function 1205972119907120597119909
2 after weintegrate onΩ then it comes out that
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909 = 0 (34)
By using theGreen formula (seeDefinition 4) and integrationby parts we get firstly
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909
= [ℎ (119909)
1205973119906
1205971199093
1205972119907
1205971199092]
120597Ω
minus int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
(35)
and secondly
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
119906
1205972119907
1205971199092119889119909
int
Ω
119906
1205972119907
1205971199092119889119909 = int
120597Ω
119906
120597119907
120597119899
119889120590 minus int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(36)
As we are in one-dimension case the normal derivative andthe gradient on the edge in the use of the Green formula arethe same that is to say
int
120597Ω
119906
120597119907
120597119899
119889120590 = int
120597Ω
119906
120597119907
120597119909
119889120590 (37)
In the following we adopt the notation 120597119907120597119899Considering 119906 isin 1198673
(Ω) such that
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
119907 isin 119881 = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(38)
it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
forall119907 isin 119881
(39)
Let us consider the following bilinear forms
119886 (119906 119907) = int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
119887 (119906 119907) = int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(40)
The addition of boundaries conditions transforms our initialproblem into the system
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
119906 (0 119909) = 1199060
in Ω
(41)
The variational formulation of our problem consequentlysubject to additional conditions that would cause a change ofspace is the following
Find 119906 isin 119881 such that 119889
119889119905
(119887 (119906 119907)) + 119886 (119906 119907) = 0
forall119907 isin 119881
(42)
where 119886 and 119887 are like in Theorem 7This kind of problem is studied in [21] However before
continuing the resolution we explore some properties of thespace 119881
431 On the Quotient Space 119881R The space 119881 is defined by119863(119886) cap 119863(119887) where 119863(119886) and 119863(119887) are the domains of 119886 and119887 Let us consider the quotient space
119881
R= 119907 + 119888 119888 isin R such that 119907 isin 1198673
(Ω)
and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(43)
and the new spaceV defined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (44)
Let 119907 isin 119881R and let 119906 be a representative of 119907 class then(120597119907120597119899)|
120597Ωis defined by
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597 (119906 + 119888)
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 with 119888 isin R (45)
ISRN Signal Processing 7
Thenullity ofintΩ119907119889119909meansexist119888 isin R constant such thatint
Ω(119906+
119888)119889119909 = 0 with 119906 one representative of 119907
if 119888 = minus 1
|Ω|
int
Ω
119906119889119909 then
int
Ω
119907119889119909 = int
Ω
(119906 + 119888) 119889119909
= int
Ω
(119906 minus
1
|Ω|
int
Ω
119906119889119909)119889119909 = 0
(46)
So for all 119907 isin 119881R we have intΩ119907119889119909 = 0 Accordingly any
element of the quotient space 119881R is on one hand in1198673(Ω)
with normal derivative on the boundaries of Ω null and onthe other hand is zero average onΩThus the quotient space119881R is identified with the space V We can work later inthe space V with zero mean The final variational problemto solve is the following
find 119906isinV such that 119889
119889119905
(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V
(47)
432 Application of the Lions Theorem To apply the Lionstheorem we need to identify the applications 119886 119887 and 119861 andthe spaces119867 and119881 And finally wemust prove that the normdefined inV which derives from the scalar product given bythe bilinear application 119886 is equivalent to the norm of1198673
(Ω)Let us define 119861 as the identity and let us define 119886 and 119887 by
|119886(119906 119907)| = (radicℎ(1205973119906120597119909
3) radicℎ(120597
3119907120597119909
3))
1198712(Ω)
and 119887(119906 119907) =intΩ(120597119906120597119909)(120597119907120597119909)119889119909 obtained from (39) (40) the spaces
119881 = V and119867 are define as follows(a) The linear operator 119861 in the variational prob-
lem (11) in the Lions theorem (119889119906119889119905 119861119907)119867
denotes ascalar product on 119867 Indeed we define 119867 = 119907 isin
1198671(Ω) such that (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 = 0
The operator 119861 = 119868 119861119907 = 119907 is the identity whichis continuous (verify the Lions conditions Theorem 7) andthe term (119889119906119889119905 119861119907)
119867= (119889119889119905)(119887(119906 119907)) is equal to
int
Ω
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
119889119909 = int
Ω
119889
119889119905
(
120597119906
120597119909
)
120597119907
120597119909
119889119909
= (
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
)
1198712(Ω)
(48)
In this identification ((120597120597119909)(119889119906119889119905) 120597119907120597119909)1198712(Ω)
is a wellscalar product in the subspace 119867 of 1198671
(Ω) formed byfunctions with zero mean Indeed according to Poincare-Wirtinger inequality (see Definition 5) in1198671
(Ω)10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119909
100381710038171003817100381710038171003817100381710038171198671(Ω)
le
1
120582
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(49)
where 120582 is the first positive eigenvalue of Laplace operator inΩ But as 119906 119907 isin V then int
Ω119906119889119909 = 0 consequently there
exists 1198622gt 0 such that
1199061198671(Ω)
le 1198622
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(50)
that is to say the 1198671(Ω) norm is equivalent to the gradient
norm inV Hence one can replace the scalar product of thegradient by1198671
(Ω) scalar productHereinafter we complete the proof that the precedent
choices allow the application of the Lions theorem in ourcontext
(b) Vis an Hilbert space with equivalent norm to 1198673(Ω)
norm
Theorem 10 The spaceV resulting from (11) andV analysisdefined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (51)
is an Hilbert space with the a norm equivalent to1198673(Ω) norm
Proof At first V = 119907 isin 1198673(Ω) | (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 =
0 is a closed vector subspace of 1198673(Ω) Note that the
condition of the nullity of the mean in V may be replacedby int
ΓΩ
119906119889119909 = 0 where ΓΩis a nonempty open subset of Ω In
this caseV becomes rather
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω verifying
int
ΓΩ
119907119889119909 = 0 and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(52)
From the fact that
0 lt 120572 le ℎ (119909) le 120573 lt 1 forall119909 isin Ω (53)
the norms built with the two bilinear applications 119886 and 119887 areequivalent to the norm constructed by the linear form notedby | |2V with 119907 isin V and given by
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(54)
Indeed | |2V is as well a norm inVThere is no difficulty toformally verify the triangular inequality If |119907|2V = 0 then120597119907120597119909
2
1198712(Ω)
= 0 and 12059731199071205971199093
2
1198712(Ω)
= 0 Consequently 119907 isnecessarily constant almost everywhere inΩ for example 119907 isequal to a constant real 119870 But as int
Ω119907119889119909 = 0 implies 119870|Ω| =
0 so119870 = 119907 = 0 We have thus built a standard scalar producton V from bilinear forms of the variational formulation Vis a Hilbert space with the scalar product associated withthe norm (54) In addition this norm is equivalent to the119867
3(Ω) norm as it shown in Lemma A2 Now we just need
to prove that the correspondence (119906 119907) isin V timesV 997891rarr 119886(119906 119907)
is continuous and coercive on the Hilbert spaceV
8 ISRN Signal Processing
433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V
|119886 (119906 119907)| =
1003816100381610038161003816100381610038161003816100381610038161003816
(radicℎ
1205973119906
1205971199093 radicℎ
1205973119907
1205971199093)
1198712(Ω)
1003816100381610038161003816100381610038161003816100381610038161003816
le 120573
100381710038171003817100381710038171003817100381710038171003817
1205973119906
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
sdot
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
le 120573|119906|V sdot |119907|V
(55)
hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem
For
119907 isin V minus1199072
1198671(Ω)
le minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(56)
hence
|119907|2
V minus 1199072
1198671(Ω)
le |119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(57)
nevertheless
119886 (119907 119907) =
100381710038171003817100381710038171003817100381710038171003817
radicℎ
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 120572
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= 120572[|119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]
(58)
Thus
119886 (119907 119907) ge 120572 [|119907|2
V minus 1199072
1198671(Ω)] forall119907 isin V (59)
We have found thus (120588 120583) = (120572 120572) such that
119886 (119907 119907) ge 120588|119907|2
V minus 1205831199072
1198671(Ω) forall119907 isin V (60)
All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows
Theorem 11 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905
119906 (0) = 1199060
119889119906
119889119905
isin 1198712(0 119879119867)
(61)
434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0 forall119907 isin V (62)
As we must consider a test function
119907 isin 1198673(Ω) such that 120597
2119907
1205971199092still in 119863 (Ω)
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(63)
Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that
int
Ω
(120601 + 119870) 119889119909 = 0 (64)
We pose the following Neumann problem
1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(65)
Its variational formulation gives a unique solution in119867 =
119908 isin 1198671(Ω) | int
Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672
(Ω)
implies the regularity of the solution which is in 119907 isin 1198673(Ω)
consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the
first term can be written as
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
(int
Ω
119906
1205972119907
1205971199092119889119909 minus int
120597Ω
119906
120597119907
120597119899
119889120590)
as 1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
and it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906 (120601 + 119870) 119889119909
=
119889
119889119905
(int
Ω
119906120601119889119909 + 119870int
Ω
119906119889119909)
(66)
However intΩ119906119889119909 = 0 subsequently
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906120601119889119909 = int
Ω
119889119906
119889119905
120601119889119909 (67)
The second term of (62) is written as follows
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
ℎ (119909)
1205973119906
1205971199093
120597120601
120597119909
119889119909 (68)
which gives in the distribution sense
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909 (69)
from which
int
Ω
119889119906
119889119905
120601119889119909+int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)
(70)
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
International Journal of
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2 ISRN Signal Processing
Perona andMalik model [8]The basic equation that governsnonlinear diffusion filtering is
119906119905(x 119905) = div (119892 (|nabla119906 (x 119905)|2) nabla119906 (x 119905)) (1)
with x = (1199091 119909
2) and where 119906(x 119905) is a filtered version of the
original image 119906(x 119905) = 1199060(x) as the initial condition with
reflecting boundary conditions In (1) 119892(sdot) is the conductivity(or diffusivity) function which is dependent (in space andtime) on the image gradient magnitude Several forms ofdiffusivity were introduced in the original paper of Peronaand Malik [8] All forms of diffusivity are chosen to be amonotonically decreasing function of the signal gradientPossible expressions for conductivity are
1198921(x 119905) = 1
1 + (|nabla119906 (x 119905)| 120573)2
1198922(x 119905) = exp (minus(|nabla119906 (x 119905)| 120573)2)
(2)
Parameter 120573 is a threshold one which influences the aniso-tropic smoothing process The nonlinear equation (1) actsas a forward parabolic equation smoothing regions whilepreserving edges Other methods based on high-order PDEare provided for image restoration like in [9ndash11] Efficientnumerical schemes were introduced in [12] based on additiveoperator splitting (AOS) scheme or based on alternatingdirection implicit (ADI) scheme See [12ndash15] for a review andextensions of these methods Unlike these methods of high-order PDE that are specially developed for denoising ourmodelwas constructed to interpolate the characteristic pointsof a signal
The major problem of nonlinear diffusion-based processis that it is generally difficult to correctly separate the highfrequency components from the low frequency ones In caseof denoising applications the objective of this process is touse the diffusivity function as a guide to retain useful dataand suppress noise
Numerous authors have proposed fourth-order PDEsfor image smoothing and denoising with the hope thatthese methods would perform better than their second-orderanalogues [16ndash18] Indeed there are good reasons to considerfourth-order equationsThefirst reason that can be retained isthe fact that fourth-order linear diffusion damps oscillationsat high frequencies (eg noise) much faster than second-order diffusion On the other hand the theory of fourth ordernonlinear PDEs is far less developed than the second orderone Also such equations often do not satisfy a maximumprinciple or comparison principle and implementation ofthe equations could thus introduce artificial singularities orother undesirable behavior In recent studies Tumblin [19]Tumblin and Turk [16] and Wei [17] proposed the followingform
119906119905(x 119905) = minus div (119892 (119898 (119906)) nablaΔ119906 (x 119905)) (3)
where 119892(sdot) = 1198921(sdot) as in (2) and 119898 is some measurement of
119906(x 119905) In [16] (3) is called a ldquolow curvature image simplifierrdquo(LCIS) and a good choice for 119898 is defined as 119898 = Δ119906 to
enforce isotropic diffusion [19] These PDE tools for digitalsignal and image processing make more reachable the 2Dextension of the 1D PDE-based method for characteristicpoints interpolation presented in this paper
212TheEmpiricalModeDecomposition Principle TheEMD[1] method decomposes iteratively a signal into amplitudemodulation-frequency modulation (AM-FM) type compo-nents called intrinsic mode functions (IMF) The underlyingprinciple of this decomposition is to locally identify in thesignal the most rapid oscillations defined as the waveforminterpolating interwoven local maxima and minima To dothis local extrema points are interpolated with a cubicspline to yield the upper and lower envelopes The meanenvelope (half sum of upper and lower envelopes) is thensubtracted from the initial signal and the same interpolationscheme is reiterated on the remainder The so-called siftingprocess stops when the mean envelope is reasonably zeroeverywhere and the resulting signal is called the first IMFThe higher-order IMFs are iteratively extracted applying thesame procedure to the initial signal after the previous IMFshave been removed
22 Some Useful Mathematical Concepts Throughout thepaperΩ denotes an open-bounded subset ofR119899 119899 = 1 or 2In the sequel we will need the following definitions andresults
Definition 1 we define one the spaces 119881 andV as follows
119881 = 119907 isin 1198673(Ω) |
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
V = 119907 isin 1198673(Ω) |
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω
open and verifying int
ΓΩ
119907119889119909 = 0 and 120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(4)
Definition 2 For a given function 119907 isin 1198671(Ω) if119863
119889(119907)(119909) and
119863119892(119907)(119909) denote respectively the right and left derivative of
119907 at 119909 the minmod derivatives of 119907 is define by
120597119907
120597119909
(119909) = minmod (119863119892(119907) (119909) 119863
119889(119907) (119909)) where
minmod (119901 119902) =
sign (119901)min (1003816100381610038161003816119901100381610038161003816100381610038161003816100381610038161199021003816100381610038161003816) if 119901119902 gt 0
0 if 119901119902 le 0
(5)
A 1198671(Ω) function being absolutely continuous admits right
and left derivatives then 1199060isin 119867
1(Ω) has obviously left
and right derivatives so that we can validate numericallycomputing of the diffusivity function 119892 defined in Section 3
ISRN Signal Processing 3
Definition 3 Let (119909119899)119899isinN be a sequence of elements in a vec-
torial normed space (119864 sdot 119864) it is said that (119909
119899)119899isinN converge
weakly [20] in 119864 and noted by 119909119899 119909 if exists an element
119909 isin 119864 such that forall119891 isin 1198641015840 lim
119899rarrinfin119891(119909
119899) = 119891(119909) where 1198641015840
denotes the set of continuous linear forms on 119864
Definition 4 (the Green formula) Let 119906 119907 isin 1198622(Ω) then
int
Ω
Δ119906119889119909 = int
120597Ω
120597119906
120597119899
119889119904
int
Ω
nabla119906nabla119907119889119909 = minusint
Ω
119906Δ119907119889119909 + int
120597Ω
120597119907
120597119899
119889119904
int
Ω
(119906Δ119907 minus 119907Δ119906) 119889119909 = int
120597Ω
119906
120597119907
120597119899
minus 119907
120597119906
120597119899
119889119904 Green formula
(6)
where 120597119906120597119899 denotes the normal derivative of 119906 on theboundary 120597Ω ofΩ
Definition 5 (the Poincare-Wirtinger inequality) LetΩ be anopen-bounded set and let 119906 isin 119867
1(Ω) then there exists a
constant 119862 gt 0 such that the norm of 119906 in 1198671(Ω) and the
norm of 119906 in 1198712(Ω) are linked by the following inequality
10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119904
100381710038171003817100381710038171003817100381710038171198671(Ω)
le 119862nabla1199061198712(Ω) (7)
where |Ω| denotes the length ofΩ The best constant 119862 in thePoincare-Wirtinger inequality is 1120582 for example the inverseof the first positive eigenvalue 120582 of the Laplace operator withhomogenous Neumann boundaries conditions In our caseΩ sub R and |Ω| is the diameter ofΩ
Theorem 6 (regularity theorem) Let
119906 isin 119906 isin 1198673(Ω) 119904119906119888ℎ 119905ℎ119886119905
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119906119889119909 = 0 (8)
then there exist two strictly positive reals 119862 and 119870 such thatfirstly
nabla1199062
1198712(Ω)
le 119862Δ1199062
1198712(Ω) (9)
and secondly
1199062
1198673(Ω)
le 119870Δ1199062
1198671(Ω) (10)
The main existence and uniqueness result of the solutionof our henceforth problem (29) is due to the application ofthe following Lions theorem [21]
Theorem 7 (the Lions theorem) Let119867 and 119881 be two Hilbertspaces with 119881 sub 119867 One considers a bilinear application 119886 on119881times119881 and 119861 an operator on119881 Under the following conditions
(1) exist120583 and 120588 gt 0 such that |119886(119906 119907)| ge 1205881199072119881minus 120583|119907|
2
119867
(2) exist 1198882gt 0 such that |(119907 119861119907)
119881| le 119888
2|119907|
2
119881
For 1199060= 119906(0) isin 119867 the problem
119906 isin 1198712(0 119879 119881) cap 119862 (0 119879119867) (11)
such that
forall119907 isin 119881 (
119889119906
119889119905
119861119907)
119867
+ 119886 (119906 119907) = 0 (12)
and 1199060= 119906(0) has a unique solution
Furthermore 119889119906119889119905 isin 1198712(0 119879119867
1015840)
3 Mathematical Modeling of the NewPDE-Interpolator
Recall that the PDE model aimed to contribute to themathematical modeling of the EMD as it is in [2] In a firststep to be in line with the classical EMD our goal is tomodel the upper or lower envelope as the asymptotic solutionof a PDE system whose initial condition is the input signalthat we want to interpolate local extrema (or more generallycharacteristic points) Initially this envelope is obtained bycubic spline interpolation Let Ω be the open domain of angiven finite energy signal 119906
0isin 119871
2(Ω) We construct an
operator 119860 of appropriate domain119863(119860) as follows
120597119906
120597119905
+ 119860119906 = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(13)
The need of asymptotic solution existence denoted by 119906infin
implies (120597119906120597119905)(infin 119909) = 0 for example according to (13) wehave 119860119906(infin 119909) = 0 Moreover the requirement of the sameregularity between the solution and a cubic spline leads to1205974119906120597119909
4(infin 119909) = 0 Thus in the first analysis we can choose
119860 = 1205974119906120597119909
4 To leave invariant features points during thediffusion process simply multiply in the expression 119860119906 theterm 120597
4119906120597119909
4 by function 119892 depending on spatial variable 119909and vanishing at characteristic points That gives
(119860119906) (119909) = 119892 (119909)
1205974119906
1205971199094 (14)
Another reminiscent form of long-range diffusion [22] is thefollowing
(119860119906) (119909) =
120597
120597119909
(119892 (119909)
1205973119906
1205971199093) (15)
Function 119892 can be interpreted as a diffusivity function whoserole is to control the diffusion process We take it necessarilypositive to have a direct diffusion and the existence ofsolution Other forms of operators are possible [3] we presentsome ones as follows for (14)
(119860119906) (119909) = 119892 (119909) [minus120579
1205972119906
1205971199092+ (1 minus 120579)
1205974119906
1205971199094] (16)
4 ISRN Signal Processing
or for (15)
(119860119906) (119909) =
120597
120597119909
[119892 (119909) (minus120579
120597119906
120597119909
+ (1 minus 120579)
1205973119906
1205971199093)] (17)
or accomplete diffusion involving a flow
(119860119906) (119909) =
120597
120597119909
[minus120579119892 (119909)
120597119906
120597119909
+ (1 minus 120579)
120597
120597119909
(119892 (119909)
1205972119906
1205971199092)]
(18)where 0 le 120579 le 1 is the tension factor which controls theregularity of the solution
Both forms allow more freedom on the regularity ofthe solution 119906 To access the mathematical properties of thesolution we chose the second form in (17) with 120579 = 0 and thecorresponding equation (15)
At the local scale between two consecutive character-istic pointsmdashtwo local maxima for examplemdashthe diffusioninduces a smoothing phenomenon that deletes the local min-imum A simple form for 119892 to calculate the envelope is givenby a positive piecewise function lower than 1 that is constantbetween two characteristic points of 119906
0and zeroed only at
these points Characteristic points are often being definedby their values and the signs of first second and third localderivatives of119906
0We characterize the function119892 as depending
on sign(1205971199060120597119909) sign(1205972119906
0120597119909
2) and sign(1205973119906
0120597119909
3)
For the purpose of existence and regularity of the solu-tion we are led to work with the regularized version of sign
sign120599(119911) =
2
120587
arctan(120587119911120599
) (19)
where 120599 is a regularization coefficientIn the following we define 119892plusmn
119890for extrema detection
119892119891for turning points detection and 119892plusmn
mc for maximum andminimum curvature points detection
119892plusmn
119890(119909) =
1
9
[
10038161003816100381610038161003816100381610038161003816
sign(120597119906
0
120597119909
)
10038161003816100381610038161003816100381610038161003816
plusmn sign(12059721199060
1205971199092) + 1]
2
(+) for maxima and (minus) for minima
119892119891(119909) = [sign
120599(
12059721199060
1205971199092)]
2
for turning points
119892plusmn
mc (119909) =1
9
[
100381610038161003816100381610038161003816100381610038161003816
sign(12059731199060
1205971199093)
100381610038161003816100381610038161003816100381610038161003816
plusmn sign(12059721199060
1205971199092) + 1]
2
(20)
(+) for maximum curvature points and (minus) for minimumcurvature points All these functions are of the form 119892(119909) =
[ℎ(119909)]2 with ℎ(119909) = 0 at characteristic points
So we have 1198921015840(119909) = 0 if 119892(119909) = 0 This property formally
allows us to cancel119860119906 and 120597119906120597119905 of the form (15) at the pointswhere 119892 is null The factor 19 is a normalization term andverifies 0 le 119892(119909) le 1 A more general diffusivity function 119892could be envisaged
Let us consider the following differential operators119860119890and
119860119891(119890 for upper and 119891 for inflexion or curvature)
forall119906 isin 119863 (119860119890) 119860
119890119906 =
120597
120597119909
(119892plusmn
119890(119909)
1205973119906
1205971199093) (21)
or
forall119906 isin 119863 (119860119891) 119860
119891119906 =
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) (22)
If 1199060isin 119862
2(Ω) as it is the case in the classical imple-
mentation EMD there is no problem for interpolation bycubic spline method But it is indeed a restriction on theregularity of 119906
0which is due to the choice of the interpolation
technique that requires a good detection of local extremafor the envelopes calculation The basic EMD principle mustapply for input functions that are not regular as functions in119862
2(Ω) for example for signal 119906
0isin 119867
1(Ω) as it is the case
in reality [23] But numerically without even a regularizationfunction sign if the derivatives are taken in the sense ofminmod called flux limiter or slope the function 119892(119909) stillhas a meaning
31 Interpretation of the Diffusivity Action in the DiffusionProcess Between two consecutivemaxima119883max119894 and119883max119894+1 the function 119892+
119890is piecewise constant and can be written as
follows
119892 (119909) =
0 at 119883max119894 119883max119894+1
4
9
at 119883inf 119894 119883inf 119894+1 119883min119894
1 on ]119883inf 119894 119883min119894[ cup ]119883min119894 119883inf 119894+1[
1
9
sur ]119883max119894 119883inf 119894[ cup ]119883inf 119894+1 119883max119894+1[
(23)
We have a diffusion in the sense that 119906119905= 120597119906120597119905 unhook
to the maximum of the curve 1199060 The reason is that the
diffusion is more pronounced in local minimum becausethe smoothing effect tends to regularize curves Thus theupper envelope 119906(119905 cong infin sdot) of 119906
0is less oscillating than
1199060is The same behavior occurs for the mean envelope We
would then try to interpret locally the relationship 119906 ge 1199060
by evoking the maximum principle [24] But this principleis not immediately checked for equations of the type (15)The justification lies in the fact that smoothing impliesnabla119906(119905 sdot)
1198712(Ω)
le nabla11990601198712(Ω)
In summary we have just built a PDE system
(1) having the input signal 1199060to decompose as an initial
value condition
(2) such that the result of a diffusion process preservescharacteristic points of 119906
0
(3) with a regularity similar to a cubic spline
32 Formulation of the Mathematical Problem Let 119860 be oneof the above constructed interpolation operators and 119879 gt 0
be a diffusion time sufficiently long To fix ideas we solvethe problem of upper envelope calculus Indeed for the otherkinds of characteristic points the same problem arises withanalogous mathematical resolution methods
ISRN Signal Processing 5
The problem posed by our mathematical modeling is tofind a solution 119906 of the system
120597119906
120597119905
+ 119860119906 = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(24)
Then the mean envelope (interpolating turning points) cal-culus is stated as follows
For given 1199060 find 119906 such that
120597119906
120597119905
+
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(25)
33 Comments on theDegeneracy in Problem (24) To preventthe zero degeneracy case in the system (25) whose solution isnot directly accessible by variational methods we are led tosolve an intermediate problem with
0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)
That is the nondegenerate problem defined hereafter
34 Formulation of the Zero Degeneracy Induced byDiffusivityFunction Thediffusivity function in (26) is very close to zeroat the characteristic points while refraining to cancel and tosolve the system (25) now just to move on to the limit when120572 tends to zero to retrieve (25) and its solution
Thus in particular we can define the function
119892119891119899(119909) = 119892
119891(119909) +
1
119899
119899 isin Nlowast (27)
which verifies the conditions of (26)The new system that we call nondegenerate problem is
fomulated as followsFind 119906
119899such that
120597119906119899
120597119905
+
120597
120597119909
(119892119899(119909)
1205973119906119899
1205971199093) = 0 in [0 119879] times Ω
119906119899(0 119909) = 119906
0in Ω
plus boundaries conditions
(28)
35 Formulation of the Nondegenerate Problem For 119899 suffi-ciently large 119906
119899is a close approximation of the envelope 119906
interpolating the characteristic pointsBy a density technique we can demonstrate that this
sequence (119906119899)119899isinN converges to the solution of the system (25)
By posing for the convenience of notation 0 lt 120572 le 119892119891119899
= ℎ le
120573 le 1 the non-degenerate problem is reworded as
Find 119906 such that
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(29)
Equation (29) is an operational differential equationstype The techniques for the resolution of this kind ofproblem are numerous we mainly refer to the variationalformulation to find weak solutions [25] and the method ofthe approximation of evolutionary operators The last allowsto work on a new elliptic problemThe solution is obtained bypassing to the limit of the approximate solution nstead of theequation
In the next section we solve (29) by a variationalapproach according to the resolution methods for parabolicproblems described in [7 20]
4 Existence and Uniqueness of Solutions forPDE Interpolator
41 The Guideline for the Resolution of the MathematicalProblem Now we summarize the general procedure to solvethe problem
(1) First step resolution of the problem (29) This stepincludes
(i) the variational formulation in Section 43(ii) the resolution of the variational problem in
appropriate functional space(iii) the converse showing that the solution of the
variational problem solves the departure prob-lem in Section 434
(2) Second step in Section 5 construction of the sequ-ence (119906
119899)119899isinN of solutions of problem (29) where 119906
119899
is the solution obtained with the diffusivity ℎ119899(119909) =
ℎ(119909) + (1119899) 119899 isin Nlowast see Section 5 Finally demon-strate that the sequence (119906
119899)119899isinN converges to a limit 119906
where 119906 is the solution of problem (25) in Section 5
42 The Main Results Formulation The main result which isgoing to be solves is the following
Theorem 8 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909+int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V
(30)
such that 119906(0) = 1199060and 119889119906119889119905 isin 1198712
(0 119879119867) Reversely let 119906be a solution of the variational problem (30) then (120597119906120597119905) +(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3)) = 0
And the complete follows result
6 ISRN Signal Processing
Theorem 9 For 1199060isin 119867
1(Ω) exists a unique solution
119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867
1(Ω)) (31)
verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(32)
43 Variational Formulation Let 119907 isin 1198673(Ω) and (to make
sense to integrals) multiply the equation
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (33)
of the system (29) by the test function 1205972119907120597119909
2 after weintegrate onΩ then it comes out that
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909 = 0 (34)
By using theGreen formula (seeDefinition 4) and integrationby parts we get firstly
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909
= [ℎ (119909)
1205973119906
1205971199093
1205972119907
1205971199092]
120597Ω
minus int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
(35)
and secondly
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
119906
1205972119907
1205971199092119889119909
int
Ω
119906
1205972119907
1205971199092119889119909 = int
120597Ω
119906
120597119907
120597119899
119889120590 minus int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(36)
As we are in one-dimension case the normal derivative andthe gradient on the edge in the use of the Green formula arethe same that is to say
int
120597Ω
119906
120597119907
120597119899
119889120590 = int
120597Ω
119906
120597119907
120597119909
119889120590 (37)
In the following we adopt the notation 120597119907120597119899Considering 119906 isin 1198673
(Ω) such that
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
119907 isin 119881 = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(38)
it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
forall119907 isin 119881
(39)
Let us consider the following bilinear forms
119886 (119906 119907) = int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
119887 (119906 119907) = int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(40)
The addition of boundaries conditions transforms our initialproblem into the system
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
119906 (0 119909) = 1199060
in Ω
(41)
The variational formulation of our problem consequentlysubject to additional conditions that would cause a change ofspace is the following
Find 119906 isin 119881 such that 119889
119889119905
(119887 (119906 119907)) + 119886 (119906 119907) = 0
forall119907 isin 119881
(42)
where 119886 and 119887 are like in Theorem 7This kind of problem is studied in [21] However before
continuing the resolution we explore some properties of thespace 119881
431 On the Quotient Space 119881R The space 119881 is defined by119863(119886) cap 119863(119887) where 119863(119886) and 119863(119887) are the domains of 119886 and119887 Let us consider the quotient space
119881
R= 119907 + 119888 119888 isin R such that 119907 isin 1198673
(Ω)
and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(43)
and the new spaceV defined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (44)
Let 119907 isin 119881R and let 119906 be a representative of 119907 class then(120597119907120597119899)|
120597Ωis defined by
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597 (119906 + 119888)
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 with 119888 isin R (45)
ISRN Signal Processing 7
Thenullity ofintΩ119907119889119909meansexist119888 isin R constant such thatint
Ω(119906+
119888)119889119909 = 0 with 119906 one representative of 119907
if 119888 = minus 1
|Ω|
int
Ω
119906119889119909 then
int
Ω
119907119889119909 = int
Ω
(119906 + 119888) 119889119909
= int
Ω
(119906 minus
1
|Ω|
int
Ω
119906119889119909)119889119909 = 0
(46)
So for all 119907 isin 119881R we have intΩ119907119889119909 = 0 Accordingly any
element of the quotient space 119881R is on one hand in1198673(Ω)
with normal derivative on the boundaries of Ω null and onthe other hand is zero average onΩThus the quotient space119881R is identified with the space V We can work later inthe space V with zero mean The final variational problemto solve is the following
find 119906isinV such that 119889
119889119905
(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V
(47)
432 Application of the Lions Theorem To apply the Lionstheorem we need to identify the applications 119886 119887 and 119861 andthe spaces119867 and119881 And finally wemust prove that the normdefined inV which derives from the scalar product given bythe bilinear application 119886 is equivalent to the norm of1198673
(Ω)Let us define 119861 as the identity and let us define 119886 and 119887 by
|119886(119906 119907)| = (radicℎ(1205973119906120597119909
3) radicℎ(120597
3119907120597119909
3))
1198712(Ω)
and 119887(119906 119907) =intΩ(120597119906120597119909)(120597119907120597119909)119889119909 obtained from (39) (40) the spaces
119881 = V and119867 are define as follows(a) The linear operator 119861 in the variational prob-
lem (11) in the Lions theorem (119889119906119889119905 119861119907)119867
denotes ascalar product on 119867 Indeed we define 119867 = 119907 isin
1198671(Ω) such that (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 = 0
The operator 119861 = 119868 119861119907 = 119907 is the identity whichis continuous (verify the Lions conditions Theorem 7) andthe term (119889119906119889119905 119861119907)
119867= (119889119889119905)(119887(119906 119907)) is equal to
int
Ω
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
119889119909 = int
Ω
119889
119889119905
(
120597119906
120597119909
)
120597119907
120597119909
119889119909
= (
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
)
1198712(Ω)
(48)
In this identification ((120597120597119909)(119889119906119889119905) 120597119907120597119909)1198712(Ω)
is a wellscalar product in the subspace 119867 of 1198671
(Ω) formed byfunctions with zero mean Indeed according to Poincare-Wirtinger inequality (see Definition 5) in1198671
(Ω)10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119909
100381710038171003817100381710038171003817100381710038171198671(Ω)
le
1
120582
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(49)
where 120582 is the first positive eigenvalue of Laplace operator inΩ But as 119906 119907 isin V then int
Ω119906119889119909 = 0 consequently there
exists 1198622gt 0 such that
1199061198671(Ω)
le 1198622
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(50)
that is to say the 1198671(Ω) norm is equivalent to the gradient
norm inV Hence one can replace the scalar product of thegradient by1198671
(Ω) scalar productHereinafter we complete the proof that the precedent
choices allow the application of the Lions theorem in ourcontext
(b) Vis an Hilbert space with equivalent norm to 1198673(Ω)
norm
Theorem 10 The spaceV resulting from (11) andV analysisdefined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (51)
is an Hilbert space with the a norm equivalent to1198673(Ω) norm
Proof At first V = 119907 isin 1198673(Ω) | (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 =
0 is a closed vector subspace of 1198673(Ω) Note that the
condition of the nullity of the mean in V may be replacedby int
ΓΩ
119906119889119909 = 0 where ΓΩis a nonempty open subset of Ω In
this caseV becomes rather
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω verifying
int
ΓΩ
119907119889119909 = 0 and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(52)
From the fact that
0 lt 120572 le ℎ (119909) le 120573 lt 1 forall119909 isin Ω (53)
the norms built with the two bilinear applications 119886 and 119887 areequivalent to the norm constructed by the linear form notedby | |2V with 119907 isin V and given by
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(54)
Indeed | |2V is as well a norm inVThere is no difficulty toformally verify the triangular inequality If |119907|2V = 0 then120597119907120597119909
2
1198712(Ω)
= 0 and 12059731199071205971199093
2
1198712(Ω)
= 0 Consequently 119907 isnecessarily constant almost everywhere inΩ for example 119907 isequal to a constant real 119870 But as int
Ω119907119889119909 = 0 implies 119870|Ω| =
0 so119870 = 119907 = 0 We have thus built a standard scalar producton V from bilinear forms of the variational formulation Vis a Hilbert space with the scalar product associated withthe norm (54) In addition this norm is equivalent to the119867
3(Ω) norm as it shown in Lemma A2 Now we just need
to prove that the correspondence (119906 119907) isin V timesV 997891rarr 119886(119906 119907)
is continuous and coercive on the Hilbert spaceV
8 ISRN Signal Processing
433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V
|119886 (119906 119907)| =
1003816100381610038161003816100381610038161003816100381610038161003816
(radicℎ
1205973119906
1205971199093 radicℎ
1205973119907
1205971199093)
1198712(Ω)
1003816100381610038161003816100381610038161003816100381610038161003816
le 120573
100381710038171003817100381710038171003817100381710038171003817
1205973119906
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
sdot
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
le 120573|119906|V sdot |119907|V
(55)
hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem
For
119907 isin V minus1199072
1198671(Ω)
le minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(56)
hence
|119907|2
V minus 1199072
1198671(Ω)
le |119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(57)
nevertheless
119886 (119907 119907) =
100381710038171003817100381710038171003817100381710038171003817
radicℎ
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 120572
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= 120572[|119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]
(58)
Thus
119886 (119907 119907) ge 120572 [|119907|2
V minus 1199072
1198671(Ω)] forall119907 isin V (59)
We have found thus (120588 120583) = (120572 120572) such that
119886 (119907 119907) ge 120588|119907|2
V minus 1205831199072
1198671(Ω) forall119907 isin V (60)
All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows
Theorem 11 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905
119906 (0) = 1199060
119889119906
119889119905
isin 1198712(0 119879119867)
(61)
434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0 forall119907 isin V (62)
As we must consider a test function
119907 isin 1198673(Ω) such that 120597
2119907
1205971199092still in 119863 (Ω)
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(63)
Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that
int
Ω
(120601 + 119870) 119889119909 = 0 (64)
We pose the following Neumann problem
1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(65)
Its variational formulation gives a unique solution in119867 =
119908 isin 1198671(Ω) | int
Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672
(Ω)
implies the regularity of the solution which is in 119907 isin 1198673(Ω)
consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the
first term can be written as
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
(int
Ω
119906
1205972119907
1205971199092119889119909 minus int
120597Ω
119906
120597119907
120597119899
119889120590)
as 1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
and it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906 (120601 + 119870) 119889119909
=
119889
119889119905
(int
Ω
119906120601119889119909 + 119870int
Ω
119906119889119909)
(66)
However intΩ119906119889119909 = 0 subsequently
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906120601119889119909 = int
Ω
119889119906
119889119905
120601119889119909 (67)
The second term of (62) is written as follows
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
ℎ (119909)
1205973119906
1205971199093
120597120601
120597119909
119889119909 (68)
which gives in the distribution sense
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909 (69)
from which
int
Ω
119889119906
119889119905
120601119889119909+int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)
(70)
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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DistributedSensor Networks
International Journal of
ISRN Signal Processing 3
Definition 3 Let (119909119899)119899isinN be a sequence of elements in a vec-
torial normed space (119864 sdot 119864) it is said that (119909
119899)119899isinN converge
weakly [20] in 119864 and noted by 119909119899 119909 if exists an element
119909 isin 119864 such that forall119891 isin 1198641015840 lim
119899rarrinfin119891(119909
119899) = 119891(119909) where 1198641015840
denotes the set of continuous linear forms on 119864
Definition 4 (the Green formula) Let 119906 119907 isin 1198622(Ω) then
int
Ω
Δ119906119889119909 = int
120597Ω
120597119906
120597119899
119889119904
int
Ω
nabla119906nabla119907119889119909 = minusint
Ω
119906Δ119907119889119909 + int
120597Ω
120597119907
120597119899
119889119904
int
Ω
(119906Δ119907 minus 119907Δ119906) 119889119909 = int
120597Ω
119906
120597119907
120597119899
minus 119907
120597119906
120597119899
119889119904 Green formula
(6)
where 120597119906120597119899 denotes the normal derivative of 119906 on theboundary 120597Ω ofΩ
Definition 5 (the Poincare-Wirtinger inequality) LetΩ be anopen-bounded set and let 119906 isin 119867
1(Ω) then there exists a
constant 119862 gt 0 such that the norm of 119906 in 1198671(Ω) and the
norm of 119906 in 1198712(Ω) are linked by the following inequality
10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119904
100381710038171003817100381710038171003817100381710038171198671(Ω)
le 119862nabla1199061198712(Ω) (7)
where |Ω| denotes the length ofΩ The best constant 119862 in thePoincare-Wirtinger inequality is 1120582 for example the inverseof the first positive eigenvalue 120582 of the Laplace operator withhomogenous Neumann boundaries conditions In our caseΩ sub R and |Ω| is the diameter ofΩ
Theorem 6 (regularity theorem) Let
119906 isin 119906 isin 1198673(Ω) 119904119906119888ℎ 119905ℎ119886119905
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119906119889119909 = 0 (8)
then there exist two strictly positive reals 119862 and 119870 such thatfirstly
nabla1199062
1198712(Ω)
le 119862Δ1199062
1198712(Ω) (9)
and secondly
1199062
1198673(Ω)
le 119870Δ1199062
1198671(Ω) (10)
The main existence and uniqueness result of the solutionof our henceforth problem (29) is due to the application ofthe following Lions theorem [21]
Theorem 7 (the Lions theorem) Let119867 and 119881 be two Hilbertspaces with 119881 sub 119867 One considers a bilinear application 119886 on119881times119881 and 119861 an operator on119881 Under the following conditions
(1) exist120583 and 120588 gt 0 such that |119886(119906 119907)| ge 1205881199072119881minus 120583|119907|
2
119867
(2) exist 1198882gt 0 such that |(119907 119861119907)
119881| le 119888
2|119907|
2
119881
For 1199060= 119906(0) isin 119867 the problem
119906 isin 1198712(0 119879 119881) cap 119862 (0 119879119867) (11)
such that
forall119907 isin 119881 (
119889119906
119889119905
119861119907)
119867
+ 119886 (119906 119907) = 0 (12)
and 1199060= 119906(0) has a unique solution
Furthermore 119889119906119889119905 isin 1198712(0 119879119867
1015840)
3 Mathematical Modeling of the NewPDE-Interpolator
Recall that the PDE model aimed to contribute to themathematical modeling of the EMD as it is in [2] In a firststep to be in line with the classical EMD our goal is tomodel the upper or lower envelope as the asymptotic solutionof a PDE system whose initial condition is the input signalthat we want to interpolate local extrema (or more generallycharacteristic points) Initially this envelope is obtained bycubic spline interpolation Let Ω be the open domain of angiven finite energy signal 119906
0isin 119871
2(Ω) We construct an
operator 119860 of appropriate domain119863(119860) as follows
120597119906
120597119905
+ 119860119906 = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(13)
The need of asymptotic solution existence denoted by 119906infin
implies (120597119906120597119905)(infin 119909) = 0 for example according to (13) wehave 119860119906(infin 119909) = 0 Moreover the requirement of the sameregularity between the solution and a cubic spline leads to1205974119906120597119909
4(infin 119909) = 0 Thus in the first analysis we can choose
119860 = 1205974119906120597119909
4 To leave invariant features points during thediffusion process simply multiply in the expression 119860119906 theterm 120597
4119906120597119909
4 by function 119892 depending on spatial variable 119909and vanishing at characteristic points That gives
(119860119906) (119909) = 119892 (119909)
1205974119906
1205971199094 (14)
Another reminiscent form of long-range diffusion [22] is thefollowing
(119860119906) (119909) =
120597
120597119909
(119892 (119909)
1205973119906
1205971199093) (15)
Function 119892 can be interpreted as a diffusivity function whoserole is to control the diffusion process We take it necessarilypositive to have a direct diffusion and the existence ofsolution Other forms of operators are possible [3] we presentsome ones as follows for (14)
(119860119906) (119909) = 119892 (119909) [minus120579
1205972119906
1205971199092+ (1 minus 120579)
1205974119906
1205971199094] (16)
4 ISRN Signal Processing
or for (15)
(119860119906) (119909) =
120597
120597119909
[119892 (119909) (minus120579
120597119906
120597119909
+ (1 minus 120579)
1205973119906
1205971199093)] (17)
or accomplete diffusion involving a flow
(119860119906) (119909) =
120597
120597119909
[minus120579119892 (119909)
120597119906
120597119909
+ (1 minus 120579)
120597
120597119909
(119892 (119909)
1205972119906
1205971199092)]
(18)where 0 le 120579 le 1 is the tension factor which controls theregularity of the solution
Both forms allow more freedom on the regularity ofthe solution 119906 To access the mathematical properties of thesolution we chose the second form in (17) with 120579 = 0 and thecorresponding equation (15)
At the local scale between two consecutive character-istic pointsmdashtwo local maxima for examplemdashthe diffusioninduces a smoothing phenomenon that deletes the local min-imum A simple form for 119892 to calculate the envelope is givenby a positive piecewise function lower than 1 that is constantbetween two characteristic points of 119906
0and zeroed only at
these points Characteristic points are often being definedby their values and the signs of first second and third localderivatives of119906
0We characterize the function119892 as depending
on sign(1205971199060120597119909) sign(1205972119906
0120597119909
2) and sign(1205973119906
0120597119909
3)
For the purpose of existence and regularity of the solu-tion we are led to work with the regularized version of sign
sign120599(119911) =
2
120587
arctan(120587119911120599
) (19)
where 120599 is a regularization coefficientIn the following we define 119892plusmn
119890for extrema detection
119892119891for turning points detection and 119892plusmn
mc for maximum andminimum curvature points detection
119892plusmn
119890(119909) =
1
9
[
10038161003816100381610038161003816100381610038161003816
sign(120597119906
0
120597119909
)
10038161003816100381610038161003816100381610038161003816
plusmn sign(12059721199060
1205971199092) + 1]
2
(+) for maxima and (minus) for minima
119892119891(119909) = [sign
120599(
12059721199060
1205971199092)]
2
for turning points
119892plusmn
mc (119909) =1
9
[
100381610038161003816100381610038161003816100381610038161003816
sign(12059731199060
1205971199093)
100381610038161003816100381610038161003816100381610038161003816
plusmn sign(12059721199060
1205971199092) + 1]
2
(20)
(+) for maximum curvature points and (minus) for minimumcurvature points All these functions are of the form 119892(119909) =
[ℎ(119909)]2 with ℎ(119909) = 0 at characteristic points
So we have 1198921015840(119909) = 0 if 119892(119909) = 0 This property formally
allows us to cancel119860119906 and 120597119906120597119905 of the form (15) at the pointswhere 119892 is null The factor 19 is a normalization term andverifies 0 le 119892(119909) le 1 A more general diffusivity function 119892could be envisaged
Let us consider the following differential operators119860119890and
119860119891(119890 for upper and 119891 for inflexion or curvature)
forall119906 isin 119863 (119860119890) 119860
119890119906 =
120597
120597119909
(119892plusmn
119890(119909)
1205973119906
1205971199093) (21)
or
forall119906 isin 119863 (119860119891) 119860
119891119906 =
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) (22)
If 1199060isin 119862
2(Ω) as it is the case in the classical imple-
mentation EMD there is no problem for interpolation bycubic spline method But it is indeed a restriction on theregularity of 119906
0which is due to the choice of the interpolation
technique that requires a good detection of local extremafor the envelopes calculation The basic EMD principle mustapply for input functions that are not regular as functions in119862
2(Ω) for example for signal 119906
0isin 119867
1(Ω) as it is the case
in reality [23] But numerically without even a regularizationfunction sign if the derivatives are taken in the sense ofminmod called flux limiter or slope the function 119892(119909) stillhas a meaning
31 Interpretation of the Diffusivity Action in the DiffusionProcess Between two consecutivemaxima119883max119894 and119883max119894+1 the function 119892+
119890is piecewise constant and can be written as
follows
119892 (119909) =
0 at 119883max119894 119883max119894+1
4
9
at 119883inf 119894 119883inf 119894+1 119883min119894
1 on ]119883inf 119894 119883min119894[ cup ]119883min119894 119883inf 119894+1[
1
9
sur ]119883max119894 119883inf 119894[ cup ]119883inf 119894+1 119883max119894+1[
(23)
We have a diffusion in the sense that 119906119905= 120597119906120597119905 unhook
to the maximum of the curve 1199060 The reason is that the
diffusion is more pronounced in local minimum becausethe smoothing effect tends to regularize curves Thus theupper envelope 119906(119905 cong infin sdot) of 119906
0is less oscillating than
1199060is The same behavior occurs for the mean envelope We
would then try to interpret locally the relationship 119906 ge 1199060
by evoking the maximum principle [24] But this principleis not immediately checked for equations of the type (15)The justification lies in the fact that smoothing impliesnabla119906(119905 sdot)
1198712(Ω)
le nabla11990601198712(Ω)
In summary we have just built a PDE system
(1) having the input signal 1199060to decompose as an initial
value condition
(2) such that the result of a diffusion process preservescharacteristic points of 119906
0
(3) with a regularity similar to a cubic spline
32 Formulation of the Mathematical Problem Let 119860 be oneof the above constructed interpolation operators and 119879 gt 0
be a diffusion time sufficiently long To fix ideas we solvethe problem of upper envelope calculus Indeed for the otherkinds of characteristic points the same problem arises withanalogous mathematical resolution methods
ISRN Signal Processing 5
The problem posed by our mathematical modeling is tofind a solution 119906 of the system
120597119906
120597119905
+ 119860119906 = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(24)
Then the mean envelope (interpolating turning points) cal-culus is stated as follows
For given 1199060 find 119906 such that
120597119906
120597119905
+
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(25)
33 Comments on theDegeneracy in Problem (24) To preventthe zero degeneracy case in the system (25) whose solution isnot directly accessible by variational methods we are led tosolve an intermediate problem with
0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)
That is the nondegenerate problem defined hereafter
34 Formulation of the Zero Degeneracy Induced byDiffusivityFunction Thediffusivity function in (26) is very close to zeroat the characteristic points while refraining to cancel and tosolve the system (25) now just to move on to the limit when120572 tends to zero to retrieve (25) and its solution
Thus in particular we can define the function
119892119891119899(119909) = 119892
119891(119909) +
1
119899
119899 isin Nlowast (27)
which verifies the conditions of (26)The new system that we call nondegenerate problem is
fomulated as followsFind 119906
119899such that
120597119906119899
120597119905
+
120597
120597119909
(119892119899(119909)
1205973119906119899
1205971199093) = 0 in [0 119879] times Ω
119906119899(0 119909) = 119906
0in Ω
plus boundaries conditions
(28)
35 Formulation of the Nondegenerate Problem For 119899 suffi-ciently large 119906
119899is a close approximation of the envelope 119906
interpolating the characteristic pointsBy a density technique we can demonstrate that this
sequence (119906119899)119899isinN converges to the solution of the system (25)
By posing for the convenience of notation 0 lt 120572 le 119892119891119899
= ℎ le
120573 le 1 the non-degenerate problem is reworded as
Find 119906 such that
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(29)
Equation (29) is an operational differential equationstype The techniques for the resolution of this kind ofproblem are numerous we mainly refer to the variationalformulation to find weak solutions [25] and the method ofthe approximation of evolutionary operators The last allowsto work on a new elliptic problemThe solution is obtained bypassing to the limit of the approximate solution nstead of theequation
In the next section we solve (29) by a variationalapproach according to the resolution methods for parabolicproblems described in [7 20]
4 Existence and Uniqueness of Solutions forPDE Interpolator
41 The Guideline for the Resolution of the MathematicalProblem Now we summarize the general procedure to solvethe problem
(1) First step resolution of the problem (29) This stepincludes
(i) the variational formulation in Section 43(ii) the resolution of the variational problem in
appropriate functional space(iii) the converse showing that the solution of the
variational problem solves the departure prob-lem in Section 434
(2) Second step in Section 5 construction of the sequ-ence (119906
119899)119899isinN of solutions of problem (29) where 119906
119899
is the solution obtained with the diffusivity ℎ119899(119909) =
ℎ(119909) + (1119899) 119899 isin Nlowast see Section 5 Finally demon-strate that the sequence (119906
119899)119899isinN converges to a limit 119906
where 119906 is the solution of problem (25) in Section 5
42 The Main Results Formulation The main result which isgoing to be solves is the following
Theorem 8 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909+int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V
(30)
such that 119906(0) = 1199060and 119889119906119889119905 isin 1198712
(0 119879119867) Reversely let 119906be a solution of the variational problem (30) then (120597119906120597119905) +(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3)) = 0
And the complete follows result
6 ISRN Signal Processing
Theorem 9 For 1199060isin 119867
1(Ω) exists a unique solution
119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867
1(Ω)) (31)
verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(32)
43 Variational Formulation Let 119907 isin 1198673(Ω) and (to make
sense to integrals) multiply the equation
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (33)
of the system (29) by the test function 1205972119907120597119909
2 after weintegrate onΩ then it comes out that
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909 = 0 (34)
By using theGreen formula (seeDefinition 4) and integrationby parts we get firstly
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909
= [ℎ (119909)
1205973119906
1205971199093
1205972119907
1205971199092]
120597Ω
minus int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
(35)
and secondly
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
119906
1205972119907
1205971199092119889119909
int
Ω
119906
1205972119907
1205971199092119889119909 = int
120597Ω
119906
120597119907
120597119899
119889120590 minus int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(36)
As we are in one-dimension case the normal derivative andthe gradient on the edge in the use of the Green formula arethe same that is to say
int
120597Ω
119906
120597119907
120597119899
119889120590 = int
120597Ω
119906
120597119907
120597119909
119889120590 (37)
In the following we adopt the notation 120597119907120597119899Considering 119906 isin 1198673
(Ω) such that
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
119907 isin 119881 = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(38)
it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
forall119907 isin 119881
(39)
Let us consider the following bilinear forms
119886 (119906 119907) = int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
119887 (119906 119907) = int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(40)
The addition of boundaries conditions transforms our initialproblem into the system
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
119906 (0 119909) = 1199060
in Ω
(41)
The variational formulation of our problem consequentlysubject to additional conditions that would cause a change ofspace is the following
Find 119906 isin 119881 such that 119889
119889119905
(119887 (119906 119907)) + 119886 (119906 119907) = 0
forall119907 isin 119881
(42)
where 119886 and 119887 are like in Theorem 7This kind of problem is studied in [21] However before
continuing the resolution we explore some properties of thespace 119881
431 On the Quotient Space 119881R The space 119881 is defined by119863(119886) cap 119863(119887) where 119863(119886) and 119863(119887) are the domains of 119886 and119887 Let us consider the quotient space
119881
R= 119907 + 119888 119888 isin R such that 119907 isin 1198673
(Ω)
and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(43)
and the new spaceV defined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (44)
Let 119907 isin 119881R and let 119906 be a representative of 119907 class then(120597119907120597119899)|
120597Ωis defined by
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597 (119906 + 119888)
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 with 119888 isin R (45)
ISRN Signal Processing 7
Thenullity ofintΩ119907119889119909meansexist119888 isin R constant such thatint
Ω(119906+
119888)119889119909 = 0 with 119906 one representative of 119907
if 119888 = minus 1
|Ω|
int
Ω
119906119889119909 then
int
Ω
119907119889119909 = int
Ω
(119906 + 119888) 119889119909
= int
Ω
(119906 minus
1
|Ω|
int
Ω
119906119889119909)119889119909 = 0
(46)
So for all 119907 isin 119881R we have intΩ119907119889119909 = 0 Accordingly any
element of the quotient space 119881R is on one hand in1198673(Ω)
with normal derivative on the boundaries of Ω null and onthe other hand is zero average onΩThus the quotient space119881R is identified with the space V We can work later inthe space V with zero mean The final variational problemto solve is the following
find 119906isinV such that 119889
119889119905
(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V
(47)
432 Application of the Lions Theorem To apply the Lionstheorem we need to identify the applications 119886 119887 and 119861 andthe spaces119867 and119881 And finally wemust prove that the normdefined inV which derives from the scalar product given bythe bilinear application 119886 is equivalent to the norm of1198673
(Ω)Let us define 119861 as the identity and let us define 119886 and 119887 by
|119886(119906 119907)| = (radicℎ(1205973119906120597119909
3) radicℎ(120597
3119907120597119909
3))
1198712(Ω)
and 119887(119906 119907) =intΩ(120597119906120597119909)(120597119907120597119909)119889119909 obtained from (39) (40) the spaces
119881 = V and119867 are define as follows(a) The linear operator 119861 in the variational prob-
lem (11) in the Lions theorem (119889119906119889119905 119861119907)119867
denotes ascalar product on 119867 Indeed we define 119867 = 119907 isin
1198671(Ω) such that (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 = 0
The operator 119861 = 119868 119861119907 = 119907 is the identity whichis continuous (verify the Lions conditions Theorem 7) andthe term (119889119906119889119905 119861119907)
119867= (119889119889119905)(119887(119906 119907)) is equal to
int
Ω
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
119889119909 = int
Ω
119889
119889119905
(
120597119906
120597119909
)
120597119907
120597119909
119889119909
= (
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
)
1198712(Ω)
(48)
In this identification ((120597120597119909)(119889119906119889119905) 120597119907120597119909)1198712(Ω)
is a wellscalar product in the subspace 119867 of 1198671
(Ω) formed byfunctions with zero mean Indeed according to Poincare-Wirtinger inequality (see Definition 5) in1198671
(Ω)10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119909
100381710038171003817100381710038171003817100381710038171198671(Ω)
le
1
120582
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(49)
where 120582 is the first positive eigenvalue of Laplace operator inΩ But as 119906 119907 isin V then int
Ω119906119889119909 = 0 consequently there
exists 1198622gt 0 such that
1199061198671(Ω)
le 1198622
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(50)
that is to say the 1198671(Ω) norm is equivalent to the gradient
norm inV Hence one can replace the scalar product of thegradient by1198671
(Ω) scalar productHereinafter we complete the proof that the precedent
choices allow the application of the Lions theorem in ourcontext
(b) Vis an Hilbert space with equivalent norm to 1198673(Ω)
norm
Theorem 10 The spaceV resulting from (11) andV analysisdefined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (51)
is an Hilbert space with the a norm equivalent to1198673(Ω) norm
Proof At first V = 119907 isin 1198673(Ω) | (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 =
0 is a closed vector subspace of 1198673(Ω) Note that the
condition of the nullity of the mean in V may be replacedby int
ΓΩ
119906119889119909 = 0 where ΓΩis a nonempty open subset of Ω In
this caseV becomes rather
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω verifying
int
ΓΩ
119907119889119909 = 0 and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(52)
From the fact that
0 lt 120572 le ℎ (119909) le 120573 lt 1 forall119909 isin Ω (53)
the norms built with the two bilinear applications 119886 and 119887 areequivalent to the norm constructed by the linear form notedby | |2V with 119907 isin V and given by
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(54)
Indeed | |2V is as well a norm inVThere is no difficulty toformally verify the triangular inequality If |119907|2V = 0 then120597119907120597119909
2
1198712(Ω)
= 0 and 12059731199071205971199093
2
1198712(Ω)
= 0 Consequently 119907 isnecessarily constant almost everywhere inΩ for example 119907 isequal to a constant real 119870 But as int
Ω119907119889119909 = 0 implies 119870|Ω| =
0 so119870 = 119907 = 0 We have thus built a standard scalar producton V from bilinear forms of the variational formulation Vis a Hilbert space with the scalar product associated withthe norm (54) In addition this norm is equivalent to the119867
3(Ω) norm as it shown in Lemma A2 Now we just need
to prove that the correspondence (119906 119907) isin V timesV 997891rarr 119886(119906 119907)
is continuous and coercive on the Hilbert spaceV
8 ISRN Signal Processing
433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V
|119886 (119906 119907)| =
1003816100381610038161003816100381610038161003816100381610038161003816
(radicℎ
1205973119906
1205971199093 radicℎ
1205973119907
1205971199093)
1198712(Ω)
1003816100381610038161003816100381610038161003816100381610038161003816
le 120573
100381710038171003817100381710038171003817100381710038171003817
1205973119906
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
sdot
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
le 120573|119906|V sdot |119907|V
(55)
hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem
For
119907 isin V minus1199072
1198671(Ω)
le minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(56)
hence
|119907|2
V minus 1199072
1198671(Ω)
le |119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(57)
nevertheless
119886 (119907 119907) =
100381710038171003817100381710038171003817100381710038171003817
radicℎ
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 120572
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= 120572[|119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]
(58)
Thus
119886 (119907 119907) ge 120572 [|119907|2
V minus 1199072
1198671(Ω)] forall119907 isin V (59)
We have found thus (120588 120583) = (120572 120572) such that
119886 (119907 119907) ge 120588|119907|2
V minus 1205831199072
1198671(Ω) forall119907 isin V (60)
All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows
Theorem 11 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905
119906 (0) = 1199060
119889119906
119889119905
isin 1198712(0 119879119867)
(61)
434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0 forall119907 isin V (62)
As we must consider a test function
119907 isin 1198673(Ω) such that 120597
2119907
1205971199092still in 119863 (Ω)
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(63)
Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that
int
Ω
(120601 + 119870) 119889119909 = 0 (64)
We pose the following Neumann problem
1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(65)
Its variational formulation gives a unique solution in119867 =
119908 isin 1198671(Ω) | int
Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672
(Ω)
implies the regularity of the solution which is in 119907 isin 1198673(Ω)
consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the
first term can be written as
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
(int
Ω
119906
1205972119907
1205971199092119889119909 minus int
120597Ω
119906
120597119907
120597119899
119889120590)
as 1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
and it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906 (120601 + 119870) 119889119909
=
119889
119889119905
(int
Ω
119906120601119889119909 + 119870int
Ω
119906119889119909)
(66)
However intΩ119906119889119909 = 0 subsequently
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906120601119889119909 = int
Ω
119889119906
119889119905
120601119889119909 (67)
The second term of (62) is written as follows
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
ℎ (119909)
1205973119906
1205971199093
120597120601
120597119909
119889119909 (68)
which gives in the distribution sense
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909 (69)
from which
int
Ω
119889119906
119889119905
120601119889119909+int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)
(70)
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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DistributedSensor Networks
International Journal of
4 ISRN Signal Processing
or for (15)
(119860119906) (119909) =
120597
120597119909
[119892 (119909) (minus120579
120597119906
120597119909
+ (1 minus 120579)
1205973119906
1205971199093)] (17)
or accomplete diffusion involving a flow
(119860119906) (119909) =
120597
120597119909
[minus120579119892 (119909)
120597119906
120597119909
+ (1 minus 120579)
120597
120597119909
(119892 (119909)
1205972119906
1205971199092)]
(18)where 0 le 120579 le 1 is the tension factor which controls theregularity of the solution
Both forms allow more freedom on the regularity ofthe solution 119906 To access the mathematical properties of thesolution we chose the second form in (17) with 120579 = 0 and thecorresponding equation (15)
At the local scale between two consecutive character-istic pointsmdashtwo local maxima for examplemdashthe diffusioninduces a smoothing phenomenon that deletes the local min-imum A simple form for 119892 to calculate the envelope is givenby a positive piecewise function lower than 1 that is constantbetween two characteristic points of 119906
0and zeroed only at
these points Characteristic points are often being definedby their values and the signs of first second and third localderivatives of119906
0We characterize the function119892 as depending
on sign(1205971199060120597119909) sign(1205972119906
0120597119909
2) and sign(1205973119906
0120597119909
3)
For the purpose of existence and regularity of the solu-tion we are led to work with the regularized version of sign
sign120599(119911) =
2
120587
arctan(120587119911120599
) (19)
where 120599 is a regularization coefficientIn the following we define 119892plusmn
119890for extrema detection
119892119891for turning points detection and 119892plusmn
mc for maximum andminimum curvature points detection
119892plusmn
119890(119909) =
1
9
[
10038161003816100381610038161003816100381610038161003816
sign(120597119906
0
120597119909
)
10038161003816100381610038161003816100381610038161003816
plusmn sign(12059721199060
1205971199092) + 1]
2
(+) for maxima and (minus) for minima
119892119891(119909) = [sign
120599(
12059721199060
1205971199092)]
2
for turning points
119892plusmn
mc (119909) =1
9
[
100381610038161003816100381610038161003816100381610038161003816
sign(12059731199060
1205971199093)
100381610038161003816100381610038161003816100381610038161003816
plusmn sign(12059721199060
1205971199092) + 1]
2
(20)
(+) for maximum curvature points and (minus) for minimumcurvature points All these functions are of the form 119892(119909) =
[ℎ(119909)]2 with ℎ(119909) = 0 at characteristic points
So we have 1198921015840(119909) = 0 if 119892(119909) = 0 This property formally
allows us to cancel119860119906 and 120597119906120597119905 of the form (15) at the pointswhere 119892 is null The factor 19 is a normalization term andverifies 0 le 119892(119909) le 1 A more general diffusivity function 119892could be envisaged
Let us consider the following differential operators119860119890and
119860119891(119890 for upper and 119891 for inflexion or curvature)
forall119906 isin 119863 (119860119890) 119860
119890119906 =
120597
120597119909
(119892plusmn
119890(119909)
1205973119906
1205971199093) (21)
or
forall119906 isin 119863 (119860119891) 119860
119891119906 =
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) (22)
If 1199060isin 119862
2(Ω) as it is the case in the classical imple-
mentation EMD there is no problem for interpolation bycubic spline method But it is indeed a restriction on theregularity of 119906
0which is due to the choice of the interpolation
technique that requires a good detection of local extremafor the envelopes calculation The basic EMD principle mustapply for input functions that are not regular as functions in119862
2(Ω) for example for signal 119906
0isin 119867
1(Ω) as it is the case
in reality [23] But numerically without even a regularizationfunction sign if the derivatives are taken in the sense ofminmod called flux limiter or slope the function 119892(119909) stillhas a meaning
31 Interpretation of the Diffusivity Action in the DiffusionProcess Between two consecutivemaxima119883max119894 and119883max119894+1 the function 119892+
119890is piecewise constant and can be written as
follows
119892 (119909) =
0 at 119883max119894 119883max119894+1
4
9
at 119883inf 119894 119883inf 119894+1 119883min119894
1 on ]119883inf 119894 119883min119894[ cup ]119883min119894 119883inf 119894+1[
1
9
sur ]119883max119894 119883inf 119894[ cup ]119883inf 119894+1 119883max119894+1[
(23)
We have a diffusion in the sense that 119906119905= 120597119906120597119905 unhook
to the maximum of the curve 1199060 The reason is that the
diffusion is more pronounced in local minimum becausethe smoothing effect tends to regularize curves Thus theupper envelope 119906(119905 cong infin sdot) of 119906
0is less oscillating than
1199060is The same behavior occurs for the mean envelope We
would then try to interpret locally the relationship 119906 ge 1199060
by evoking the maximum principle [24] But this principleis not immediately checked for equations of the type (15)The justification lies in the fact that smoothing impliesnabla119906(119905 sdot)
1198712(Ω)
le nabla11990601198712(Ω)
In summary we have just built a PDE system
(1) having the input signal 1199060to decompose as an initial
value condition
(2) such that the result of a diffusion process preservescharacteristic points of 119906
0
(3) with a regularity similar to a cubic spline
32 Formulation of the Mathematical Problem Let 119860 be oneof the above constructed interpolation operators and 119879 gt 0
be a diffusion time sufficiently long To fix ideas we solvethe problem of upper envelope calculus Indeed for the otherkinds of characteristic points the same problem arises withanalogous mathematical resolution methods
ISRN Signal Processing 5
The problem posed by our mathematical modeling is tofind a solution 119906 of the system
120597119906
120597119905
+ 119860119906 = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(24)
Then the mean envelope (interpolating turning points) cal-culus is stated as follows
For given 1199060 find 119906 such that
120597119906
120597119905
+
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(25)
33 Comments on theDegeneracy in Problem (24) To preventthe zero degeneracy case in the system (25) whose solution isnot directly accessible by variational methods we are led tosolve an intermediate problem with
0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)
That is the nondegenerate problem defined hereafter
34 Formulation of the Zero Degeneracy Induced byDiffusivityFunction Thediffusivity function in (26) is very close to zeroat the characteristic points while refraining to cancel and tosolve the system (25) now just to move on to the limit when120572 tends to zero to retrieve (25) and its solution
Thus in particular we can define the function
119892119891119899(119909) = 119892
119891(119909) +
1
119899
119899 isin Nlowast (27)
which verifies the conditions of (26)The new system that we call nondegenerate problem is
fomulated as followsFind 119906
119899such that
120597119906119899
120597119905
+
120597
120597119909
(119892119899(119909)
1205973119906119899
1205971199093) = 0 in [0 119879] times Ω
119906119899(0 119909) = 119906
0in Ω
plus boundaries conditions
(28)
35 Formulation of the Nondegenerate Problem For 119899 suffi-ciently large 119906
119899is a close approximation of the envelope 119906
interpolating the characteristic pointsBy a density technique we can demonstrate that this
sequence (119906119899)119899isinN converges to the solution of the system (25)
By posing for the convenience of notation 0 lt 120572 le 119892119891119899
= ℎ le
120573 le 1 the non-degenerate problem is reworded as
Find 119906 such that
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(29)
Equation (29) is an operational differential equationstype The techniques for the resolution of this kind ofproblem are numerous we mainly refer to the variationalformulation to find weak solutions [25] and the method ofthe approximation of evolutionary operators The last allowsto work on a new elliptic problemThe solution is obtained bypassing to the limit of the approximate solution nstead of theequation
In the next section we solve (29) by a variationalapproach according to the resolution methods for parabolicproblems described in [7 20]
4 Existence and Uniqueness of Solutions forPDE Interpolator
41 The Guideline for the Resolution of the MathematicalProblem Now we summarize the general procedure to solvethe problem
(1) First step resolution of the problem (29) This stepincludes
(i) the variational formulation in Section 43(ii) the resolution of the variational problem in
appropriate functional space(iii) the converse showing that the solution of the
variational problem solves the departure prob-lem in Section 434
(2) Second step in Section 5 construction of the sequ-ence (119906
119899)119899isinN of solutions of problem (29) where 119906
119899
is the solution obtained with the diffusivity ℎ119899(119909) =
ℎ(119909) + (1119899) 119899 isin Nlowast see Section 5 Finally demon-strate that the sequence (119906
119899)119899isinN converges to a limit 119906
where 119906 is the solution of problem (25) in Section 5
42 The Main Results Formulation The main result which isgoing to be solves is the following
Theorem 8 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909+int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V
(30)
such that 119906(0) = 1199060and 119889119906119889119905 isin 1198712
(0 119879119867) Reversely let 119906be a solution of the variational problem (30) then (120597119906120597119905) +(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3)) = 0
And the complete follows result
6 ISRN Signal Processing
Theorem 9 For 1199060isin 119867
1(Ω) exists a unique solution
119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867
1(Ω)) (31)
verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(32)
43 Variational Formulation Let 119907 isin 1198673(Ω) and (to make
sense to integrals) multiply the equation
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (33)
of the system (29) by the test function 1205972119907120597119909
2 after weintegrate onΩ then it comes out that
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909 = 0 (34)
By using theGreen formula (seeDefinition 4) and integrationby parts we get firstly
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909
= [ℎ (119909)
1205973119906
1205971199093
1205972119907
1205971199092]
120597Ω
minus int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
(35)
and secondly
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
119906
1205972119907
1205971199092119889119909
int
Ω
119906
1205972119907
1205971199092119889119909 = int
120597Ω
119906
120597119907
120597119899
119889120590 minus int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(36)
As we are in one-dimension case the normal derivative andthe gradient on the edge in the use of the Green formula arethe same that is to say
int
120597Ω
119906
120597119907
120597119899
119889120590 = int
120597Ω
119906
120597119907
120597119909
119889120590 (37)
In the following we adopt the notation 120597119907120597119899Considering 119906 isin 1198673
(Ω) such that
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
119907 isin 119881 = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(38)
it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
forall119907 isin 119881
(39)
Let us consider the following bilinear forms
119886 (119906 119907) = int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
119887 (119906 119907) = int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(40)
The addition of boundaries conditions transforms our initialproblem into the system
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
119906 (0 119909) = 1199060
in Ω
(41)
The variational formulation of our problem consequentlysubject to additional conditions that would cause a change ofspace is the following
Find 119906 isin 119881 such that 119889
119889119905
(119887 (119906 119907)) + 119886 (119906 119907) = 0
forall119907 isin 119881
(42)
where 119886 and 119887 are like in Theorem 7This kind of problem is studied in [21] However before
continuing the resolution we explore some properties of thespace 119881
431 On the Quotient Space 119881R The space 119881 is defined by119863(119886) cap 119863(119887) where 119863(119886) and 119863(119887) are the domains of 119886 and119887 Let us consider the quotient space
119881
R= 119907 + 119888 119888 isin R such that 119907 isin 1198673
(Ω)
and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(43)
and the new spaceV defined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (44)
Let 119907 isin 119881R and let 119906 be a representative of 119907 class then(120597119907120597119899)|
120597Ωis defined by
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597 (119906 + 119888)
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 with 119888 isin R (45)
ISRN Signal Processing 7
Thenullity ofintΩ119907119889119909meansexist119888 isin R constant such thatint
Ω(119906+
119888)119889119909 = 0 with 119906 one representative of 119907
if 119888 = minus 1
|Ω|
int
Ω
119906119889119909 then
int
Ω
119907119889119909 = int
Ω
(119906 + 119888) 119889119909
= int
Ω
(119906 minus
1
|Ω|
int
Ω
119906119889119909)119889119909 = 0
(46)
So for all 119907 isin 119881R we have intΩ119907119889119909 = 0 Accordingly any
element of the quotient space 119881R is on one hand in1198673(Ω)
with normal derivative on the boundaries of Ω null and onthe other hand is zero average onΩThus the quotient space119881R is identified with the space V We can work later inthe space V with zero mean The final variational problemto solve is the following
find 119906isinV such that 119889
119889119905
(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V
(47)
432 Application of the Lions Theorem To apply the Lionstheorem we need to identify the applications 119886 119887 and 119861 andthe spaces119867 and119881 And finally wemust prove that the normdefined inV which derives from the scalar product given bythe bilinear application 119886 is equivalent to the norm of1198673
(Ω)Let us define 119861 as the identity and let us define 119886 and 119887 by
|119886(119906 119907)| = (radicℎ(1205973119906120597119909
3) radicℎ(120597
3119907120597119909
3))
1198712(Ω)
and 119887(119906 119907) =intΩ(120597119906120597119909)(120597119907120597119909)119889119909 obtained from (39) (40) the spaces
119881 = V and119867 are define as follows(a) The linear operator 119861 in the variational prob-
lem (11) in the Lions theorem (119889119906119889119905 119861119907)119867
denotes ascalar product on 119867 Indeed we define 119867 = 119907 isin
1198671(Ω) such that (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 = 0
The operator 119861 = 119868 119861119907 = 119907 is the identity whichis continuous (verify the Lions conditions Theorem 7) andthe term (119889119906119889119905 119861119907)
119867= (119889119889119905)(119887(119906 119907)) is equal to
int
Ω
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
119889119909 = int
Ω
119889
119889119905
(
120597119906
120597119909
)
120597119907
120597119909
119889119909
= (
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
)
1198712(Ω)
(48)
In this identification ((120597120597119909)(119889119906119889119905) 120597119907120597119909)1198712(Ω)
is a wellscalar product in the subspace 119867 of 1198671
(Ω) formed byfunctions with zero mean Indeed according to Poincare-Wirtinger inequality (see Definition 5) in1198671
(Ω)10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119909
100381710038171003817100381710038171003817100381710038171198671(Ω)
le
1
120582
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(49)
where 120582 is the first positive eigenvalue of Laplace operator inΩ But as 119906 119907 isin V then int
Ω119906119889119909 = 0 consequently there
exists 1198622gt 0 such that
1199061198671(Ω)
le 1198622
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(50)
that is to say the 1198671(Ω) norm is equivalent to the gradient
norm inV Hence one can replace the scalar product of thegradient by1198671
(Ω) scalar productHereinafter we complete the proof that the precedent
choices allow the application of the Lions theorem in ourcontext
(b) Vis an Hilbert space with equivalent norm to 1198673(Ω)
norm
Theorem 10 The spaceV resulting from (11) andV analysisdefined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (51)
is an Hilbert space with the a norm equivalent to1198673(Ω) norm
Proof At first V = 119907 isin 1198673(Ω) | (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 =
0 is a closed vector subspace of 1198673(Ω) Note that the
condition of the nullity of the mean in V may be replacedby int
ΓΩ
119906119889119909 = 0 where ΓΩis a nonempty open subset of Ω In
this caseV becomes rather
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω verifying
int
ΓΩ
119907119889119909 = 0 and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(52)
From the fact that
0 lt 120572 le ℎ (119909) le 120573 lt 1 forall119909 isin Ω (53)
the norms built with the two bilinear applications 119886 and 119887 areequivalent to the norm constructed by the linear form notedby | |2V with 119907 isin V and given by
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(54)
Indeed | |2V is as well a norm inVThere is no difficulty toformally verify the triangular inequality If |119907|2V = 0 then120597119907120597119909
2
1198712(Ω)
= 0 and 12059731199071205971199093
2
1198712(Ω)
= 0 Consequently 119907 isnecessarily constant almost everywhere inΩ for example 119907 isequal to a constant real 119870 But as int
Ω119907119889119909 = 0 implies 119870|Ω| =
0 so119870 = 119907 = 0 We have thus built a standard scalar producton V from bilinear forms of the variational formulation Vis a Hilbert space with the scalar product associated withthe norm (54) In addition this norm is equivalent to the119867
3(Ω) norm as it shown in Lemma A2 Now we just need
to prove that the correspondence (119906 119907) isin V timesV 997891rarr 119886(119906 119907)
is continuous and coercive on the Hilbert spaceV
8 ISRN Signal Processing
433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V
|119886 (119906 119907)| =
1003816100381610038161003816100381610038161003816100381610038161003816
(radicℎ
1205973119906
1205971199093 radicℎ
1205973119907
1205971199093)
1198712(Ω)
1003816100381610038161003816100381610038161003816100381610038161003816
le 120573
100381710038171003817100381710038171003817100381710038171003817
1205973119906
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
sdot
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
le 120573|119906|V sdot |119907|V
(55)
hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem
For
119907 isin V minus1199072
1198671(Ω)
le minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(56)
hence
|119907|2
V minus 1199072
1198671(Ω)
le |119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(57)
nevertheless
119886 (119907 119907) =
100381710038171003817100381710038171003817100381710038171003817
radicℎ
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 120572
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= 120572[|119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]
(58)
Thus
119886 (119907 119907) ge 120572 [|119907|2
V minus 1199072
1198671(Ω)] forall119907 isin V (59)
We have found thus (120588 120583) = (120572 120572) such that
119886 (119907 119907) ge 120588|119907|2
V minus 1205831199072
1198671(Ω) forall119907 isin V (60)
All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows
Theorem 11 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905
119906 (0) = 1199060
119889119906
119889119905
isin 1198712(0 119879119867)
(61)
434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0 forall119907 isin V (62)
As we must consider a test function
119907 isin 1198673(Ω) such that 120597
2119907
1205971199092still in 119863 (Ω)
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(63)
Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that
int
Ω
(120601 + 119870) 119889119909 = 0 (64)
We pose the following Neumann problem
1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(65)
Its variational formulation gives a unique solution in119867 =
119908 isin 1198671(Ω) | int
Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672
(Ω)
implies the regularity of the solution which is in 119907 isin 1198673(Ω)
consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the
first term can be written as
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
(int
Ω
119906
1205972119907
1205971199092119889119909 minus int
120597Ω
119906
120597119907
120597119899
119889120590)
as 1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
and it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906 (120601 + 119870) 119889119909
=
119889
119889119905
(int
Ω
119906120601119889119909 + 119870int
Ω
119906119889119909)
(66)
However intΩ119906119889119909 = 0 subsequently
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906120601119889119909 = int
Ω
119889119906
119889119905
120601119889119909 (67)
The second term of (62) is written as follows
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
ℎ (119909)
1205973119906
1205971199093
120597120601
120597119909
119889119909 (68)
which gives in the distribution sense
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909 (69)
from which
int
Ω
119889119906
119889119905
120601119889119909+int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)
(70)
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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DistributedSensor Networks
International Journal of
ISRN Signal Processing 5
The problem posed by our mathematical modeling is tofind a solution 119906 of the system
120597119906
120597119905
+ 119860119906 = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(24)
Then the mean envelope (interpolating turning points) cal-culus is stated as follows
For given 1199060 find 119906 such that
120597119906
120597119905
+
120597
120597119909
(119892119891(119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(25)
33 Comments on theDegeneracy in Problem (24) To preventthe zero degeneracy case in the system (25) whose solution isnot directly accessible by variational methods we are led tosolve an intermediate problem with
0 lt 120572 le 119892119891120572(119909) le 120573 le 1 (26)
That is the nondegenerate problem defined hereafter
34 Formulation of the Zero Degeneracy Induced byDiffusivityFunction Thediffusivity function in (26) is very close to zeroat the characteristic points while refraining to cancel and tosolve the system (25) now just to move on to the limit when120572 tends to zero to retrieve (25) and its solution
Thus in particular we can define the function
119892119891119899(119909) = 119892
119891(119909) +
1
119899
119899 isin Nlowast (27)
which verifies the conditions of (26)The new system that we call nondegenerate problem is
fomulated as followsFind 119906
119899such that
120597119906119899
120597119905
+
120597
120597119909
(119892119899(119909)
1205973119906119899
1205971199093) = 0 in [0 119879] times Ω
119906119899(0 119909) = 119906
0in Ω
plus boundaries conditions
(28)
35 Formulation of the Nondegenerate Problem For 119899 suffi-ciently large 119906
119899is a close approximation of the envelope 119906
interpolating the characteristic pointsBy a density technique we can demonstrate that this
sequence (119906119899)119899isinN converges to the solution of the system (25)
By posing for the convenience of notation 0 lt 120572 le 119892119891119899
= ℎ le
120573 le 1 the non-degenerate problem is reworded as
Find 119906 such that
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
119906 (0 119909) = 1199060
in Ω
plus boundaries conditions
(29)
Equation (29) is an operational differential equationstype The techniques for the resolution of this kind ofproblem are numerous we mainly refer to the variationalformulation to find weak solutions [25] and the method ofthe approximation of evolutionary operators The last allowsto work on a new elliptic problemThe solution is obtained bypassing to the limit of the approximate solution nstead of theequation
In the next section we solve (29) by a variationalapproach according to the resolution methods for parabolicproblems described in [7 20]
4 Existence and Uniqueness of Solutions forPDE Interpolator
41 The Guideline for the Resolution of the MathematicalProblem Now we summarize the general procedure to solvethe problem
(1) First step resolution of the problem (29) This stepincludes
(i) the variational formulation in Section 43(ii) the resolution of the variational problem in
appropriate functional space(iii) the converse showing that the solution of the
variational problem solves the departure prob-lem in Section 434
(2) Second step in Section 5 construction of the sequ-ence (119906
119899)119899isinN of solutions of problem (29) where 119906
119899
is the solution obtained with the diffusivity ℎ119899(119909) =
ℎ(119909) + (1119899) 119899 isin Nlowast see Section 5 Finally demon-strate that the sequence (119906
119899)119899isinN converges to a limit 119906
where 119906 is the solution of problem (25) in Section 5
42 The Main Results Formulation The main result which isgoing to be solves is the following
Theorem 8 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909+int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909=0 119891119900119903 119886119899119910 119907 isin V
(30)
such that 119906(0) = 1199060and 119889119906119889119905 isin 1198712
(0 119879119867) Reversely let 119906be a solution of the variational problem (30) then (120597119906120597119905) +(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3)) = 0
And the complete follows result
6 ISRN Signal Processing
Theorem 9 For 1199060isin 119867
1(Ω) exists a unique solution
119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867
1(Ω)) (31)
verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(32)
43 Variational Formulation Let 119907 isin 1198673(Ω) and (to make
sense to integrals) multiply the equation
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (33)
of the system (29) by the test function 1205972119907120597119909
2 after weintegrate onΩ then it comes out that
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909 = 0 (34)
By using theGreen formula (seeDefinition 4) and integrationby parts we get firstly
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909
= [ℎ (119909)
1205973119906
1205971199093
1205972119907
1205971199092]
120597Ω
minus int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
(35)
and secondly
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
119906
1205972119907
1205971199092119889119909
int
Ω
119906
1205972119907
1205971199092119889119909 = int
120597Ω
119906
120597119907
120597119899
119889120590 minus int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(36)
As we are in one-dimension case the normal derivative andthe gradient on the edge in the use of the Green formula arethe same that is to say
int
120597Ω
119906
120597119907
120597119899
119889120590 = int
120597Ω
119906
120597119907
120597119909
119889120590 (37)
In the following we adopt the notation 120597119907120597119899Considering 119906 isin 1198673
(Ω) such that
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
119907 isin 119881 = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(38)
it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
forall119907 isin 119881
(39)
Let us consider the following bilinear forms
119886 (119906 119907) = int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
119887 (119906 119907) = int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(40)
The addition of boundaries conditions transforms our initialproblem into the system
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
119906 (0 119909) = 1199060
in Ω
(41)
The variational formulation of our problem consequentlysubject to additional conditions that would cause a change ofspace is the following
Find 119906 isin 119881 such that 119889
119889119905
(119887 (119906 119907)) + 119886 (119906 119907) = 0
forall119907 isin 119881
(42)
where 119886 and 119887 are like in Theorem 7This kind of problem is studied in [21] However before
continuing the resolution we explore some properties of thespace 119881
431 On the Quotient Space 119881R The space 119881 is defined by119863(119886) cap 119863(119887) where 119863(119886) and 119863(119887) are the domains of 119886 and119887 Let us consider the quotient space
119881
R= 119907 + 119888 119888 isin R such that 119907 isin 1198673
(Ω)
and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(43)
and the new spaceV defined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (44)
Let 119907 isin 119881R and let 119906 be a representative of 119907 class then(120597119907120597119899)|
120597Ωis defined by
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597 (119906 + 119888)
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 with 119888 isin R (45)
ISRN Signal Processing 7
Thenullity ofintΩ119907119889119909meansexist119888 isin R constant such thatint
Ω(119906+
119888)119889119909 = 0 with 119906 one representative of 119907
if 119888 = minus 1
|Ω|
int
Ω
119906119889119909 then
int
Ω
119907119889119909 = int
Ω
(119906 + 119888) 119889119909
= int
Ω
(119906 minus
1
|Ω|
int
Ω
119906119889119909)119889119909 = 0
(46)
So for all 119907 isin 119881R we have intΩ119907119889119909 = 0 Accordingly any
element of the quotient space 119881R is on one hand in1198673(Ω)
with normal derivative on the boundaries of Ω null and onthe other hand is zero average onΩThus the quotient space119881R is identified with the space V We can work later inthe space V with zero mean The final variational problemto solve is the following
find 119906isinV such that 119889
119889119905
(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V
(47)
432 Application of the Lions Theorem To apply the Lionstheorem we need to identify the applications 119886 119887 and 119861 andthe spaces119867 and119881 And finally wemust prove that the normdefined inV which derives from the scalar product given bythe bilinear application 119886 is equivalent to the norm of1198673
(Ω)Let us define 119861 as the identity and let us define 119886 and 119887 by
|119886(119906 119907)| = (radicℎ(1205973119906120597119909
3) radicℎ(120597
3119907120597119909
3))
1198712(Ω)
and 119887(119906 119907) =intΩ(120597119906120597119909)(120597119907120597119909)119889119909 obtained from (39) (40) the spaces
119881 = V and119867 are define as follows(a) The linear operator 119861 in the variational prob-
lem (11) in the Lions theorem (119889119906119889119905 119861119907)119867
denotes ascalar product on 119867 Indeed we define 119867 = 119907 isin
1198671(Ω) such that (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 = 0
The operator 119861 = 119868 119861119907 = 119907 is the identity whichis continuous (verify the Lions conditions Theorem 7) andthe term (119889119906119889119905 119861119907)
119867= (119889119889119905)(119887(119906 119907)) is equal to
int
Ω
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
119889119909 = int
Ω
119889
119889119905
(
120597119906
120597119909
)
120597119907
120597119909
119889119909
= (
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
)
1198712(Ω)
(48)
In this identification ((120597120597119909)(119889119906119889119905) 120597119907120597119909)1198712(Ω)
is a wellscalar product in the subspace 119867 of 1198671
(Ω) formed byfunctions with zero mean Indeed according to Poincare-Wirtinger inequality (see Definition 5) in1198671
(Ω)10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119909
100381710038171003817100381710038171003817100381710038171198671(Ω)
le
1
120582
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(49)
where 120582 is the first positive eigenvalue of Laplace operator inΩ But as 119906 119907 isin V then int
Ω119906119889119909 = 0 consequently there
exists 1198622gt 0 such that
1199061198671(Ω)
le 1198622
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(50)
that is to say the 1198671(Ω) norm is equivalent to the gradient
norm inV Hence one can replace the scalar product of thegradient by1198671
(Ω) scalar productHereinafter we complete the proof that the precedent
choices allow the application of the Lions theorem in ourcontext
(b) Vis an Hilbert space with equivalent norm to 1198673(Ω)
norm
Theorem 10 The spaceV resulting from (11) andV analysisdefined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (51)
is an Hilbert space with the a norm equivalent to1198673(Ω) norm
Proof At first V = 119907 isin 1198673(Ω) | (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 =
0 is a closed vector subspace of 1198673(Ω) Note that the
condition of the nullity of the mean in V may be replacedby int
ΓΩ
119906119889119909 = 0 where ΓΩis a nonempty open subset of Ω In
this caseV becomes rather
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω verifying
int
ΓΩ
119907119889119909 = 0 and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(52)
From the fact that
0 lt 120572 le ℎ (119909) le 120573 lt 1 forall119909 isin Ω (53)
the norms built with the two bilinear applications 119886 and 119887 areequivalent to the norm constructed by the linear form notedby | |2V with 119907 isin V and given by
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(54)
Indeed | |2V is as well a norm inVThere is no difficulty toformally verify the triangular inequality If |119907|2V = 0 then120597119907120597119909
2
1198712(Ω)
= 0 and 12059731199071205971199093
2
1198712(Ω)
= 0 Consequently 119907 isnecessarily constant almost everywhere inΩ for example 119907 isequal to a constant real 119870 But as int
Ω119907119889119909 = 0 implies 119870|Ω| =
0 so119870 = 119907 = 0 We have thus built a standard scalar producton V from bilinear forms of the variational formulation Vis a Hilbert space with the scalar product associated withthe norm (54) In addition this norm is equivalent to the119867
3(Ω) norm as it shown in Lemma A2 Now we just need
to prove that the correspondence (119906 119907) isin V timesV 997891rarr 119886(119906 119907)
is continuous and coercive on the Hilbert spaceV
8 ISRN Signal Processing
433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V
|119886 (119906 119907)| =
1003816100381610038161003816100381610038161003816100381610038161003816
(radicℎ
1205973119906
1205971199093 radicℎ
1205973119907
1205971199093)
1198712(Ω)
1003816100381610038161003816100381610038161003816100381610038161003816
le 120573
100381710038171003817100381710038171003817100381710038171003817
1205973119906
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
sdot
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
le 120573|119906|V sdot |119907|V
(55)
hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem
For
119907 isin V minus1199072
1198671(Ω)
le minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(56)
hence
|119907|2
V minus 1199072
1198671(Ω)
le |119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(57)
nevertheless
119886 (119907 119907) =
100381710038171003817100381710038171003817100381710038171003817
radicℎ
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 120572
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= 120572[|119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]
(58)
Thus
119886 (119907 119907) ge 120572 [|119907|2
V minus 1199072
1198671(Ω)] forall119907 isin V (59)
We have found thus (120588 120583) = (120572 120572) such that
119886 (119907 119907) ge 120588|119907|2
V minus 1205831199072
1198671(Ω) forall119907 isin V (60)
All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows
Theorem 11 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905
119906 (0) = 1199060
119889119906
119889119905
isin 1198712(0 119879119867)
(61)
434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0 forall119907 isin V (62)
As we must consider a test function
119907 isin 1198673(Ω) such that 120597
2119907
1205971199092still in 119863 (Ω)
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(63)
Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that
int
Ω
(120601 + 119870) 119889119909 = 0 (64)
We pose the following Neumann problem
1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(65)
Its variational formulation gives a unique solution in119867 =
119908 isin 1198671(Ω) | int
Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672
(Ω)
implies the regularity of the solution which is in 119907 isin 1198673(Ω)
consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the
first term can be written as
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
(int
Ω
119906
1205972119907
1205971199092119889119909 minus int
120597Ω
119906
120597119907
120597119899
119889120590)
as 1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
and it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906 (120601 + 119870) 119889119909
=
119889
119889119905
(int
Ω
119906120601119889119909 + 119870int
Ω
119906119889119909)
(66)
However intΩ119906119889119909 = 0 subsequently
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906120601119889119909 = int
Ω
119889119906
119889119905
120601119889119909 (67)
The second term of (62) is written as follows
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
ℎ (119909)
1205973119906
1205971199093
120597120601
120597119909
119889119909 (68)
which gives in the distribution sense
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909 (69)
from which
int
Ω
119889119906
119889119905
120601119889119909+int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)
(70)
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
6 ISRN Signal Processing
Theorem 9 For 1199060isin 119867
1(Ω) exists a unique solution
119906 isin 1198712(0 119879V) cap 119862 ([0 119879] 119867
1(Ω)) (31)
verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(32)
43 Variational Formulation Let 119907 isin 1198673(Ω) and (to make
sense to integrals) multiply the equation
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (33)
of the system (29) by the test function 1205972119907120597119909
2 after weintegrate onΩ then it comes out that
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909 = 0 (34)
By using theGreen formula (seeDefinition 4) and integrationby parts we get firstly
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)
1205972119907
1205971199092119889119909
= [ℎ (119909)
1205973119906
1205971199093
1205972119907
1205971199092]
120597Ω
minus int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
(35)
and secondly
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
119906
1205972119907
1205971199092119889119909
int
Ω
119906
1205972119907
1205971199092119889119909 = int
120597Ω
119906
120597119907
120597119899
119889120590 minus int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(36)
As we are in one-dimension case the normal derivative andthe gradient on the edge in the use of the Green formula arethe same that is to say
int
120597Ω
119906
120597119907
120597119899
119889120590 = int
120597Ω
119906
120597119907
120597119909
119889120590 (37)
In the following we adopt the notation 120597119907120597119899Considering 119906 isin 1198673
(Ω) such that
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
119907 isin 119881 = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(38)
it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
forall119907 isin 119881
(39)
Let us consider the following bilinear forms
119886 (119906 119907) = int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909
119887 (119906 119907) = int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909
(40)
The addition of boundaries conditions transforms our initialproblem into the system
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 in [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (on the edge of Ω)
119906 (0 119909) = 1199060
in Ω
(41)
The variational formulation of our problem consequentlysubject to additional conditions that would cause a change ofspace is the following
Find 119906 isin 119881 such that 119889
119889119905
(119887 (119906 119907)) + 119886 (119906 119907) = 0
forall119907 isin 119881
(42)
where 119886 and 119887 are like in Theorem 7This kind of problem is studied in [21] However before
continuing the resolution we explore some properties of thespace 119881
431 On the Quotient Space 119881R The space 119881 is defined by119863(119886) cap 119863(119887) where 119863(119886) and 119863(119887) are the domains of 119886 and119887 Let us consider the quotient space
119881
R= 119907 + 119888 119888 isin R such that 119907 isin 1198673
(Ω)
and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(43)
and the new spaceV defined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (44)
Let 119907 isin 119881R and let 119906 be a representative of 119907 class then(120597119907120597119899)|
120597Ωis defined by
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597 (119906 + 119888)
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
=
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 with 119888 isin R (45)
ISRN Signal Processing 7
Thenullity ofintΩ119907119889119909meansexist119888 isin R constant such thatint
Ω(119906+
119888)119889119909 = 0 with 119906 one representative of 119907
if 119888 = minus 1
|Ω|
int
Ω
119906119889119909 then
int
Ω
119907119889119909 = int
Ω
(119906 + 119888) 119889119909
= int
Ω
(119906 minus
1
|Ω|
int
Ω
119906119889119909)119889119909 = 0
(46)
So for all 119907 isin 119881R we have intΩ119907119889119909 = 0 Accordingly any
element of the quotient space 119881R is on one hand in1198673(Ω)
with normal derivative on the boundaries of Ω null and onthe other hand is zero average onΩThus the quotient space119881R is identified with the space V We can work later inthe space V with zero mean The final variational problemto solve is the following
find 119906isinV such that 119889
119889119905
(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V
(47)
432 Application of the Lions Theorem To apply the Lionstheorem we need to identify the applications 119886 119887 and 119861 andthe spaces119867 and119881 And finally wemust prove that the normdefined inV which derives from the scalar product given bythe bilinear application 119886 is equivalent to the norm of1198673
(Ω)Let us define 119861 as the identity and let us define 119886 and 119887 by
|119886(119906 119907)| = (radicℎ(1205973119906120597119909
3) radicℎ(120597
3119907120597119909
3))
1198712(Ω)
and 119887(119906 119907) =intΩ(120597119906120597119909)(120597119907120597119909)119889119909 obtained from (39) (40) the spaces
119881 = V and119867 are define as follows(a) The linear operator 119861 in the variational prob-
lem (11) in the Lions theorem (119889119906119889119905 119861119907)119867
denotes ascalar product on 119867 Indeed we define 119867 = 119907 isin
1198671(Ω) such that (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 = 0
The operator 119861 = 119868 119861119907 = 119907 is the identity whichis continuous (verify the Lions conditions Theorem 7) andthe term (119889119906119889119905 119861119907)
119867= (119889119889119905)(119887(119906 119907)) is equal to
int
Ω
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
119889119909 = int
Ω
119889
119889119905
(
120597119906
120597119909
)
120597119907
120597119909
119889119909
= (
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
)
1198712(Ω)
(48)
In this identification ((120597120597119909)(119889119906119889119905) 120597119907120597119909)1198712(Ω)
is a wellscalar product in the subspace 119867 of 1198671
(Ω) formed byfunctions with zero mean Indeed according to Poincare-Wirtinger inequality (see Definition 5) in1198671
(Ω)10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119909
100381710038171003817100381710038171003817100381710038171198671(Ω)
le
1
120582
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(49)
where 120582 is the first positive eigenvalue of Laplace operator inΩ But as 119906 119907 isin V then int
Ω119906119889119909 = 0 consequently there
exists 1198622gt 0 such that
1199061198671(Ω)
le 1198622
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(50)
that is to say the 1198671(Ω) norm is equivalent to the gradient
norm inV Hence one can replace the scalar product of thegradient by1198671
(Ω) scalar productHereinafter we complete the proof that the precedent
choices allow the application of the Lions theorem in ourcontext
(b) Vis an Hilbert space with equivalent norm to 1198673(Ω)
norm
Theorem 10 The spaceV resulting from (11) andV analysisdefined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (51)
is an Hilbert space with the a norm equivalent to1198673(Ω) norm
Proof At first V = 119907 isin 1198673(Ω) | (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 =
0 is a closed vector subspace of 1198673(Ω) Note that the
condition of the nullity of the mean in V may be replacedby int
ΓΩ
119906119889119909 = 0 where ΓΩis a nonempty open subset of Ω In
this caseV becomes rather
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω verifying
int
ΓΩ
119907119889119909 = 0 and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(52)
From the fact that
0 lt 120572 le ℎ (119909) le 120573 lt 1 forall119909 isin Ω (53)
the norms built with the two bilinear applications 119886 and 119887 areequivalent to the norm constructed by the linear form notedby | |2V with 119907 isin V and given by
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(54)
Indeed | |2V is as well a norm inVThere is no difficulty toformally verify the triangular inequality If |119907|2V = 0 then120597119907120597119909
2
1198712(Ω)
= 0 and 12059731199071205971199093
2
1198712(Ω)
= 0 Consequently 119907 isnecessarily constant almost everywhere inΩ for example 119907 isequal to a constant real 119870 But as int
Ω119907119889119909 = 0 implies 119870|Ω| =
0 so119870 = 119907 = 0 We have thus built a standard scalar producton V from bilinear forms of the variational formulation Vis a Hilbert space with the scalar product associated withthe norm (54) In addition this norm is equivalent to the119867
3(Ω) norm as it shown in Lemma A2 Now we just need
to prove that the correspondence (119906 119907) isin V timesV 997891rarr 119886(119906 119907)
is continuous and coercive on the Hilbert spaceV
8 ISRN Signal Processing
433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V
|119886 (119906 119907)| =
1003816100381610038161003816100381610038161003816100381610038161003816
(radicℎ
1205973119906
1205971199093 radicℎ
1205973119907
1205971199093)
1198712(Ω)
1003816100381610038161003816100381610038161003816100381610038161003816
le 120573
100381710038171003817100381710038171003817100381710038171003817
1205973119906
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
sdot
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
le 120573|119906|V sdot |119907|V
(55)
hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem
For
119907 isin V minus1199072
1198671(Ω)
le minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(56)
hence
|119907|2
V minus 1199072
1198671(Ω)
le |119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(57)
nevertheless
119886 (119907 119907) =
100381710038171003817100381710038171003817100381710038171003817
radicℎ
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 120572
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= 120572[|119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]
(58)
Thus
119886 (119907 119907) ge 120572 [|119907|2
V minus 1199072
1198671(Ω)] forall119907 isin V (59)
We have found thus (120588 120583) = (120572 120572) such that
119886 (119907 119907) ge 120588|119907|2
V minus 1205831199072
1198671(Ω) forall119907 isin V (60)
All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows
Theorem 11 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905
119906 (0) = 1199060
119889119906
119889119905
isin 1198712(0 119879119867)
(61)
434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0 forall119907 isin V (62)
As we must consider a test function
119907 isin 1198673(Ω) such that 120597
2119907
1205971199092still in 119863 (Ω)
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(63)
Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that
int
Ω
(120601 + 119870) 119889119909 = 0 (64)
We pose the following Neumann problem
1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(65)
Its variational formulation gives a unique solution in119867 =
119908 isin 1198671(Ω) | int
Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672
(Ω)
implies the regularity of the solution which is in 119907 isin 1198673(Ω)
consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the
first term can be written as
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
(int
Ω
119906
1205972119907
1205971199092119889119909 minus int
120597Ω
119906
120597119907
120597119899
119889120590)
as 1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
and it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906 (120601 + 119870) 119889119909
=
119889
119889119905
(int
Ω
119906120601119889119909 + 119870int
Ω
119906119889119909)
(66)
However intΩ119906119889119909 = 0 subsequently
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906120601119889119909 = int
Ω
119889119906
119889119905
120601119889119909 (67)
The second term of (62) is written as follows
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
ℎ (119909)
1205973119906
1205971199093
120597120601
120597119909
119889119909 (68)
which gives in the distribution sense
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909 (69)
from which
int
Ω
119889119906
119889119905
120601119889119909+int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)
(70)
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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DistributedSensor Networks
International Journal of
ISRN Signal Processing 7
Thenullity ofintΩ119907119889119909meansexist119888 isin R constant such thatint
Ω(119906+
119888)119889119909 = 0 with 119906 one representative of 119907
if 119888 = minus 1
|Ω|
int
Ω
119906119889119909 then
int
Ω
119907119889119909 = int
Ω
(119906 + 119888) 119889119909
= int
Ω
(119906 minus
1
|Ω|
int
Ω
119906119889119909)119889119909 = 0
(46)
So for all 119907 isin 119881R we have intΩ119907119889119909 = 0 Accordingly any
element of the quotient space 119881R is on one hand in1198673(Ω)
with normal derivative on the boundaries of Ω null and onthe other hand is zero average onΩThus the quotient space119881R is identified with the space V We can work later inthe space V with zero mean The final variational problemto solve is the following
find 119906isinV such that 119889
119889119905
(119887 (119906 119907))+119886 (119906 119907)=0 forall119907 isin V
(47)
432 Application of the Lions Theorem To apply the Lionstheorem we need to identify the applications 119886 119887 and 119861 andthe spaces119867 and119881 And finally wemust prove that the normdefined inV which derives from the scalar product given bythe bilinear application 119886 is equivalent to the norm of1198673
(Ω)Let us define 119861 as the identity and let us define 119886 and 119887 by
|119886(119906 119907)| = (radicℎ(1205973119906120597119909
3) radicℎ(120597
3119907120597119909
3))
1198712(Ω)
and 119887(119906 119907) =intΩ(120597119906120597119909)(120597119907120597119909)119889119909 obtained from (39) (40) the spaces
119881 = V and119867 are define as follows(a) The linear operator 119861 in the variational prob-
lem (11) in the Lions theorem (119889119906119889119905 119861119907)119867
denotes ascalar product on 119867 Indeed we define 119867 = 119907 isin
1198671(Ω) such that (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 = 0
The operator 119861 = 119868 119861119907 = 119907 is the identity whichis continuous (verify the Lions conditions Theorem 7) andthe term (119889119906119889119905 119861119907)
119867= (119889119889119905)(119887(119906 119907)) is equal to
int
Ω
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
119889119909 = int
Ω
119889
119889119905
(
120597119906
120597119909
)
120597119907
120597119909
119889119909
= (
120597
120597119909
(
119889119906
119889119905
)
120597119907
120597119909
)
1198712(Ω)
(48)
In this identification ((120597120597119909)(119889119906119889119905) 120597119907120597119909)1198712(Ω)
is a wellscalar product in the subspace 119867 of 1198671
(Ω) formed byfunctions with zero mean Indeed according to Poincare-Wirtinger inequality (see Definition 5) in1198671
(Ω)10038171003817100381710038171003817100381710038171003817
119906 minus
1
|Ω|
int
Ω
119906119889119909
100381710038171003817100381710038171003817100381710038171198671(Ω)
le
1
120582
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(49)
where 120582 is the first positive eigenvalue of Laplace operator inΩ But as 119906 119907 isin V then int
Ω119906119889119909 = 0 consequently there
exists 1198622gt 0 such that
1199061198671(Ω)
le 1198622
10038171003817100381710038171003817100381710038171003817
120597119906
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
(50)
that is to say the 1198671(Ω) norm is equivalent to the gradient
norm inV Hence one can replace the scalar product of thegradient by1198671
(Ω) scalar productHereinafter we complete the proof that the precedent
choices allow the application of the Lions theorem in ourcontext
(b) Vis an Hilbert space with equivalent norm to 1198673(Ω)
norm
Theorem 10 The spaceV resulting from (11) andV analysisdefined by
V = 119907 isin 1198673(Ω) |
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 int
Ω
119907119889119909 = 0 (51)
is an Hilbert space with the a norm equivalent to1198673(Ω) norm
Proof At first V = 119907 isin 1198673(Ω) | (120597119907120597119899)|
120597Ω= 0 int
Ω119907119889119909 =
0 is a closed vector subspace of 1198673(Ω) Note that the
condition of the nullity of the mean in V may be replacedby int
ΓΩ
119906119889119909 = 0 where ΓΩis a nonempty open subset of Ω In
this caseV becomes rather
V119890= 119907 isin 119867
3(Ω) such that existΓ
Ωsub Ω verifying
int
ΓΩ
119907119889119909 = 0 and 120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(52)
From the fact that
0 lt 120572 le ℎ (119909) le 120573 lt 1 forall119909 isin Ω (53)
the norms built with the two bilinear applications 119886 and 119887 areequivalent to the norm constructed by the linear form notedby | |2V with 119907 isin V and given by
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(54)
Indeed | |2V is as well a norm inVThere is no difficulty toformally verify the triangular inequality If |119907|2V = 0 then120597119907120597119909
2
1198712(Ω)
= 0 and 12059731199071205971199093
2
1198712(Ω)
= 0 Consequently 119907 isnecessarily constant almost everywhere inΩ for example 119907 isequal to a constant real 119870 But as int
Ω119907119889119909 = 0 implies 119870|Ω| =
0 so119870 = 119907 = 0 We have thus built a standard scalar producton V from bilinear forms of the variational formulation Vis a Hilbert space with the scalar product associated withthe norm (54) In addition this norm is equivalent to the119867
3(Ω) norm as it shown in Lemma A2 Now we just need
to prove that the correspondence (119906 119907) isin V timesV 997891rarr 119886(119906 119907)
is continuous and coercive on the Hilbert spaceV
8 ISRN Signal Processing
433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V
|119886 (119906 119907)| =
1003816100381610038161003816100381610038161003816100381610038161003816
(radicℎ
1205973119906
1205971199093 radicℎ
1205973119907
1205971199093)
1198712(Ω)
1003816100381610038161003816100381610038161003816100381610038161003816
le 120573
100381710038171003817100381710038171003817100381710038171003817
1205973119906
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
sdot
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
le 120573|119906|V sdot |119907|V
(55)
hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem
For
119907 isin V minus1199072
1198671(Ω)
le minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(56)
hence
|119907|2
V minus 1199072
1198671(Ω)
le |119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(57)
nevertheless
119886 (119907 119907) =
100381710038171003817100381710038171003817100381710038171003817
radicℎ
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 120572
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= 120572[|119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]
(58)
Thus
119886 (119907 119907) ge 120572 [|119907|2
V minus 1199072
1198671(Ω)] forall119907 isin V (59)
We have found thus (120588 120583) = (120572 120572) such that
119886 (119907 119907) ge 120588|119907|2
V minus 1205831199072
1198671(Ω) forall119907 isin V (60)
All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows
Theorem 11 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905
119906 (0) = 1199060
119889119906
119889119905
isin 1198712(0 119879119867)
(61)
434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0 forall119907 isin V (62)
As we must consider a test function
119907 isin 1198673(Ω) such that 120597
2119907
1205971199092still in 119863 (Ω)
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(63)
Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that
int
Ω
(120601 + 119870) 119889119909 = 0 (64)
We pose the following Neumann problem
1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(65)
Its variational formulation gives a unique solution in119867 =
119908 isin 1198671(Ω) | int
Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672
(Ω)
implies the regularity of the solution which is in 119907 isin 1198673(Ω)
consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the
first term can be written as
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
(int
Ω
119906
1205972119907
1205971199092119889119909 minus int
120597Ω
119906
120597119907
120597119899
119889120590)
as 1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
and it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906 (120601 + 119870) 119889119909
=
119889
119889119905
(int
Ω
119906120601119889119909 + 119870int
Ω
119906119889119909)
(66)
However intΩ119906119889119909 = 0 subsequently
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906120601119889119909 = int
Ω
119889119906
119889119905
120601119889119909 (67)
The second term of (62) is written as follows
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
ℎ (119909)
1205973119906
1205971199093
120597120601
120597119909
119889119909 (68)
which gives in the distribution sense
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909 (69)
from which
int
Ω
119889119906
119889119905
120601119889119909+int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)
(70)
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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Navigation and Observation
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DistributedSensor Networks
International Journal of
8 ISRN Signal Processing
433 The Application 119886( sdot sdot ) Is Continuous and Coercive For119906 119907 isin V
|119886 (119906 119907)| =
1003816100381610038161003816100381610038161003816100381610038161003816
(radicℎ
1205973119906
1205971199093 radicℎ
1205973119907
1205971199093)
1198712(Ω)
1003816100381610038161003816100381610038161003816100381610038161003816
le 120573
100381710038171003817100381710038171003817100381710038171003817
1205973119906
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
sdot
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
le 120573|119906|V sdot |119907|V
(55)
hence the continuity of119886(sdot sdot) is onVtimesV In order to completethe application conditions of Theorem 7 we only check thefirst condition of the same theorem
For
119907 isin V minus1199072
1198671(Ω)
le minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(56)
hence
|119907|2
V minus 1199072
1198671(Ω)
le |119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(57)
nevertheless
119886 (119907 119907) =
100381710038171003817100381710038171003817100381710038171003817
radicℎ
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 120572
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= 120572[|119907|2
V minus
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]
(58)
Thus
119886 (119907 119907) ge 120572 [|119907|2
V minus 1199072
1198671(Ω)] forall119907 isin V (59)
We have found thus (120588 120583) = (120572 120572) such that
119886 (119907 119907) ge 120588|119907|2
V minus 1205831199072
1198671(Ω) forall119907 isin V (60)
All the conditions of Theorem 7 are acquired now the mainexistence result for solution of system (41) can be enunciatedas follows
Theorem 11 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) of the problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0
119891119900119903 119886119899119910 119907 isin V 119904119906119888ℎ 119905ℎ119886119905
119906 (0) = 1199060
119889119906
119889119905
isin 1198712(0 119879119867)
(61)
434 Equivalence with the Initial Problem Reversely let 119906 bea solution of the variational problem
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 + int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = 0 forall119907 isin V (62)
As we must consider a test function
119907 isin 1198673(Ω) such that 120597
2119907
1205971199092still in 119863 (Ω)
120597119907
120597120584
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(63)
Let us take a function test 120601 isin 119863(Ω) and119870 isin R such that
int
Ω
(120601 + 119870) 119889119909 = 0 (64)
We pose the following Neumann problem
1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
(65)
Its variational formulation gives a unique solution in119867 =
119908 isin 1198671(Ω) | int
Ω119908119889119909 = 0 and the regularity of 120601 isin 1198672
(Ω)
implies the regularity of the solution which is in 119907 isin 1198673(Ω)
consequently 119907 isin VReturning to (62) with 119907 solution of the problem (65) the
first term can be written as
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
(int
Ω
119906
1205972119907
1205971199092119889119909 minus int
120597Ω
119906
120597119907
120597119899
119889120590)
as 1205972119907
1205971199092= 120601 + 119870
120597119907
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0
and it comes out that
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906 (120601 + 119870) 119889119909
=
119889
119889119905
(int
Ω
119906120601119889119909 + 119870int
Ω
119906119889119909)
(66)
However intΩ119906119889119909 = 0 subsequently
119889
119889119905
int
Ω
120597119906
120597119909
120597119907
120597119909
119889119909 =
119889
119889119905
int
Ω
119906120601119889119909 = int
Ω
119889119906
119889119905
120601119889119909 (67)
The second term of (62) is written as follows
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
ℎ (119909)
1205973119906
1205971199093
120597120601
120597119909
119889119909 (68)
which gives in the distribution sense
int
Ω
ℎ (119909)
1205973119906
1205971199093
1205973119907
1205971199093119889119909 = int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909 (69)
from which
int
Ω
119889119906
119889119905
120601119889119909+int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)120601119889119909=0 forall120601 isin 119863 (Ω)
(70)
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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DistributedSensor Networks
International Journal of
ISRN Signal Processing 9
We deduce that
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 (71)
We verify now that (12059731199061205971199093)|
120597Ω= 0 Returning to (70) with
120601 = 1205972119907120597119909
2 and according to the Green formula we have
int
Ω
[
119889119906
119889119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)]
1205972119907
1205971199092119889119909
minus int
120597Ω
ℎ (119909)
1205973119906
1205971199093
120597
120597119899
(
120597119907
120597119909
) sdot 119899119889120590 = 0
forall119907 isin V
(72)
But as (120597120597119899)(120597119907120597119909) sdot 119899 = (120597120597119909)(120597119907120597119909) = 12059721199071205971199092 it givesaccording to (71) int
120597Ωℎ(119909)(120597
3119906120597119909
3)(120597
2119907120597119909
2)119889120590 = 0 forall119907 isin
V and consequently ℎ(119909)(12059731199061205971199093) = 0 on boundaries ofΩ
or as ℎ(119909) = 0 12059731199061205971199093= 0 on boundaries ofΩ
In conclusion 119906 is solution of the initial equation (41)then we enunciate the following theorem
Theorem 12 For 1199060isin 119867
1(Ω) exists a unique solution 119906 isin
1198712(0 119879V) cap 119862([0 119879]119867
1(Ω)) verifying
120597119906
120597119905
+
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093) = 0 119894119899 [0 119879] times Ω
1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
120597119906
120597119899
10038161003816100381610038161003816100381610038161003816120597Ω
= 0 (119900119899 119887119900119906119899119889119886119903119894119890119904 119900119891 Ω)
119906 (0 119909) = 1199060
119894119899 Ω
(73)
435 Some Remarks on the Modeling Space The nullitycondition of the average for functions inV has been decisivein the construction of the norm on this space This spaceis prima facie clearly not natural for EMD because of entryfunction119906
0that is not necessarily zeromean But at the cost of
working with an initial condition 1199070isin 119867
1(Ω) such that 119907
0=
1199060minus (1|Ω|) int
Ω1199060119889119909 we can consider119867 as1198671
(Ω) functionswith null local mean
The Nullity of the Mean Envelope A Characteristic of V andEMD Algorithm This condition far from being superfluouswas even expected because the space V is the space of themean envelopes which must be zero mean in EMD stiftingprocess for mode extraction If it were to calculate the upperor lower envelopes it should be noted that the conditionof the mean nullity raised a query In EMD algorithm thisnullity condition is affected on the mean envelope and not onupper or lower envelopes This question does not arise in themodel giving the envelope interpolating inflection points ofa signal The elements ofV
119890are not zero mean on Ω but on
a non-empty subset ΓΩofΩ
The nullity of the mean of the signal on a ΓΩ
is notimmediately visible andwarranty to decompose any function
We show in Lemma A1 that each relevant function to bedecomposed by EMD is an integrable function possessing atleast three extrema belonging to V
119890 The modeling space of
the extrema interpolation isV119890defined in (52)
Let 119891 be a function admitting at least three extrema wecan consider that 119891 passes at least once a zero at a point 119909
119911
in its definition domain Otherwise we just work with thetranslatory (as our model is invariant by translation of theinput function) vector which is equals to the half amplitudeof 119891
5 Convergence of the Sequence (119906119899)119899isinN of
Solutions of (41) to a Solution of theCalled Degenerate Problem
In the following sections we demonstrate that the sequence(119906
119899)119899isinN of solutions of the non degenerate problem is
bounded Next we prove that there exists a subsequence of(119906
119899)119899isinN which converges weakly to an element 119906 isin V and
finally that this element is solution of the degenerate initialproblem
51 Some Estimations From (28)we have in the distributionssense (multiplying by 1205972119906
119899120597119909
2)
int
Ω
120597119906119899
120597119905
1205972119906119899
1205971199092119889119909 + int
Ω
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092119889119909 = 0 (74)
which is equivalent to
⟨
120597119906119899
120597119905
1205972119906119899
1205971199092⟩
1198811015840119881
+⟨
120597
120597119909
(ℎ119899(119909)
1205973119906119899
1205971199093)
1205972119906119899
1205971199092⟩
1198811015840119881
=0
(75)
After integration by parts it comes out that
1
2
int
Ω
120597
120597119905
(
10038161003816100381610038161003816100381610038161003816
120597119906119899
120597119909
10038161003816100381610038161003816100381610038161003816
2
)119889119909 + int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 = 0 (76)
then by integrating with respect to the variable 119905 isin]0 119879[ itcomes out that
1
2
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
minus
1
2
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904 = 0
(77)
which implies that
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
=
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(78)
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
International Journal of
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 ISRN Signal Processing
But as ℎ119899(119909) ge 120572 we have
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
int
Ω
(
1205973119906119899
1205971199093)
2
119889119909 119889119904
(79)
or
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2int
119905
0
int
Ω
ℎ119899(119909) (
1205973119906119899
1205971199093)
2
119889119909 119889119904
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
100381710038171003817100381710038171003817100381710038171003817
1205973119906119899
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904
(80)
from which10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+ 2120572int
119905
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904
(81)
Thus on one hand10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge
10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(82)
But10038171003817100381710038171003817100381710038171003817
120597119906119899(119905 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
ge 12058221003817100381710038171003817119906119899
1003817100381710038171003817
2
1198671(Ω) (83)
where 120582 comes from inequality (49) which implies that
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
le
1
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(84)
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879119867
1(Ω))
le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(85)
Then the sequence (119906119899)119899isinN is bounded in119871infin
(0 1198791198671(Ω)) and
in 1198712(0 119879119867
1(Ω)) And on the other hand
2120572int
119879
0
[1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
Vminus
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
]119889119904 le
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(86)
which gives
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(87)
But as10038171003817100381710038171003817100381710038171003817
120597119906119899(119904 sdot)
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω) (88)
therefore
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le int
119879
0
1003817100381710038171003817119906119899(119904 sdot)
1003817100381710038171003817
2
1198671(Ω)119889119904 +
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(89)
Thus
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le 119879
1003817100381710038171003817119906119899
1003817100381710038171003817
2
119871infin(0119879119867
1(Ω))
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(90)
which implies according to (84) that
int
119879
0
1003816100381610038161003816
1003817100381710038171003817119906119899
1003817100381710038171003817
1003816100381610038161003816
2
V119889119904 le
119879
1205822
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2120572
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(91)
We finally deduced the second estimate
1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V)
=1003817100381710038171003817119906119899
1003817100381710038171003817
2
1198712(0119879V119890)
le (
119879
1205822+
1
2120572
)
10038171003817100381710038171003817100381710038171003817
1205971199060
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(92)
Then the sequence (119906119899)119899isinN is bounded in 1198712
(0 119879V)
52 Weak Convergence of the Subsequence of (119906119899)119899isinN The
sequence (119906119899)119899isinN is bounded in the space 1198712
(0 119879V) Thenthere exists a subsequence (119906
119899119896)119896isinN
that converge weakly to 119906in 1198712
(0 119879V) with 119889119906119889119905 isin 1198712(0 119879 119881
1015840)
Leting 119907 isin V be a test function we have the following
Firstly As 119906119899119896rarr 119906weakly in 1198712
(0 119879 119881) therefore 119906119899119896rarr 119906
weakly in 1198712(0 119879119867
1(Ω))
Thus in the distributions sense
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 =
119889
119889119905
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 (93)
by integrating on [0 119905] we have
int
119905
0
int
Ω
119889119906119899119896
119889119904
1205972119907
1205971199092119889119909 119889119904 = minusint
119905
0
119889
119889119904
int
Ω
120597119906119899119896
120597119909
120597119907
120597119909
119889119909 119889119904
= minusint
119905
0
⟨
119889119906119899119896
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
= minusint
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909
(94)
By integrating by parts again the last equation we have
int
Ω
(119906119899119896(119905) minus 119906
119899119896(0))
1205972119907
1205971199092119889119909 = int
Ω
120597119906119899119896(119905)
120597119909
120597119907
120597119909
119889119909
minus int
Ω
120597119906119899119896(0)
120597119909
120597119907
120597119909
119889119909
(95)
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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DistributedSensor Networks
International Journal of
ISRN Signal Processing 11
This expression converges to
int
Ω
120597119906 (119905)
120597119909
120597119907
120597119909
119889119909 minus int
Ω
1205971199060
120597119909
120597119907
120597119909
119889119909 = int
Ω
[
120597119906 (119905)
120597119909
minus
1205971199060
120597119909
]
120597119907
120597119909
119889119909
= int
Ω
[119906 (119905) minus 1199060]
1205972119907
1205971199092119889119909
= int
119905
0
⟨
119889119906
119889119904
1205972119907
1205971199092⟩
1198811015840119881
119889119904
(96)
Therefore int119905
0intΩ(119889119906
119899119896119889119904)(120597
2119907120597119909
2)119889119909119889119904 converges when
119899119896tends to infinity to
int
119905
0
int
Ω
119889119906
119889119904
1205972119907
1205971199092119889119909 119889119904 (97)
Consequently 120597119906119899119896120597119905 rarr 120597119906120597119905 weakly in 1198712
(0 119879V1015840)
Secondly As ℎ119899(119909) = ℎ(119909) + (1119899)
we have 120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093minus ℎ (119909)
1205973119906
1205971199093)
=
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) +
1
119899119896
1205974119906119899119896
1205971199094
However intΩ
1205974119906119899119896
1205971199094119907119889119909 = int
120597Ω
1205973119906
1205971199093119907 sdot 119899119889120590
minus int
Ω
120597119907
120597119909
1205973119906119899119896
1205971199093119889119909
And because 1205973119906
1205971199093
100381610038161003816100381610038161003816100381610038161003816120597Ω
= 0
1205973119906119899119896
1205971199093
converges to 1205973119906
1205971199093
in 1198712(0 119879 119871
2(Ω))
(98)
then
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205974119906119899119896
1205971199094119907119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
=
1
119899119896
1003816100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973119906119899119896
1205971199093
120597119907
120597119909
119889119909
1003816100381610038161003816100381610038161003816100381610038161003816
le
1
119899119896
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
1003817100381710038171003817100381710038171003817100381710038171003817
1205973119906119899119896
1205971199093
10038171003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(99)
converges to zero when 119899119896tends to infinity
On the other hand from (98) we have100381610038161003816100381610038161003816100381610038161003816
int
Ω
120597
120597119909
(ℎ (119909)
1205973
1205971199093(119906
119899119896minus 119906)) 119907119889119909
100381610038161003816100381610038161003816100381610038161003816
le 120573
100381610038161003816100381610038161003816100381610038161003816
int
Ω
1205973
1205971199093(119906
119899119896minus 119906)
120597119907
120597119909
119889119909
100381610038161003816100381610038161003816100381610038161003816
(100)
which converges to zero in 1198712(0 119879 119871
2(Ω)) according to weak
convergence (see Definition 3) in 1198673(Ω) of 1205973119906
119899119896120597119909
3 to
1205973119906120597119909
3 Thus we have shown that in the distributions senseof the one hand
int
Ω
119889119906119899119896
119889119905
1205972119907
1205971199092119889119909 (101)
converges to
int
Ω
119889119906
119889119905
1205972119907
1205971199092119889119909 (102)
and on the other hand
int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (103)
converges to
int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (104)
Consequently for each 119907 isin 1198712(0 119879 119881
1015840)
int
Ω
119889119906119899119896
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ119899119896
1205973119906119899119896
1205971199093)119907119889119909 (105)
converges in the distributions sense to
int
Ω
119889119906
119889119905
119907119889119909 + int
Ω
120597
120597119909
(ℎ (119909)
1205973119906
1205971199093)119907119889119909 (106)
that is to say ⟨(119889119906119899119896119889119905) 119907⟩
1198811015840119881+ ⟨(120597120597119909)(ℎ
119899119896(119909)(120597
3119906119899119896120597119909
3))
119907⟩1198811015840119881convergeswhen 119899
119896tends to infinity to ⟨(119889119906119889119905) 119907⟩
1198811015840119881+
⟨(120597120597119909)(ℎ(119909)(1205973119906120597119909
3)) 119907⟩
1198811015840119881 It remains to show that the
limit 119906 is a solution of the degenerate problem ButintΩ(119889119906
119899119896119889119905)119907119889119909 + int
Ω(120597120597119909)(ℎ
119899119896(120597
3119906119899119896120597119909
3))119907119889119909 = 0
which implies in the distributions sense intΩ(119889119906119889119905)119907119889119909 +
intΩ(120597120597119909)(ℎ(119909)(120597
3119906120597119909
3))119907119889119909 = 0 In other words the
weak limit of the sequence (119906119899119896)119896isinN
is weak solution of theinitial degenerate problem Moreover this solution is uniqueand is in 119871
2(0 119879V) cap 119862([0 119879]119867
1(Ω)) Better using the
compact inclusion of V (which behaves like 1198673(Ω) because
of the equivalence of the norms) in 1198672(Ω) we conclude
that the sequence of solutions of non-degenerate problem(approximated problem) converges strongly in1198672
(Ω) to thesolution of the initial degenerate problem The space 1198672
(Ω)
is moreover the space of strong solutions of the initial system(25)
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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International Journal of
12 ISRN Signal Processing
6 Numerical Implementationand Applications
Deliberately we present the numerical implementation in thecase of dimension 119899 = 2 The reason is that we give examplesboth on 1D and 2D signals
In the sequel we adopt the following notation The meshof the domainΩ times [0 119879] is given by
(119905119896 119909
119894) = (119896Δ119905 119894Δ119909) for 119896 ge 0 119894 isin N (107)
We note by 119880119896
119894the approximate value of the solution
119906(119905 119909) at the point (119896Δ119905 119894Δ119909) = (119896Δ119905 119894)The discrete solution at the iteration 119896 is obtained at the
discrete time 119905119896and is denoted by 119880119896 An approximation of
the temporal derivative is 119906119905= 120597119906120597119905 at this same time 119905
119896=
119896Δ119905 is then (119880119896+1minus 119880
119896)Δ119905
61 Numerical Schemes for the PDE Resolution The discretemodel calculating the mean envelope can be written asfollows
119906119905= minus
2
sum
119894119895=1
1205971
119909119894(119892
1198941198951205971
1199091198941205972
119909119895119906) (108)
where 120597119902119909119894denotes the derivative of order 119902 isin 1 2 relatively
to the spatial coordinate 119909119894
To take into account the discontinuity of the diffusivityfunction we use the harmonic mean of 119892
119894119895defined by
119892119894+12119895
= [
119892minus1
119894119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894minus12119895
= [
119892minus1
119894minus1119895+ 119892
minus1
119894+1119895
2
]
minus1
119892119894119895+12
= [
119892minus1
119894119895+ 119892
minus1
119894119895+1
2
]
minus1
119892119894119895minus12
= [
119892minus1
119894119895+ 119892
minus1
119894119895minus1
2
]
minus1
(109)
Recall that 119892 is strictly positive which allows its inversion inthe previous system
We define also the numerical derivatives of 119907 by
119863+
119894119907 (119894 119895) = 119907 (119894 + 1 119895) minus 119907 (119894 119895)
forward difference on the 119909119894-direction
119863minus
119894119907 (119894 119895) = 119907 (119894 119895) minus 119907 (119894 minus 1 119895)
backward difference on the 119909119894-direction
1198630
119894119907 (119894 119895) = 119907 (119894 +
1
2
119895) minus 119907 (119894 minus
1
2
119895)
central difference on the 119909119894-direction
119863+
119895119907 (119894 119895) = 119907 (119894 119895 + 1) minus 119907 (119894 119895)
forward difference on the 119909119895-direction
119863minus
119895119907 (119894 119895) = 119907 (119894 119895 minus 1) minus 119907 (119894 119895)
backward difference on the 119909119895-direction
(110)
and finally we pose 119908119894119895119906(119894 119895) = 119863
+
119894119863
minus
119895119863
+
119895119906(119894 119895) For posi-
tive integer 119894 and 119895 the numerical approximation 119871119894119895119906 of
1198921198941198951205971
1199091198941205972
119909119895119906 in (108) is given by
119871119894119895= 119892
119894+12119895119863
+
119894119908
119894119895minus 119892
119894minus12119895119863
minus
119894119908
119894119895
+ 119892119894119895+12
119863+
119895119908
119895119894minus 119892
119894119895minus12119863
minus
119895119908
119895119894
(111)
We present the first natural explicit scheme
611 An Explicit Scheme An explicit scheme to solve thePDE is the following (119880119896+1
minus 119880119896)Δ119905 = minussum
2
119894119895=1119871
119894119895119880
119896 or119880
119896+1= (119868minusΔ119905sum
2
119894119895=1119871
119894119895)119880
119896 Let us poseD = 119868minusΔ119905sum2
119894119895=1119871
119894119895
Thus
119880119896+1
= D119880119896 with 119880
0= 119906
0 (112)
If 119873 is the image or the signal size the matrix D is a sparsematrix of order1198732
However this explicit scheme requires for its stabilityΔ119905 very small To overcome this drawback other numericalschemes can be used such as Du Fort and Frankel [22] whichis unconditionally stable but conditionally consisting of timesteps Δ119905 not too great to satisfy the condition of numericalstability of Courant-Friedrich-Levy [26] schemes Anothervery effective scheme is proposed by Vogel and Oman in [27]it is based on the principle of the fixed point It allows us toaddress the problem stationarywith the result of the existenceof the asymptotic solution With this scheme the solution isan eigenvector of the operator represented by the matrixD
612 The Additive Operator Splitting Scheme We can use theadditive operator splitting scheme (AOS) which is written asfollows
119880119896+1
=
1
2
2
sum
119901=1
(119868 + 2Δ119905119871119901119901)
minus1
(119868 minus Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (113)
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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DistributedSensor Networks
International Journal of
ISRN Signal Processing 13
This scheme is more stable but requires a matrix inversioneven if the matrix is pentadiagonal This inversion may beaffected by theThomas [28] algorithm modified In (113) thereverse of the matrix is independent of the iteration 119896 and iscarried out once only for 119896 = 0
613 The Alternate Direction Implicit Scheme (ADI) Simpli-fication which reduces the complexity of the computationof the solution can be made in the implementation of thenumerical equation (108) combining an implicit form for halfthe computation time and an explicit form for the remainder
It is more difficult to implement this scheme ADI for thefourth-order PDE This is due to cross derivatives in (108)Witelski and Bowen [29] suggest a readjustment of the ADIin which a mixed derivative is calculated explicitly in (108) bythe following
119880119896+1
= (
2
prod
119901=1
(119868 minus Δ119905119871119901119901))
minus1
(119868 + Δ119905
2
sum
119894=1
sum
119895 = 119894
119871119894119895)119880
119896 (114)
Besides the approximation errors inherent in all numericalscheme the fundamental difference between theses methodsremains the computation time
62 The Boundary Conditions The boundary conditionsused are those of system (41) In fact the solution is apolynomial of order 3 and can be written as 119906(119909) = 119886
31199093+
11988621199092+ 119886
1119909 + 119886
0 The nullity of its derivatives of orders 3 and
1 on the edges leads to 1198863= 0 120597119906120597119909 = 2119886
2119909 + 119886
1= 0 on
the boundary Hence 1198862= 119886
1= 0 which finally gives 119906(119909) =
1198860 We have taken as boundary conditions that outside the
domain the signal is constant This means into practice thecancellation of all the derivatives of order less than or equalto 3 For diffusivity function at the edges it is constantWhenthis constant is null we join a nonnatural manner to care ofedge effects which was to take the signal ends at the sametime as local maximum and local minimum We do not thishere
63 Test Results of PDE Interpolator on Signal In our numer-ical tests we used the implicit scheme of Crank-Nikelsonwhich gives good results with Δ119905 that can take very largevalues In the following we give some examples of calculationof upper and lower envelopes and for the mean envelopeinterpolating the inflection points of the signal This schemein consistent of order 1 both in space and in time
631 Some Examples First we present in Figure 1 a patho-logical case of a signal without local extremum when wecannot calculate the envelopes by spline interpolation of theextrema Our PDE interpolator works well with the max-mincurvature points detection in the diffusivity function
Figure 2 represents the evolution of the PDE solutionwiththe form given by (14) Upon iteration 40000 the solutionalmost evolves addition it is potentially an envelope
In Figure 3 we represent the envelope calculation of thesame function with the form given by (16) for different
15 20 25 30 35 40 45145514601465147014751480148514901495
Mean envelopeUpper envelopeLower envelope
Signal s
Figure 1 PDE interpolator for signal without local extrema Theinput signal is in blue unbroken line and the envelopes are in dottedlines
tension factor 120579 We can see that the regularity of envelopes isno longer for values of 120579 close to zero
In Figure 4 we represent the envelope calculation withthe form given by (17) for different tension factor 120579 = 0
In Figure 5 we make a comparison between our PDEinterpolator and the cubic spline interpolation for an inputsignal 119904 that equal to the superposition of two sinusoidalcomponents with different amplitudes and frequencies 119904
1
and 1199041 When it comes to interpolate the classical local
extrema we have almost the same result the two not beingsatisfactory which certainly requires a bit more iterationsin the sifting process In contrast by our interpolator whichpasses through the points of maximum and minimum localcurvature gives the correct envelopes quite properly
632 The Specific Case of the PDE Interpolator of the TurningPoints In the case of the PDE interpolator of inflectionpoints we have the advantage of not only calculating asymp-totic solution of PDE but also dividing the computing timealmost in half Only we must numerically have a goodestimate of the envelope to guard against a sampling of theinput signal which must also be quite dense This modelasks more numerically regularity and in some cases an over-sampling
In Figures 6 and 7 we proceed to calculate the solution ofthe PDE interpolator passing through the inflection pointsfor different sampling for the same composite signal 119904 =
1199041+ 119904
2 The convergence is slower if the number of sampling
points is higher but the solution is a good estimate of themean envelope In Figure 6 with a more dense samplingconvergence is slower and we get a mean envelope judgedmore suitable than 128 sampling points In Figure 7 withadequate samplingwe presented themean envelope expectedtrend that is the component 119904
1of the signal 119904 To capture
the inflection points a sampling of four once of the smallestlocal period of the signal is generally eligible for a suitable toestimate the envelope This is a consequence of the theory ofShannon sampling
633 Test on 2D Signal Finally in Figure 8 we use the 2Dversion of the PDE interpolator for image restoration The
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
14 ISRN Signal Processing
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
4000 iterations
(a) Input signal is in blue color In red and green colors are theenvelopes upon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes upon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes upon iteration 400000
Figure 2 In (a) input signal is in blue color In red and green colorsare the envelopes calculus upon iteration 4000 In (b) the envelopescalculus upon iteration 40000 In (c) the envelopes calculus uponiteration 400000
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(a) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 005
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(b) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 05
200 250 300 350 400 450 500
minus06
minus04
minus02
0
02
04
06
(c) Input signal is in blue color In red and green colors are theenvelopes at the convergence for 120579 = 1
Figure 3 In (a b and c) input signal is in blue color In red andgreen colors are the envelopes at the convergence of the evolutivePDE
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
International Journal of
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RotatingMachinery
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Electrical and Computer Engineering
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
ISRN Signal Processing 15
200 250 300 350 400 450 500
4000 iterations
minus06
minus04
minus02
0
02
04
06
(a) Input signal in blue color In red and green colors the envelopesupon iteration 4000
200 250 300 350 400 450 500
40000 iterations
minus06
minus04
minus02
0
02
04
06
(b) Input signal in blue color In red and green colors the envelopesupon iteration 40000
200 250 300 350 400 450 500
400000 iterations
minus06
minus04
minus02
0
02
04
06
(c) Input signal in blue color In red and green colors the envelopesupon iteration 400000
Figure 4 Envelope calculus by (17) with 120579 = 0 In (a) at thisstage the two envelopes overlap by location In (b) less overlap andpractically the asymptotic solution are almost reached In (c) thestability of the evolution leading to the solution
corrupted image taken as an input 2D signal in our modelwhich proceeds by edge propagating provides an interpolatedversion of the data that is clean enough
7 Conclusion
In this paper we model a new PDE interpolator whichpermits to calculate the envelopes of a signal The existenceand uniqueness of the solution to the mathematical problemare determined by a variational approach This theoreticalframework contributes to the mathematical modeling ofEMD algorithm We have shown in particular how the Vspace reminds the extraction conditions of the EMD modesand we prove that the set of eligible functions to be decom-posed by EMD are in 119867
1(Ω) The PDE interpolator is not
based on any prior knowledge on the noise level as opposedto the total variation method in [11] The tests in both signaland image processing demonstrate the effectiveness of thenew interpolator and provide an insight into opportunities inmultiscale analysis of multidimensional signal
Appendix
LemmaA1 Let119891 isin 1198711(Ω) having at least three local extrema
and null at least at one point of Ω One can assume that 119891vanishes by changing sign Then there exist 120575 gt 0 and 119909
119888isin Ω
such that the local mean at 119909119888is null for example
119872loc120575[119891] (119909
119888) =
1
2120575
int
119909119888+120575
119909119888minus120575
119891 (119904) 119889119904 = 0 (A1)
Proof Let 119909119888and 120598 gt 0 such that 119891(119909
119888) = 0 119891(119909
119888minus 120598) lt
0 and 119891(119909119888+ 120598) gt 0 Proceed by contradiction as the
local mean is continuous and 119872loc120575[119891] not null implies
that it keeps a constant sign on Ω Suppose that forall120575 119909 gt
0 119872loc120575[119891](119909) gt 0 In particular the positive sequence
(119872loc1119899[119891](119909
119888minus 120598))
119899isinNlowastconverges to 119891(119909
119888minus 120598) ge 0 This is
absurd because 119891(119909119888minus 120598) lt 0 Thus the proof of Lemma A1
is performed
From this lemma any eligible function for EMD is inV119890
Lemma A2 There exist two constants 119862119881 119862
119867gt 0 such that
forall119907 isin V
119862119881119907
2
1198673(Ω)
le |119907|2
V le 119862119867119907
2
1198673(Ω) (A2)
Proof of Lemma A2Step 1 (second part of (A2)) Rewriting first the norms119907
2
1198673(Ω)
= sum3
119904=0120597
119904119907120597119909
1199042
1198712
(Ω)
that is to say
1199072
1198673(Ω)
= 1199072
1198712(Ω)
+
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A3)
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
16 ISRN Signal Processing
100 200 300 400 500 600 700 800 900 1000
0
2
0
1
280 300 320 340 360 380 400 420 440 460 480
Env infEnv sup
minus2
minus1
0
1
minus1
0
1
minus1
Spline min-max
PDE interpolator min-max
PDE interpolator courbure
119904
Figure 5 Comparison between PDE interpolator and cubic spline The input signal is in blue color and in red and black colors are theenvelopes
80 100 120 140 160 180
256 points-40 iterations
minus1
minus05
0
05
1
(a) Envelope after 40 iterations
80 100 120 140 160 180
256 points-400 iterations
minus1
minus05
0
05
1
(b) Envelope after 400 iterations
Figure 6 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 256 points
and secondly
|119907|2
V =
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A4)
we have immediately |119907|2V le 1199072
1198673(Ω)
posing 119862119867= 1
Step 2 (first part of (A2)) Firstly
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
= int
Ω
(
1205972119907
1205971199092)
2
119889119909 (A5)
and by integration by parts
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = int
120597Ω
120597119907
120597119899
1205972119907
1205971199092119889120590 minus int
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A6)
As 119907 isin V we have
int
Ω
(
1205972119907
1205971199092)
2
119889119909 = minusint
Ω
120597119907
120597119909
1205973119907
1205971199093119889119909 (A7)
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
ISRN Signal Processing 17
350 400 450 500 550 600 650 700
1024 points-40000 iterations
minus1
minus05
0
05
1
(a) Envelope after 40000 iterations
350 400 450 500 550 600 650 700
1024 points-400000 iterations
minus1
minus05
0
05
1
(b) Envelope after 400000 iterations
Figure 7 The envelope of the signal test 119904 = 1199041+ 119904
2sampling with 1024 points
(a) (b) (c)
Figure 8 PDE interpolator for image restoration An original image (a) is corrupted (b) and restored (c) by our 2DPDE-interpolator Originalimage
and therefore the use of Schwarz inegality gives100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
100381710038171003817100381710038171003817100381710038171198712(Ω)
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
1003817100381710038171003817100381710038171003817100381710038171198712(Ω)
(A8)
then we deduce100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A9)
More intΩ119907119889119909 = 0 then according to Poincare-Wirtinger
theorem [20] it comes out that 11990721198671(Ω)
le 119862Ω120597119907120597119909
2
1198712(Ω)
Therefore
1199072
1198673(Ω)
= 1199072
1198671(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205972119907
1205971199092
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
le 119862Ω
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
1
2
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
(A10)
Thus
1199072
1198673(Ω)
le max(12
119862Ω
3
2
)(
10038171003817100381710038171003817100381710038171003817
120597119907
120597119909
10038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
+
100381710038171003817100381710038171003817100381710038171003817
1205973119907
1205971199093
100381710038171003817100381710038171003817100381710038171003817
2
1198712(Ω)
)
(A11)
If we pose
119862119881=
1
(max ((12) + 119862Ω 32))
(A12)
we have
119862119881119907
2
1198673(Ω)
le |119907|2
V (A13)
The inegality (A2) is established and the lemmarsquos proof iscompleted
References
[1] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and Hilbert spectrum for nonlinear and non-stationary time series analysisrdquo Proceedings of the Royal SocietyA vol 545 no 1971 pp 903ndash995 1998
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
18 ISRN Signal Processing
[2] E Delechelle J Lemoine and O Niang ldquoEmpirical modedecomposition an analytical approach for sifting processrdquo IEEESignal Processing Letters vol 12 no 11 pp 764ndash767 2005
[3] O Niang Empirical mode decomposition contribution a lamode-lisation mathematique et application en traitement dusignal et lrsquoimage [PhD thesis] University Paris-Est CreteilParis France 2007
[4] O Niang E Delechelle and J Lemoine ldquoA spectral approachfor sifting process in empirical mode decompositionrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5612ndash56232010
[5] O Niang A Thioune E Delechelle and J Lemoine ldquoSpectralintrinsic decomposition method for adaptive signal represen-tationrdquo ISRN Signal Processing vol 2012 Article ID 457152 10pages 2012
[6] O Niang A Thioune M C El Gueirea E Delechelle andJ Lemoine ldquoPartial differential equation-based approach forempirical mode decomposition application on image analysisrdquoIEEE Transactions on Image Processing vol 21 no 9 pp 3991ndash4001 2012
[7] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992
[8] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[9] Y MeyerOscillating Patterns in Image Processing and NonlinearEvolution Equations vol 22 of University Lecture Series AMSProvidence RI USA 2002
[10] S Osher A Sole and L Vese ldquoImage decomposition and re-storation using total variation minimization and the H1 normrdquoMultiscale Modeling and Simulation A SIAM InterdisciplinaryJournal vol 1 no 3 pp 349ndash3370 2003
[11] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003
[12] J Weickert B M ter Haar Romeny and M A ViergeverldquoEfficient and reliable schemes for nonlinear diffusion filteringrdquoIEEE Transactions on Image Processing vol 7 no 3 pp 398ndash4101998
[13] J Weickert ldquoA review of nonlinear diffusion filteringrdquo in Scale-Space Theory for Computer Vision B H Romeny Ed vol 1252of Lecture Notes in Computer Science pp 3ndash28 Springer NewYork NY USA 1997
[14] D W Peaceman and H H Rachford ldquoThe numerical solutionof parabolic and elliptic differential equationsrdquo Journal of theSociety For Industrial and Applied Mathematics vol 3 no 1 pp28ndash41 1955
[15] D Barash and R Kimmel An Accurate Operator SplittingScheme for Nonlinear Diffusion Filter HP Company 2000
[16] J Tumblin and G Turk ldquoLCIS a boundary hierarchy for detail-preserving contrast reductionrdquo inProceedings of the SIGGRAPH1999 Annual Conference on Computer Graphics Los AngelesCalif USA 1999
[17] G W Wei ldquoGeneralized Perona-Malik equation for imagerestorationrdquo IEEE Signal Processing Letters vol 6 no 7 pp 165ndash167 1999
[18] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000
[19] J Tumblin Private Communication 2003[20] H BrezisAnalyse fonctionnelleTheorie et ApplicationsMasson
Paris France 1983[21] J L Lions Equations Differentielles Operationnelles et Problemes
aux Limites Springer Berlin Germany 1961[22] J D MurrayMathematical Biology Springer Berlin Germany
1993[23] P Flandrin G Rilling and P Goncalves ldquoEmpirical mode
decomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004
[24] R SperbMaximum Principle andTheir Applications vol 157 ofMathematics in Science and Engineering Series Academic PressNew York NY USA 1981
[25] H Brezis Operateurs Maximaux monotones et semigropes decontraction dans les espaces de Hilbert North-Holland Amster-dam The Netherlands 1972
[26] G Aubert and P KornprobstMathematical Problems in ImagesProcessing Partial Differential Equations and the Calculus ofVariations vol 147 of Applied Mathematical Sciences SeriesSpringer Berlin Germany 2002
[27] C Vogel and M Oman ldquoIteration methods for total variationdenoisingrdquo SIAM Journal on Scientific Computing vol 17 no 1pp 227ndash238 1996
[28] G Engeln-Muellges and F UhligNumerical Algorithms with Cchapter 4 Springer Berlin Germany 1996
[29] T P Witelski and M Bowen ldquoADI schemes for higher-ordernonlinear diffusion equationsrdquoAppliedNumericalMathematicsvol 45 no 2-3 pp 331ndash351 2003
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of