Research ArticleOn Surface Completion and Image Inpainting byBiharmonic Functions Numerical Aspects

S B Damelin 1 and N S Hoang2

1Mathematical Reviews The American Mathematical Society 416 Fourth Street Ann Arbor MI 48104 USA2Department of Mathematics University of Oklahoma Norman OK 73019-3103 USA

Correspondence should be addressed to S B Damelin damelinumichedu

Received 22 July 2017 Accepted 11 January 2018 Published 27 February 2018

Academic Editor Irena Lasiecka

Copyright copy 2018 S BDamelin andN SHoangThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Numerical experiments with smooth surface extension and image inpainting using harmonic and biharmonic functions are carriedout The boundary data used for constructing biharmonic functions are the values of the Laplacian and normal derivatives of thefunctions on the boundary Finite difference schemes for solving these harmonic functions are discussed in detail

1 Introduction

Thesmooth function extension problem is a classical problemthat has been studied extensively in the literature fromvarious viewpoints Some of the well-known results includeUrysohnrsquos Lemma the Tietze Extension and WhitneyrsquosExtensionTheorem (see eg [1ndash9])

Inpainting was first introduced in [10] and then has beenstudied extensively by several authors (see eg [11 12])Although smooth image inpainting is a smooth functionextension problem the most common approach in imageinpainting so far is to use the solution to some PDE whichare obtained fromminimum-energy models as the recoveredimage The most commonly used density function for theseenergy models is total derivation

In [12] by considering smooth inpainting as a smoothsurface extension problem the author studied methods forlinear inpainting and cubic inpainting Error bounds forthese inpainting methods are derived in [12] In [13] severalsurface completion methods have been studied An optimalbound for the errors of the cubic inpainting method in[12] is given Applications to smooth inpainting have alsobeen discussed in [13] There error bounds of completionmethods are derived in terms of the radius of the domainson which the functions are completed In one of the methodsin [13] the author proposed to use 119901-harmonic functions forsmooth surface completion and smooth surface inpaintingLater in [14] 119901-harmonic functions are also studied for

smooth surface completion and inpainting The differencesof the method using 119901-harmonic functions in [13 14] are asfollows the method in [13] uses Δ119894119906|119878 119894 = 0 1 119901 minus 1as boundary data while the method in [14] uses (120597119894120597119873119894)119906|

119878119894 = 0 1 119901minus1 as boundary data to solve for a 119901-harmonic

function Here 119906 is the function to be inpainted and 119878 is theboundary of the inpainted region

The goals of this paper are to implement and comparethe performance of the two surface completion schemes in[13 14] In particular we focus our study on smooth surfacecompletion and smooth surface inpainting by biharmonicfunctions

2 Surface Completion byBiharmonic Functions

Let119863 be a simply connected region inR2 with 1198621-boundary119878 = 120597119863 and 119889 be the diameter of 119863 Let 1199060be a

smooth function on any region containing 119863 Assume that1199060is known on a neighbourhood outside 119863 The surface

completion problem consists of finding a function 119906 on aregion containing119863 such that119906 = 119906

0outside 119863 (1)

There are several ways to construct the function 119906 so that (1)holds For smooth surface completion one is often interestedin finding a sufficiently smooth function 119906 satisfying (1)

HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2018 Article ID 3950312 8 pageshttpsdoiorg10115520183950312

2 International Journal of Mathematics and Mathematical Sciences

An application of smooth surface completion is insmooth image inpainting In smooth image inpainting onehas a smooth image 119906

0which is known in a neighbourhood

outside of a region 119863 while the data inside 119863 is missing Thegoal of image inpainting is to extend the function 119906 over theregion 119863 in such a way that the extension over the missingregion is not noticeable with human eyes

In image inpainting an inpainting scheme is said to be oflinear order or simply linear inpainting if for any smoothtest image 1199060 as the diameter 119889 of the inpainting region 119863shrinks to 0 one has1003817100381710038171003817119906 minus 119906

0

1003817100381710038171003817119863 = 119874 (1198892) (2)

where 119906 is the image obtained from the inpainting schemeHere sdot

119863 denotes the 119871infin(119863) norm Here and throughout119891 = 119874(119892) if |119891||119892| is bounded uniformly by some constant119862 gt 0Note that harmonic inpainting that is the extension

found from the equationΔ119906 = 0 in 119863119906|119878 = 1199060

1003816100381610038161003816119878 (3)

is a linear inpainting scheme [12]In [12] the following result for cubic inpainting is proved

Theorem1 (cubic inpaintingTheorem 65 [12]) Let1199061be the

harmonic inpainting of 1199060 Let 119906

ℓbe a linear inpainting of Δ119906

0

on119863 (not necessarily harmonic) and let 119906(119909) be defined by119906 (119909) = 1199061 (119909) + 119906

2 (119909) 119909 isin 119863 (4)

where 1199062solves Poissonrsquos equationΔ119906

2= 119906ℓ 119894119899 11986311990621003816100381610038161003816119878 = 0 (5)

Then 119906 defines a cubic inpainting of 1199060 that is1003817100381710038171003817119906 minus 11990601003817100381710038171003817119863 = 119874 (1198894) (6)

Remark 2 If 119906ℓ is the harmonic inpainting of Δ1199060 in 119863 that

is 119906ℓ solves the equationΔ119906ℓ= 0 in 119863119906ℓ1003816100381610038161003816119878 = Δ11990601003816100381610038161003816119878 (7)

then the element 119906 defined by (4) is a biharmonic functionwhich solves the following problemΔ2119906 = 0Δ119906|119878 = Δ119906

0

1003816100381610038161003816119878 119906|119878 = 1199060

1003816100381610038161003816119878 (8)

In [13] this result is generalized to a multiresolutionapproximation extension scheme in which the Laplacian

is replaced by more general lagged diffusivity anisotropicoperators It is proved in [13] that if 119906 solves the equationΔ119899119906 = 0 in 119863Δ11989411990610038161003816100381610038161003816119878 = Δ119894119906010038161003816100381610038161003816119878 119894 = 0 1 119899 minus 1 (9)

then 1003817100381710038171003817119906 minus 11990601003817100381710038171003817119863 = 119874 (1198892119899) (10)

A sharper error bound than (10) is obtained in [13]Equation (9) can be written as a system of Poissonrsquos

equations as followsV0= 0 in 119863ΔV119894= V119894minus1

in 119863V1198941003816100381610038161003816119878 = Δ119899minus119894119906010038161003816100381610038161003816119878 119894 = 1 2 119899119906 = V119899

(11)

Thus the problem of solving (9) is reduced to the problem ofsolving Poissonrsquos equations of the formΔ119906 = 119891 in 119863119906|119878 = 119892 (12)

Numerical methods for solving (12) have been extensivelystudied in the literature

Note that the normal derivatives (120597119894120597119873119894)1199060|119878 are notpresented in (9) Thus extension using (9) may not be differ-entiable across the boundary 119878 To improve the smoothnessof the extension across 119878 it is suggested to find 119906 from theequation Δ119899119906 = 0 in 119863120597119894120597119873119894 119906 = 120597119894120597119873119894 1199060 119894 = 0 1 119899 minus 1 (13)

It is proved in [14] that if 119906 is the solution to (13) then (10)also holds

Equation (13) cannot be reduced to a system of Poissonrsquosequations as (9) In fact to solve (13) one often uses afinite difference approach which consists of finding discreteapproximations to the operators Δ119899 and 120597119894120597119873119894 119894 = 1 119899minus1 For ldquolargerdquo 119899 it is quite complicated though possible toobtain these approximations

As we can see from the above discussions (9) is easierto solve numerically than (13) However scheme (9) doesnot use any information about the normal derivatives ofthe surface on 119878 Thus the extension surface obtained fromscheme (9) may not be smooth across the boundary 119878 Onthe contrary (13) uses normal derivatives as boundary dataand therefore is expected to yield better results than scheme(9) does

In the next section we will implement and comparethe two surface completion schemes using (9) and (13) Inparticular we focus our study on biharmonic functionswhichare solutions to (9) and (13) for 119899 = 2

International Journal of Mathematics and Mathematical Sciences 3

3 Implementation

Let us discuss a numerical method for solving the equation

Δ2119906 = 0 in 119863Δ119906|119878 = 119891119906|119878 = 119892 (14)

To solve this equation one often defines V = Δ119906 and solvesfor 119906 from the following systemΔV = 0

V|119878 = 119891Δ119906 = V119906|119878 = 119892(15)

Thus the problem of solving (15) is reduced to the problemof solving the following Poissonrsquos equationΔ119906 = 119891

1 in 119863119906|119878 = 1198921 (16)

To solve (16) we use a 5-point finite difference scheme toapproximate the Laplacian operator This 5-point scheme isbased on the following well-known formula

Δ 5119906 fl1ℎ2 ( 11 minus4 11 )119906 = Δ119906 + 119874 (ℎ2) (17)

Here ℎ is the discretization step size This scheme is welldefined at points 119875 whose nearest neighbours are interiorpoints of119863 If119875 has a neighbour119876 isin 120597119863 thenwe use a stencilof the form

Δ5119906 (119875) fl 1ℎ2 ( 1minus4 11 )119906 (119875) + 119906 (119876)ℎ2

= Δ119906 (119875) + 119874 (ℎ2) (18)

In the above formula119876 is the nearest neighbour to the left of119875 Similar formulae when 119876 is the nearest neighbour on thetop in the bottom and to the right of119875 can be obtained easily

In our experiments we choose 119863 as a square and thesolution 119906 on the computation grid is presented as a vectorUsing the 5-point finite difference scheme (16) is reduced tothe following algebraic system

119860119906 = (1198911minus 1198921ℎ2) (19)

where 119860 is the 5-point finite difference approximation to theLaplacian and 119892

1is a vector containing boundary values of119906 on 119878 at suitable entries Matrix 119860 is a tridiagonal matrix

that is all nonzero elements of 119860 lie on the main diagonaland the first diagonals above and below the main diagonalMatrix minusℎ2119860 can be obtained by the function delsq availablein Matlab

Let us discuss a numerical method for solving the equa-tion Δ2119906 = 01199061198731003816100381610038161003816119878 = 119891119906|119878 = 119892 (20)

For a discrete approximation to the bi-Laplacian we use a 13-point finite difference schemewhich is based on the followingformula (see [15])

Δ213119906 = 1ℎ4((

(

12 minus8 21 minus8 20 minus8 12 minus8 21)))

119906= Δ2119906 + 119874 (ℎ2)

(21)

This stencil is well defined for a grid point 119875 if all its nearestneighbours are in the interior of the domain If 119875 has aneighbour 119876 isin 120597119863 and 119876 is on the left of 119875 then we use thefollowing formula

Δ213119906 (119875) fl 1ℎ4((

(

12 minus8 2minus8 20 minus8 12 minus8 21)))

119906(119875) + 119906119873 (119876)ℎ3

+ 119906 (119876)ℎ4 = Δ2119906 + 119874 (ℎminus1) (22)

Using the above finite difference scheme (20) is reducedto a linear algebraic system of the form 119860119906 = 119887 where 119860 isa five-diagonal matrix Numerical solutions to 119906 on the gridcan be obtained by solving this linear algebraic system

Before we proceed with numerical experiments we needthe following

31 Quantitative Comparisons It is constructive to providequantitative correlations between original and processedimages and in particular code to compare figures such asthose below In order to calculate these required correla-tions (and many are provided) we refer the reader to afree access code for our method in scikit-image processingin python unit completely at the disposal of the readerwhich readily provides quantitative correlationsmdashin partic-ular for the figures The scikit-image processing package

4 International Journal of Mathematics and Mathematical Sciences

20

4010

2030

4050

Original function

20

4010

2030

4050

Error of harmonic interpolation

minus1

0

1

2

3

0

02

04

06

Figure 1

is at httpscikit-imageorg and is a collection of openaccess algorithms for image processing with peer-reviewedcode

For our method see httpscikit-imageorgdocsstableapiskimagerestorationhtmlhighlight=biharmonicskimagerestorationinpaint-biharmonic

Moreover for the benefit of the reader many comparisonmethods with associated code are given in this unit see somebelow but see the full package for a longer list with references

(i) Denoise-bilateral (see [16])(ii) Denoise-nl-means (see [17ndash19])(iii) Denoise-tv-bregman denoise-tv-chambolle denoise-

wavelet estimate-sigma inpaint-biharmonic (thepresent paper) nl-means-denoising richardson-lucyunsupervised-wiener unwrap-phase Wiener-Huntdeconvolution

4 Numerical Experiments

41 Smooth Surface Completion Let us first do some numer-ical experiments with smooth surface completion In ourexperiments we compare numerical solutions from the threesurface completion methods the method with harmonicfunctions the method with biharmonic functions in [12 13]and the method with biharmonic functions in [14]

The function 119906 to be completed in our first experiment is119906 (119909 119910) = 119909119910 + 1199092 (119910 + 1) 119909 119910 isin 119863 = [minus1 1] (23)

Note that this function is a biharmonic functionThe domain119863 is discretized by a grid of size (119899+1)times (119899+1) points In thefirst experiment we used 119899 = 50

In our numerical experiments we denote by 119906119867

theextension by a harmonic function by 119906

119871the biharmonic

extension from [13] and by 119906119873

the biharmonic extensionfrom [14]

Figure 1 plots the original function and the error ofreconstruction by a harmonic function Figure 2 plots theerrors of surface reconstructions by biharmonic functionsfrom [13 14]

From Figures 1 and 2 one can see that the biharmonicreconstructions from [13 14] are much better than thereconstruction by harmonic functions The method in [13]in this experiment yields numerical results with accuracy abit higher than the method in [14] However this does notimply that the method in [13] is better in terms of accuracythan the method in [14] In the condition number of Athe finite difference approximation to the bi-Laplacian inthis experiment is larger than that of the finite differenceapproximation to the Laplacian Due to these conditionnumbers the algorithmusing themethod in [13] yields resultswith higher accuracy than the algorithm using the methodin [14] As we can see in later experiments the methodin [14] often gives better results than the method in [13]The conclusion from this example is that both methods in[13 14] yield numerical solutions at very high accuracy Theharmonic reconstruction in this experiment is not very goodThis stems from the fact that the function to be reconstructedis not harmonic

In the next experiment the function to be reconstructedis chosen by

119906 (119909 119910) = (1 + cos (119909)) (1 + cos (119910))4 119909 119910 isin [minus1 1] (24)

This function 119906(119909 119910) is not a biharmonic function

International Journal of Mathematics and Mathematical Sciences 3

3 Implementation

Let us discuss a numerical method for solving the equation

Δ2119906 = 0 in 119863Δ119906|119878 = 119891119906|119878 = 119892 (14)

To solve this equation one often defines V = Δ119906 and solvesfor 119906 from the following systemΔV = 0

V|119878 = 119891Δ119906 = V119906|119878 = 119892(15)

Thus the problem of solving (15) is reduced to the problemof solving the following Poissonrsquos equationΔ119906 = 119891

1 in 119863119906|119878 = 1198921 (16)

To solve (16) we use a 5-point finite difference scheme toapproximate the Laplacian operator This 5-point scheme isbased on the following well-known formula

Δ 5119906 fl1ℎ2 ( 11 minus4 11 )119906 = Δ119906 + 119874 (ℎ2) (17)

Here ℎ is the discretization step size This scheme is welldefined at points 119875 whose nearest neighbours are interiorpoints of119863 If119875 has a neighbour119876 isin 120597119863 thenwe use a stencilof the form

Δ5119906 (119875) fl 1ℎ2 ( 1minus4 11 )119906 (119875) + 119906 (119876)ℎ2

= Δ119906 (119875) + 119874 (ℎ2) (18)

In the above formula119876 is the nearest neighbour to the left of119875 Similar formulae when 119876 is the nearest neighbour on thetop in the bottom and to the right of119875 can be obtained easily

In our experiments we choose 119863 as a square and thesolution 119906 on the computation grid is presented as a vectorUsing the 5-point finite difference scheme (16) is reduced tothe following algebraic system

119860119906 = (1198911minus 1198921ℎ2) (19)

where 119860 is the 5-point finite difference approximation to theLaplacian and 119892

1is a vector containing boundary values of119906 on 119878 at suitable entries Matrix 119860 is a tridiagonal matrix

that is all nonzero elements of 119860 lie on the main diagonaland the first diagonals above and below the main diagonalMatrix minusℎ2119860 can be obtained by the function delsq availablein Matlab

Let us discuss a numerical method for solving the equa-tion Δ2119906 = 01199061198731003816100381610038161003816119878 = 119891119906|119878 = 119892 (20)

For a discrete approximation to the bi-Laplacian we use a 13-point finite difference schemewhich is based on the followingformula (see [15])

Δ213119906 = 1ℎ4((

(

12 minus8 21 minus8 20 minus8 12 minus8 21)))

119906= Δ2119906 + 119874 (ℎ2)

(21)

This stencil is well defined for a grid point 119875 if all its nearestneighbours are in the interior of the domain If 119875 has aneighbour 119876 isin 120597119863 and 119876 is on the left of 119875 then we use thefollowing formula

Δ213119906 (119875) fl 1ℎ4((

(

12 minus8 2minus8 20 minus8 12 minus8 21)))

119906(119875) + 119906119873 (119876)ℎ3

+ 119906 (119876)ℎ4 = Δ2119906 + 119874 (ℎminus1) (22)

Using the above finite difference scheme (20) is reducedto a linear algebraic system of the form 119860119906 = 119887 where 119860 isa five-diagonal matrix Numerical solutions to 119906 on the gridcan be obtained by solving this linear algebraic system

Before we proceed with numerical experiments we needthe following

31 Quantitative Comparisons It is constructive to providequantitative correlations between original and processedimages and in particular code to compare figures such asthose below In order to calculate these required correla-tions (and many are provided) we refer the reader to afree access code for our method in scikit-image processingin python unit completely at the disposal of the readerwhich readily provides quantitative correlationsmdashin partic-ular for the figures The scikit-image processing package

4 International Journal of Mathematics and Mathematical Sciences

20

4010

2030

4050

Original function

20

4010

2030

4050

Error of harmonic interpolation

minus1

0

1

2

3

0

02

04

06

Figure 1

is at httpscikit-imageorg and is a collection of openaccess algorithms for image processing with peer-reviewedcode

For our method see httpscikit-imageorgdocsstableapiskimagerestorationhtmlhighlight=biharmonicskimagerestorationinpaint-biharmonic

Moreover for the benefit of the reader many comparisonmethods with associated code are given in this unit see somebelow but see the full package for a longer list with references

(i) Denoise-bilateral (see [16])(ii) Denoise-nl-means (see [17ndash19])(iii) Denoise-tv-bregman denoise-tv-chambolle denoise-

wavelet estimate-sigma inpaint-biharmonic (thepresent paper) nl-means-denoising richardson-lucyunsupervised-wiener unwrap-phase Wiener-Huntdeconvolution

4 Numerical Experiments

41 Smooth Surface Completion Let us first do some numer-ical experiments with smooth surface completion In ourexperiments we compare numerical solutions from the threesurface completion methods the method with harmonicfunctions the method with biharmonic functions in [12 13]and the method with biharmonic functions in [14]

The function 119906 to be completed in our first experiment is119906 (119909 119910) = 119909119910 + 1199092 (119910 + 1) 119909 119910 isin 119863 = [minus1 1] (23)

Note that this function is a biharmonic functionThe domain119863 is discretized by a grid of size (119899+1)times (119899+1) points In thefirst experiment we used 119899 = 50

In our numerical experiments we denote by 119906119867

theextension by a harmonic function by 119906

119871the biharmonic

extension from [13] and by 119906119873

the biharmonic extensionfrom [14]

Figure 1 plots the original function and the error ofreconstruction by a harmonic function Figure 2 plots theerrors of surface reconstructions by biharmonic functionsfrom [13 14]

From Figures 1 and 2 one can see that the biharmonicreconstructions from [13 14] are much better than thereconstruction by harmonic functions The method in [13]in this experiment yields numerical results with accuracy abit higher than the method in [14] However this does notimply that the method in [13] is better in terms of accuracythan the method in [14] In the condition number of Athe finite difference approximation to the bi-Laplacian inthis experiment is larger than that of the finite differenceapproximation to the Laplacian Due to these conditionnumbers the algorithmusing themethod in [13] yields resultswith higher accuracy than the algorithm using the methodin [14] As we can see in later experiments the methodin [14] often gives better results than the method in [13]The conclusion from this example is that both methods in[13 14] yield numerical solutions at very high accuracy Theharmonic reconstruction in this experiment is not very goodThis stems from the fact that the function to be reconstructedis not harmonic

In the next experiment the function to be reconstructedis chosen by

119906 (119909 119910) = (1 + cos (119909)) (1 + cos (119910))4 119909 119910 isin [minus1 1] (24)

This function 119906(119909 119910) is not a biharmonic function

4 International Journal of Mathematics and Mathematical Sciences

20

4010

2030

4050

Original function

20

4010

2030

4050

Error of harmonic interpolation

minus1

0

1

2

3

0

02

04

06

Figure 1

is at httpscikit-imageorg and is a collection of openaccess algorithms for image processing with peer-reviewedcode

For our method see httpscikit-imageorgdocsstableapiskimagerestorationhtmlhighlight=biharmonicskimagerestorationinpaint-biharmonic

Moreover for the benefit of the reader many comparisonmethods with associated code are given in this unit see somebelow but see the full package for a longer list with references

(i) Denoise-bilateral (see [16])(ii) Denoise-nl-means (see [17ndash19])(iii) Denoise-tv-bregman denoise-tv-chambolle denoise-

wavelet estimate-sigma inpaint-biharmonic (thepresent paper) nl-means-denoising richardson-lucyunsupervised-wiener unwrap-phase Wiener-Huntdeconvolution

4 Numerical Experiments

41 Smooth Surface Completion Let us first do some numer-ical experiments with smooth surface completion In ourexperiments we compare numerical solutions from the threesurface completion methods the method with harmonicfunctions the method with biharmonic functions in [12 13]and the method with biharmonic functions in [14]

The function 119906 to be completed in our first experiment is119906 (119909 119910) = 119909119910 + 1199092 (119910 + 1) 119909 119910 isin 119863 = [minus1 1] (23)

Note that this function is a biharmonic functionThe domain119863 is discretized by a grid of size (119899+1)times (119899+1) points In thefirst experiment we used 119899 = 50

In our numerical experiments we denote by 119906119867

theextension by a harmonic function by 119906

119871the biharmonic

extension from [13] and by 119906119873

the biharmonic extensionfrom [14]

Figure 1 plots the original function and the error ofreconstruction by a harmonic function Figure 2 plots theerrors of surface reconstructions by biharmonic functionsfrom [13 14]

From Figures 1 and 2 one can see that the biharmonicreconstructions from [13 14] are much better than thereconstruction by harmonic functions The method in [13]in this experiment yields numerical results with accuracy abit higher than the method in [14] However this does notimply that the method in [13] is better in terms of accuracythan the method in [14] In the condition number of Athe finite difference approximation to the bi-Laplacian inthis experiment is larger than that of the finite differenceapproximation to the Laplacian Due to these conditionnumbers the algorithmusing themethod in [13] yields resultswith higher accuracy than the algorithm using the methodin [14] As we can see in later experiments the methodin [14] often gives better results than the method in [13]The conclusion from this example is that both methods in[13 14] yield numerical solutions at very high accuracy Theharmonic reconstruction in this experiment is not very goodThis stems from the fact that the function to be reconstructedis not harmonic

In the next experiment the function to be reconstructedis chosen by

119906 (119909 119910) = (1 + cos (119909)) (1 + cos (119910))4 119909 119910 isin [minus1 1] (24)

This function 119906(119909 119910) is not a biharmonic function

International Journal of Mathematics and Mathematical Sciences 5

20

4020

40

Error of Chui method

20

4020

40

Error of ChuiminusMhaskar method

0

05

1

15

2

0

05

1

times10minus14 times10

minus13

Figure 2

1020

3040

501020

3040

50

Original function

1020

3040

501020

3040

50

Error of harmonic interpolation

0

005

01

015

02

025

06

07

08

09

Figure 3

Figures 3 and 4 plot the errors of harmonic and bihar-monic reconstructions From these figures it is clear that themethod in [14] yields the best approximation The harmonicreconstruction is the worst amongst the 3 methods in thisexperiment

42 Image Inpainting Let us do some numerical experimentswith image inpainting

Figure 5 plots a damaged image and a reconstructedimage by harmonic functions Figure 6 plots restored imagesby biharmonic functions following themethods from [13 14]

It can be seen from Figures 5 and 6 that the biharmonicextension method from [14] yields the best reconstructionAlthough the biharmonic extension method from [13] isbetter than harmonic extension in our experiments withsmooth surface completion it is not as good as the harmonicextension in this experimentThis is understandable since ourimage contains edges and is not a smooth function It canbe seen from the restored image by the method in [13] inFigure 6 that the reconstruction may not be smooth or evendifferentiable across the boundary

6 International Journal of Mathematics and Mathematical Sciences

1020

3040

501020

3040

50

Error of Chui method

1020

3040

501020

3040

50

Error of ChuiminusMhaskar method

0

002

004

006

008

0

0005

001

0015

002

0025

Figure 4

Damaged image Harmonic inpainting

Figure 5 Damaged image and restored image by harmonic functions

ChuiminusMhaskar inpainting Chui inpainting

Figure 6 Restored image by biharmonic functions

International Journal of Mathematics and Mathematical Sciences 7

Damaged image Harmonic inpainting

Figure 7 Damaged image and restored image by harmonic functions

ChuiminusMhaskar inpainting Chui inpainting

Figure 8 Restored image by biharmonic functions

Table 1 Results for119863119894= [minus2minus119894 2minus119894] times [minus2minus119894 2minus119894] for 119894 = 0 9119894 log

2

1003817100381710038171003817119906119867 minus 1199061003817100381710038171003817infin log2

1003817100381710038171003817119906119871 minus 1199061003817100381710038171003817infin log2

1003817100381710038171003817119906119873 minus 1199061003817100381710038171003817infin0 236 037 minus1461 050 minus348 minus5342 minus146 minus744 minus9323 minus345 minus1144 minus13314 minus545 minus1543 minus17315 minus745 minus1943 minus21316 minus945 minus2343 minus25317 minus1145 minus2743 minus29318 minus1345 minus3143 minus33349 minus1545 minus3546 minus3854Appendix

Table 1 presents numerical results for the function 119906 definedby (24) and on the domain 119863

119894= [minus2minus119894 2minus119894] times [minus2minus119894 2minus119894]119894 = 0 9 The diameter of 119863

119894is 119889119894= 21minus119894 From Table 1

one can see that the harmonic reconstruction has an order of

accuracy of 2 while the biharmonic reconstruction methodshave an order of accuracy of 4This agreeswith the theoreticalestimates in [12ndash14]

Figures 7 and 8 plot a damaged picture of peppersand reconstructed images by the 3 methods From thesefigures one gets the same conclusion as in the previousexperimentThe biharmonic reconstruction in [14] yields thebest restoration while the biharmonic reconstruction in [13]yields the worst reconstruction

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] P Shvartsman ldquoThe Whitney extension problem and Lipschitzselections of set-valued mappings in jet-spacesrdquo Transactions ofthe American Mathematical Society vol 360 no 10 pp 5529ndash5550 2008

8 International Journal of Mathematics and Mathematical Sciences

[2] P Shvartsman ldquoWhitney-type extension theorems for jetsgenerated by Sobolev functionsrdquo Advances in Mathematics vol313 pp 379ndash469 2017

[3] Y Brudnyi and P Shvartsman ldquoWhitneyrsquos extension problemfor multivariate C1120596-functionsrdquo Transactions of the AmericanMathematical Society vol 353 no 6 pp 2487ndash2512 2001

[4] S B Damelin andC FeffermanOn SmoothWhitney Extensionsof almost isometries with small distortion Interpolation andAlignment in RD-Part 1 arxiv 14112451

[5] S B Damelin and C FeffermanOn SmoothWhitney extensionsof almost isometries with small distortion in Cm(Rn) arxiv150506950

[6] C Fefferman S B Damelin and W Glover ldquoA BMO theoremfor 120576 and an application to comparing manifolds of speech andsound Involverdquo A Journal of Mathematics vol 5 no 2 pp 159ndash172 2012

[7] C Fefferman ldquoWhitneyrsquos extension problems and interpolationof datardquo Bulletin (New Series) of the American MathematicalSociety vol 46 no 2 pp 207ndash220 2009

[8] E J McShane ldquoExtension of range of functionsrdquo Bulletin (NewSeries) of the American Mathematical Society vol 40 no 12 pp837ndash842 1934

[9] H Whitney ldquoAnalytic extensions of differentiable functionsdefined in closed setsrdquo Transactions of the AmericanMathemat-ical Society vol 36 no 1 pp 63ndash89 1934

[10] M Bertalmio G Sapiro C Ballester and V Caselles ldquoImageInpaintingrdquo in Proceedings of the 27th annual conference onComputer graphics and interactive techniques (SIGGRAPH rsquo00)pp 417ndash424 July 2000

[11] M Bertalmio A Bertozzi and G Sapiro ldquoNavier-stokes fluiddynamics and image and video inpaintingrdquo in Proceedings ofthe 2001 IEEE Computer Society Conference on Computer Visionand Pattern Recognition CVPR 2001 pp I-355ndashI-362 KauaiHIUSA

[12] T F Chan and J Shen Image Processing and Analysis SIAMPhiladelphia Pa USA 2005

[13] C K Chui ldquoAn MRA approach to surface completion andimage inpaintingrdquo Applied and Computational Harmonic Anal-ysis vol 26 no 2 pp 270ndash276 2009

[14] C K Chui and H N Mhaskar ldquoMRA contextual-recoveryextension of smooth functions on manifoldsrdquo Applied andComputational Harmonic Analysis vol 28 no 1 pp 104ndash1132010

[15] P Bjorstad ldquoFast numerical solution of the biharmonic Dirich-let problemon rectanglesrdquo SIAM Journal onNumerical Analysisvol 20 no 1 pp 59ndash71 1983

[16] C Tomasi and R Manduchi ldquoBilateral filtering for gray andcolor imagesrdquo in Proceedings of the 6th International Conferenceon Computer Vision (ICCV rsquo98) pp 839ndash846 Bombay IndiaJanuary 1998

[17] A Buades B Coll and J-M Morel ldquoA non-local algorithm forimage denoisingrdquo in Proceedings of the 2005 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2005 pp 60ndash65 usa June 2005

[18] J Darbon A Cunha T F Chan S Osher and G J JensenldquoFast nonlocal filtering applied to electron cryomicroscopyrdquoin Proceedings of the 5th IEEE International Symposium onBiomedical Imaging From Nano to Macro (ISBI rsquo08) pp 1331ndash1334 IEEE Paris France May 2008

[19] J Froment ldquoParameter-Free Fast Pixelwise Non-Local MeansDenoisingrdquo Image Processing On Line vol 4 pp 300ndash326 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 International Journal of Mathematics and Mathematical Sciences

1020

3040

501020

3040

50

Error of Chui method

1020

3040

501020

3040

50

Error of ChuiminusMhaskar method

0

002

004

006

008

0

0005

001

0015

002

0025

Figure 4

Damaged image Harmonic inpainting

Figure 5 Damaged image and restored image by harmonic functions

ChuiminusMhaskar inpainting Chui inpainting

Figure 6 Restored image by biharmonic functions

International Journal of Mathematics and Mathematical Sciences 7

Damaged image Harmonic inpainting

Figure 7 Damaged image and restored image by harmonic functions

ChuiminusMhaskar inpainting Chui inpainting

Figure 8 Restored image by biharmonic functions

Table 1 Results for119863119894= [minus2minus119894 2minus119894] times [minus2minus119894 2minus119894] for 119894 = 0 9119894 log

2

1003817100381710038171003817119906119867 minus 1199061003817100381710038171003817infin log2

1003817100381710038171003817119906119871 minus 1199061003817100381710038171003817infin log2

1003817100381710038171003817119906119873 minus 1199061003817100381710038171003817infin0 236 037 minus1461 050 minus348 minus5342 minus146 minus744 minus9323 minus345 minus1144 minus13314 minus545 minus1543 minus17315 minus745 minus1943 minus21316 minus945 minus2343 minus25317 minus1145 minus2743 minus29318 minus1345 minus3143 minus33349 minus1545 minus3546 minus3854Appendix

Table 1 presents numerical results for the function 119906 definedby (24) and on the domain 119863

119894= [minus2minus119894 2minus119894] times [minus2minus119894 2minus119894]119894 = 0 9 The diameter of 119863

119894is 119889119894= 21minus119894 From Table 1

one can see that the harmonic reconstruction has an order of

accuracy of 2 while the biharmonic reconstruction methodshave an order of accuracy of 4This agreeswith the theoreticalestimates in [12ndash14]

Figures 7 and 8 plot a damaged picture of peppersand reconstructed images by the 3 methods From thesefigures one gets the same conclusion as in the previousexperimentThe biharmonic reconstruction in [14] yields thebest restoration while the biharmonic reconstruction in [13]yields the worst reconstruction

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] P Shvartsman ldquoThe Whitney extension problem and Lipschitzselections of set-valued mappings in jet-spacesrdquo Transactions ofthe American Mathematical Society vol 360 no 10 pp 5529ndash5550 2008

8 International Journal of Mathematics and Mathematical Sciences

[2] P Shvartsman ldquoWhitney-type extension theorems for jetsgenerated by Sobolev functionsrdquo Advances in Mathematics vol313 pp 379ndash469 2017

[3] Y Brudnyi and P Shvartsman ldquoWhitneyrsquos extension problemfor multivariate C1120596-functionsrdquo Transactions of the AmericanMathematical Society vol 353 no 6 pp 2487ndash2512 2001

[4] S B Damelin andC FeffermanOn SmoothWhitney Extensionsof almost isometries with small distortion Interpolation andAlignment in RD-Part 1 arxiv 14112451

[5] S B Damelin and C FeffermanOn SmoothWhitney extensionsof almost isometries with small distortion in Cm(Rn) arxiv150506950

[6] C Fefferman S B Damelin and W Glover ldquoA BMO theoremfor 120576 and an application to comparing manifolds of speech andsound Involverdquo A Journal of Mathematics vol 5 no 2 pp 159ndash172 2012

[7] C Fefferman ldquoWhitneyrsquos extension problems and interpolationof datardquo Bulletin (New Series) of the American MathematicalSociety vol 46 no 2 pp 207ndash220 2009

[8] E J McShane ldquoExtension of range of functionsrdquo Bulletin (NewSeries) of the American Mathematical Society vol 40 no 12 pp837ndash842 1934

[9] H Whitney ldquoAnalytic extensions of differentiable functionsdefined in closed setsrdquo Transactions of the AmericanMathemat-ical Society vol 36 no 1 pp 63ndash89 1934

[10] M Bertalmio G Sapiro C Ballester and V Caselles ldquoImageInpaintingrdquo in Proceedings of the 27th annual conference onComputer graphics and interactive techniques (SIGGRAPH rsquo00)pp 417ndash424 July 2000

[11] M Bertalmio A Bertozzi and G Sapiro ldquoNavier-stokes fluiddynamics and image and video inpaintingrdquo in Proceedings ofthe 2001 IEEE Computer Society Conference on Computer Visionand Pattern Recognition CVPR 2001 pp I-355ndashI-362 KauaiHIUSA

[12] T F Chan and J Shen Image Processing and Analysis SIAMPhiladelphia Pa USA 2005

[13] C K Chui ldquoAn MRA approach to surface completion andimage inpaintingrdquo Applied and Computational Harmonic Anal-ysis vol 26 no 2 pp 270ndash276 2009

[14] C K Chui and H N Mhaskar ldquoMRA contextual-recoveryextension of smooth functions on manifoldsrdquo Applied andComputational Harmonic Analysis vol 28 no 1 pp 104ndash1132010

[15] P Bjorstad ldquoFast numerical solution of the biharmonic Dirich-let problemon rectanglesrdquo SIAM Journal onNumerical Analysisvol 20 no 1 pp 59ndash71 1983

[16] C Tomasi and R Manduchi ldquoBilateral filtering for gray andcolor imagesrdquo in Proceedings of the 6th International Conferenceon Computer Vision (ICCV rsquo98) pp 839ndash846 Bombay IndiaJanuary 1998

[17] A Buades B Coll and J-M Morel ldquoA non-local algorithm forimage denoisingrdquo in Proceedings of the 2005 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2005 pp 60ndash65 usa June 2005

[18] J Darbon A Cunha T F Chan S Osher and G J JensenldquoFast nonlocal filtering applied to electron cryomicroscopyrdquoin Proceedings of the 5th IEEE International Symposium onBiomedical Imaging From Nano to Macro (ISBI rsquo08) pp 1331ndash1334 IEEE Paris France May 2008

[19] J Froment ldquoParameter-Free Fast Pixelwise Non-Local MeansDenoisingrdquo Image Processing On Line vol 4 pp 300ndash326 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

International Journal of Mathematics and Mathematical Sciences 7

Damaged image Harmonic inpainting

Figure 7 Damaged image and restored image by harmonic functions

ChuiminusMhaskar inpainting Chui inpainting

Figure 8 Restored image by biharmonic functions

Table 1 Results for119863119894= [minus2minus119894 2minus119894] times [minus2minus119894 2minus119894] for 119894 = 0 9119894 log

2

1003817100381710038171003817119906119867 minus 1199061003817100381710038171003817infin log2

1003817100381710038171003817119906119871 minus 1199061003817100381710038171003817infin log2

1003817100381710038171003817119906119873 minus 1199061003817100381710038171003817infin0 236 037 minus1461 050 minus348 minus5342 minus146 minus744 minus9323 minus345 minus1144 minus13314 minus545 minus1543 minus17315 minus745 minus1943 minus21316 minus945 minus2343 minus25317 minus1145 minus2743 minus29318 minus1345 minus3143 minus33349 minus1545 minus3546 minus3854Appendix

Table 1 presents numerical results for the function 119906 definedby (24) and on the domain 119863

119894= [minus2minus119894 2minus119894] times [minus2minus119894 2minus119894]119894 = 0 9 The diameter of 119863

119894is 119889119894= 21minus119894 From Table 1

one can see that the harmonic reconstruction has an order of

accuracy of 2 while the biharmonic reconstruction methodshave an order of accuracy of 4This agreeswith the theoreticalestimates in [12ndash14]

Figures 7 and 8 plot a damaged picture of peppersand reconstructed images by the 3 methods From thesefigures one gets the same conclusion as in the previousexperimentThe biharmonic reconstruction in [14] yields thebest restoration while the biharmonic reconstruction in [13]yields the worst reconstruction

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] P Shvartsman ldquoThe Whitney extension problem and Lipschitzselections of set-valued mappings in jet-spacesrdquo Transactions ofthe American Mathematical Society vol 360 no 10 pp 5529ndash5550 2008

8 International Journal of Mathematics and Mathematical Sciences

[2] P Shvartsman ldquoWhitney-type extension theorems for jetsgenerated by Sobolev functionsrdquo Advances in Mathematics vol313 pp 379ndash469 2017

[3] Y Brudnyi and P Shvartsman ldquoWhitneyrsquos extension problemfor multivariate C1120596-functionsrdquo Transactions of the AmericanMathematical Society vol 353 no 6 pp 2487ndash2512 2001

[4] S B Damelin andC FeffermanOn SmoothWhitney Extensionsof almost isometries with small distortion Interpolation andAlignment in RD-Part 1 arxiv 14112451

[5] S B Damelin and C FeffermanOn SmoothWhitney extensionsof almost isometries with small distortion in Cm(Rn) arxiv150506950

[6] C Fefferman S B Damelin and W Glover ldquoA BMO theoremfor 120576 and an application to comparing manifolds of speech andsound Involverdquo A Journal of Mathematics vol 5 no 2 pp 159ndash172 2012

[7] C Fefferman ldquoWhitneyrsquos extension problems and interpolationof datardquo Bulletin (New Series) of the American MathematicalSociety vol 46 no 2 pp 207ndash220 2009

[8] E J McShane ldquoExtension of range of functionsrdquo Bulletin (NewSeries) of the American Mathematical Society vol 40 no 12 pp837ndash842 1934

[9] H Whitney ldquoAnalytic extensions of differentiable functionsdefined in closed setsrdquo Transactions of the AmericanMathemat-ical Society vol 36 no 1 pp 63ndash89 1934

[10] M Bertalmio G Sapiro C Ballester and V Caselles ldquoImageInpaintingrdquo in Proceedings of the 27th annual conference onComputer graphics and interactive techniques (SIGGRAPH rsquo00)pp 417ndash424 July 2000

[11] M Bertalmio A Bertozzi and G Sapiro ldquoNavier-stokes fluiddynamics and image and video inpaintingrdquo in Proceedings ofthe 2001 IEEE Computer Society Conference on Computer Visionand Pattern Recognition CVPR 2001 pp I-355ndashI-362 KauaiHIUSA

[12] T F Chan and J Shen Image Processing and Analysis SIAMPhiladelphia Pa USA 2005

[13] C K Chui ldquoAn MRA approach to surface completion andimage inpaintingrdquo Applied and Computational Harmonic Anal-ysis vol 26 no 2 pp 270ndash276 2009

[14] C K Chui and H N Mhaskar ldquoMRA contextual-recoveryextension of smooth functions on manifoldsrdquo Applied andComputational Harmonic Analysis vol 28 no 1 pp 104ndash1132010

[15] P Bjorstad ldquoFast numerical solution of the biharmonic Dirich-let problemon rectanglesrdquo SIAM Journal onNumerical Analysisvol 20 no 1 pp 59ndash71 1983

[16] C Tomasi and R Manduchi ldquoBilateral filtering for gray andcolor imagesrdquo in Proceedings of the 6th International Conferenceon Computer Vision (ICCV rsquo98) pp 839ndash846 Bombay IndiaJanuary 1998

[17] A Buades B Coll and J-M Morel ldquoA non-local algorithm forimage denoisingrdquo in Proceedings of the 2005 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2005 pp 60ndash65 usa June 2005

[18] J Darbon A Cunha T F Chan S Osher and G J JensenldquoFast nonlocal filtering applied to electron cryomicroscopyrdquoin Proceedings of the 5th IEEE International Symposium onBiomedical Imaging From Nano to Macro (ISBI rsquo08) pp 1331ndash1334 IEEE Paris France May 2008

[19] J Froment ldquoParameter-Free Fast Pixelwise Non-Local MeansDenoisingrdquo Image Processing On Line vol 4 pp 300ndash326 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 International Journal of Mathematics and Mathematical Sciences

[2] P Shvartsman ldquoWhitney-type extension theorems for jetsgenerated by Sobolev functionsrdquo Advances in Mathematics vol313 pp 379ndash469 2017

[3] Y Brudnyi and P Shvartsman ldquoWhitneyrsquos extension problemfor multivariate C1120596-functionsrdquo Transactions of the AmericanMathematical Society vol 353 no 6 pp 2487ndash2512 2001

[4] S B Damelin andC FeffermanOn SmoothWhitney Extensionsof almost isometries with small distortion Interpolation andAlignment in RD-Part 1 arxiv 14112451

[5] S B Damelin and C FeffermanOn SmoothWhitney extensionsof almost isometries with small distortion in Cm(Rn) arxiv150506950

[6] C Fefferman S B Damelin and W Glover ldquoA BMO theoremfor 120576 and an application to comparing manifolds of speech andsound Involverdquo A Journal of Mathematics vol 5 no 2 pp 159ndash172 2012

[7] C Fefferman ldquoWhitneyrsquos extension problems and interpolationof datardquo Bulletin (New Series) of the American MathematicalSociety vol 46 no 2 pp 207ndash220 2009

[8] E J McShane ldquoExtension of range of functionsrdquo Bulletin (NewSeries) of the American Mathematical Society vol 40 no 12 pp837ndash842 1934

[9] H Whitney ldquoAnalytic extensions of differentiable functionsdefined in closed setsrdquo Transactions of the AmericanMathemat-ical Society vol 36 no 1 pp 63ndash89 1934

[10] M Bertalmio G Sapiro C Ballester and V Caselles ldquoImageInpaintingrdquo in Proceedings of the 27th annual conference onComputer graphics and interactive techniques (SIGGRAPH rsquo00)pp 417ndash424 July 2000

[11] M Bertalmio A Bertozzi and G Sapiro ldquoNavier-stokes fluiddynamics and image and video inpaintingrdquo in Proceedings ofthe 2001 IEEE Computer Society Conference on Computer Visionand Pattern Recognition CVPR 2001 pp I-355ndashI-362 KauaiHIUSA

[12] T F Chan and J Shen Image Processing and Analysis SIAMPhiladelphia Pa USA 2005

[13] C K Chui ldquoAn MRA approach to surface completion andimage inpaintingrdquo Applied and Computational Harmonic Anal-ysis vol 26 no 2 pp 270ndash276 2009

[14] C K Chui and H N Mhaskar ldquoMRA contextual-recoveryextension of smooth functions on manifoldsrdquo Applied andComputational Harmonic Analysis vol 28 no 1 pp 104ndash1132010

[15] P Bjorstad ldquoFast numerical solution of the biharmonic Dirich-let problemon rectanglesrdquo SIAM Journal onNumerical Analysisvol 20 no 1 pp 59ndash71 1983

[16] C Tomasi and R Manduchi ldquoBilateral filtering for gray andcolor imagesrdquo in Proceedings of the 6th International Conferenceon Computer Vision (ICCV rsquo98) pp 839ndash846 Bombay IndiaJanuary 1998

[17] A Buades B Coll and J-M Morel ldquoA non-local algorithm forimage denoisingrdquo in Proceedings of the 2005 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2005 pp 60ndash65 usa June 2005

[18] J Darbon A Cunha T F Chan S Osher and G J JensenldquoFast nonlocal filtering applied to electron cryomicroscopyrdquoin Proceedings of the 5th IEEE International Symposium onBiomedical Imaging From Nano to Macro (ISBI rsquo08) pp 1331ndash1334 IEEE Paris France May 2008

[19] J Froment ldquoParameter-Free Fast Pixelwise Non-Local MeansDenoisingrdquo Image Processing On Line vol 4 pp 300ndash326 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom