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Wavelet Analysis in Signal and Image Processing

Jean-Pierre Gazeau

Laboratoire Astroparticules et CosmologieCNRS–Universite Diderot Paris 7,

[email protected]

University of PalermoJanuary 14, 2010


1. Introduction to wavelet analysis

(a) Hilbert and Fourier : notations(b) Time-frequency representation : the windowed Fourier or continuous

Gabor transform (1D CGT)(c) One-dimensional continuous wavelet transform (1D CWT)(d) Implementation and interpretation(e) About the discretization problem(f) One-dimensional discrete wavelet transform (1D DWT)(g) Multiresolution analysis

2. Wavelet analysis and image processing

(a) Two-dimensional continuous wavelet transform (2D CWT)(b) Two-dimensional discrete wavelet transform (2D DWT)


Lab session

1. 1-D Transform : time-frequency, time-scale

2. 2-D Transform : space parameters, angle-scale


REFERENCES

1. Stephane Mallat A Wavelet Tour of Signal Processing Academic Press; 2ndedition (1999)

2. Ingrid Daubechies Ten Lectures on Wavelets SIAM 1992

3. S.T. Ali, J.P. Antoine, and JPG, Coherent states and wavelets, a mathemat-ical overview ,Graduate Textbooks in Contemporary Physics (Springer, New York) (2000)

4. Matlab Wavelet Toolbox


FOURIER SIGNAL ANALYSIS

Signal : t“time”

→ s(t) ∈ C

Finite energy signal :

∞∫−∞

|s(t)|2 dt <∞,

i.e. s ∈ L2(R)

L2(R) : Hilbert space with scalar product

〈s1|s2〉 =

∞∫−∞

s1(t)s2(t) dt

Fourier transform : frequency content of signal

s(ω)(≡ (Fs)(ω)) =1√2π

∞∫−∞

e−iωts(t) dt


RECONSTRUCTION

s(t)(≡ (F−1s)(t)) =1√2π

∞∫−∞

eiωts(ω) dω

PLANCHEREL

〈s1|s2〉 =

∞∫−∞

s1(t)s2(t) dt =

∞∫−∞

s1(ω)s2(ω) dω = 〈s1|s2〉

⇒ ENERGY CONSERVATION

‖s‖2 ≡ 〈s|s〉 =

∞∫−∞

|s(t)|2 dt =

∞∫−∞

|s(ω)|2 dω = ‖s‖2


LIKE IN EUCLIDEAN GEOMETRY ...Signal : s(t)≡ vector |s〉 ou |s(t)〉

Elementary signal or atom : 1√2π e

iωt ≡ continuous orthonormal basis vector

〈 1√2π

eiωt| 1√2π

eiω′t〉 =

1

2π

∞∫−∞

ei(ω′−ω)t dt

= δ(ω′ − ω) [orthonormality]

I =

∞∫−∞

| 1√2π

eiωt〉〈 1√2π

eiωt| dω [basis]

Euclidean decomposition in elementary signals :

|s(t)〉 =

∞∫−∞

〈 1√2π

eiωt|s(t)〉︸︷︷︸Fourier transform

| 1√2π

eiωt〉 dω


Example of signal

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TIME-FREQUENCY REPRESENTATIONS (“Gaborets”)Ingredients: translation + modulation.

It is also called the windowed Fourier transform.One chooses a probe or window ψ which is well localized in time and

frequency at once, and which is normalized, ‖ψ‖ = 1. The probe is thentranslated in time and frequency, but its size is not modified (in modulus)

ψ(t)→ ψb,ω= 1a(t) = ei

taψ(t− b) = eiωtψ(t− b)

The time-frequency transform is then :

s(t)→ S(b, ω) = 〈ψb,ω|s〉 =

∫ +∞

−∞e−iωtψ(t− b)s(t) dt.

It is easy to prove that there is conservation of the energy :

‖s‖2 =

∫ +∞

−∞|s(t)|2 dt =

∫ +∞

−∞

∫ +∞

−∞|S(b, ω)|2 db dω

2πdef= ‖S‖2,

and so the reciprocity or reconstruction formula:

s(t)) =

∫ +∞

−∞

∫ +∞

−∞S(b, ω)eiωtψ(t− b) db dω

2π.


Vostok temperatures

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Time variations of Vostok temperatures

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Fou

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Time−frequency representation of Vostok temperature500 1000 1500 2000 2500 3000

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Gabor Transform of the signal superpos.mat

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CONTINUOUS WAVELET TRANSFORMIngredients (transport + zoom)

1. A mother wavelet or probe ψ(t) ∈ L2(R)

(a) well localized

(b) zero average :∞∫−∞

ψ(t) dt = 0

more precisely :

0 < cψ ≡∞∫

−∞

|ψ(ω)|2 dω|ω|

=

∞∫−∞

|ψ(−ω)|2 dω|ω|

<∞

2. The (continuous) family of translated-dilated-contracted versions of theprobe ψ:

1√aψ

(t− ba

)a∈R?

+,b∈R


Then :

1.

1√aψ(t−ba

)a∈R?

+,b∈Rforms an overcomplete family in L2(R), which means

that any signal decomposes as

s(t) =1

cψ

∞∫−∞

db

∞∫0

da

a2S(b, a)

1√aψ

(t− ba

)

2. The “coefficient” S(b, a), as a function of the two continuous variables b(time) and a (scale), is the wavelet transform of the signal :

S(b, a) = 〈 1√aψ

(t− ba

)|s〉

=

∞∫−∞

1√aψ

(t− ba

)s(t) dt

=

∞∫−∞

√aeiωbψ(aω)s(ω) dω


1. Equivalent to 1 and 2 : resolution of the unity

I =1

cψ

∞∫−∞

db

∞∫0

da

a2

∣∣∣∣ 1√aψ

(t− ba

)⟩⟨1√aψ

(t− ba

)∣∣∣∣2. Equivalent to 1 and 2 and 3 : energy conservation

‖s‖2 =1

cψ

∞∫−∞

db

∞∫0

da

a2|S(b, a)|2


In practice, one imposes additional constraints on ψ :

1. restrictions on the supports of ψ or ψ

2. vanishing of higher order moments :

∞∫−∞

tpψ(t) dt = 0, p = 0, 1, . . . , pmax

Then the wavelet transform ignores polynomial components of the signal(i.e. most regular or smoother parts ) in order to enhance the most singularaspects.

WAVELET = SINGULARITY DETECTOR


EXAMPLES OF WAVELETS

1. Morlet wavelet : modulated gaussian ψM(t) = π−14

(eiωmt − e−ω2

m2

)e−

t2

2 ,

ψM(ω) = π−14

(e−

(ω−ωm)2

2 − e−ω2

2 e−ω2m2

),

2. Mexican hat : second derivative of the gaussian, two first moments vanishψH(t) = (1− t2)e− t2

2 = − d2

dt2e−

t2

2 ,

ψH(ω) = ω2e−ω2

2 .


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CWT of the Vostok temperature data: 2sd der. of Gaussian, width=0.5500 1000 1500 2000 2500 3000

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INTERPRETATION

Efficiency of the wavelet transform due to

• condition(s) of admissibility :∞∫−∞

tpψ(t) dt = 0,

• constraints on support of ψ. Indeed, if

– ψ has support of length≈ T around 0

– ψ has support of length≈ Ω around ω0

Then TΩ = cste (Fourier-Heisenberg) and

–∣∣∣ 1√

aψ( t−b

a)∣∣∣ has support of length≈ aT around b,

–∣∣∣√aψ(aω)

∣∣∣ a support of length ≈ Ω/a around ω0/a (relative band-

width ∆ω/ω ≈ cste).


Consequences

• if a 1, then

– 1√aψ( t−b

a) is wide window,

– e−iωb√aψ(aω) is very sharp,

and the CWT reacts mainly to low frequencies (low-band filter),

• if a 1, then

– 1√aψ( t−b

a) is narrow window,

– e−iωb√aψ(aω) is wide window,

and the CWT reacts mainly to high frequencies (high-band filter) whileoffering an efficient temporal localization.


CONCLUSION

The wavelet transform

s(t)→ S(b, a) =

∞∫−∞

1√aψ( t−b

a)s(t) dt

∞∫−∞

√aeiωbψ(aω)s(ω) dω

acts as a local filter, for time and scale at once : it selects the part of the signalpossibly concentrated around instant b and scale a.Furthermore, if the wavelet ψ is well localized, then the energy density|S(b, a)|2 of the CWT will be concentrated on those parts of the signal whichare the most significant in terms of information.

The CWT acts as amathematical microscope :

• ψ ≡ optics

• b≡ position

• 1/a≡ global magnification


DISCRETIZATION PROBLEMS

Redundancy due to continuum of the wavelet representation

s( t︸︷︷︸1D

) =1

cψ

∞∫−∞

db

∞∫0

da

a2S( b, a︸︷︷︸

2D

)1√aψ

(t− ba

)The CWT unmixes parts of signal which live at at same instants, but at differentscales


Discretization : might eliminate this redundancy through the choice of aminimal grid Γ = (bk, aj), j, k ∈ Z in the half-plane time-scale

R× R?+ = (b, a)

s(t) =∑j,k∈Z

〈ψbkaj|s〉 ψbkaj

(t)︸︷︷︸dual frame

A good grid is that one for which there exists 0 < m ≤M s.t.

m‖s‖2 ≤∑j,k

|〈ψbkaj|s〉|2 ≤M‖s‖2

One speaks of discrete frame, of resolving power

M −mM + m

.

If m = M , then the frame is tight. It is orthonormal basis if m = M = 1.


EXAMPLE

Choice induced by the non-euclidean geometry (Lobatchevski) of the time-scalehalf-plane :

aj = aj0, bk = kb0aj0, a0 > 0, k, j ∈ Z

ψbkaj(t) = a

−j/20 ψ(a−j0 t− kb0)

Dyadic wavelets are obtained with : a0 = 2, b0 = 1 (but the approach is totallydifferent from Discrete Wavelet Transform)


1D-DISCRETE WAVELET ANALYSIS


FIRST STEP IN DISCRETE WAVELETS (HAAR) :REFINING PERIODIC SAMPLINGS

Smoothing + sampling of signal s(t) at the integer scale n ∈ Z (scale “zero”)

(Π0s)(t) ≡ s0(t) =∑n

〈ϕ(t− n)|s〉︸︷︷︸ϕ?s(n)

ϕ(t− n) ∈ V0

where

• ϕ(t) : characteristic function of [0, 1] (scaling function or father wavelet)

• ϕ(t)def= ϕ(−t)

• f ∗ g(t)def=∫ +∞−∞ f (u− t)g(t) dt (convolution)

• Also: 〈ϕ(t− n)|s〉 =∫ +∞−∞ ϕ(t− n)s(t) dt =

∫ n+1n

s(t) dt is the averageos the signal on the interval [n, n + 1]

• ϕ(t− n)n∈Z : orthonormal system spans subspace V0 ' l2(Z) of signalconstant on intervals [n, n + 1].


Smoothing + finer sampling of signal s(t)at the half-integer scale ( scale 1), i.e. on V1 ' l2(Z/2) :

(Π1s)(t) =∑n

〈√

2ϕ(2t− n)|s〉︸︷︷︸ϕ1?s(

n2 )

√2ϕ(2t− n) ∈ V1


SECOND STEP IN DISCRETE WAVELETS:HAAR WAVELET

• Information at scale one= Information at scale zero or Approximation + Details :

(Π1s)(t) = (Π0s)(t) + (∆0s)(t) ⇔ V1 = V0 ⊕W0

• Details :(∆0s)(t) =

∑n

〈ψ(t− n)|s〉︸︷︷︸d0,n

ψ(t− n) ∈ W0

• Mother wavelet or Haar wavelet :

ψ(t) = ϕ(2t)− ϕ(2t− 1)

• ψ(t − n)n∈Z : orthonormal system spans W0 of details at scale 1, or-thogonal complement of V0 in V1


THIRD STEP IN DISCRETE WAVELETS:HAAR TRANSFORM AND MULTIRESOLUTION ANALYSIS

Information (or Tendency) at scale j + 1= Tendency at j + Fluctuations (Details) :

(Πj+1s)(t) = (Πjs)(t) + (∆js)(t)⇔ Vj+1 = Vj ⊕Wj

(Πjs)(t) =∑n

〈2j/2ϕ(2jt− n)|s〉︸︷︷︸ϕj?s(n/2j)

2j/2ϕ(2jt− n) ∈ Vj

(∆js)(t) =∑n

〈2j/2ψ(2jt− n)|s〉︸︷︷︸dj,n

2j/2ψ(2jt− n) ∈ Wj

dj,n =∞∫−∞

2j/2ψ(2jt− n)s(t) dt : Wavelet Transform of signal

Multiresolution analysis of L2(R):· · ·Vj−1 ⊂ Vj ⊂ Vj+1 · · ·

2j/2ψ(2jt− n)j,n∈Z : orthonormal basis of

L2(R) =⊕

jWj


SUMMARY :Original Signal (or smoothing + sampling on fine grid Z/2N ):

(ΠNs)(t) ≡ sN(t) ∈ l2(Z/2N) ≡ VN

VN = VN−1 ⊕WN−1 = VN−2 ⊕ [WN−2 ⊕WN−1] =

· · · = V0︸︷︷︸tendency

⊕ [W0 ⊕W1 + · · · +⊕WN−1]︸︷︷︸fluctuation

Corresponds to analysis :

sN = s0 + [r0 + r1 + · · · rN−1] ≡ (s0, r0, r1, . . . , rN−1)

rj = ∆js =∑n

dj,n2j/2ψ(2jt− n) ≡ (dj,n)n∈Z

Discrete wavelet transform :

dj,n =∞∫−∞

2j/2ψ(2jt− n)s(t) dt =∞∫−∞

2j/2ψ( t−2−jn2−j )s(t) dt

2−j : dilation (j > 0) or contraction (j < 0) parameter (scale)2−jn : translation parameter (localization)


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Ingredients of multiresolution analysis :1. A scaling function ϕ(t) ∈ L2(R) s.t.ϕ(t− n)n∈Z is orthonormal system

2. Space V0 linear span of ϕ(t− n)n∈Z

3. Sequence · · ·Vj−1 ⊂ Vj ⊂ Vj+1 · · · defined by f (t) ∈ V0 ⇔ f (2jt) ∈ Vjand s.t. ∩jVj = 0, ∪jVj dense in L2(R)

4. A wavelet, i.e. a function ψ(t) s. t. ψ(t − n)n∈Z spans the orthogonalcomplement W0 of V0 in V1 = V0 ⊕W0

Then :

1. 2j/2ψ(2jt− n)j,n∈Z orthonormal basis of L2(R)

2. Any signal s(t) decomposes ass(t) =

∑j,n dj,n2

j/2ψ(2jt− n)

3. The coefficient dj,n =∞∫−∞

2j/2ψ(2jt− n)s(t) dt = 〈2j/2ψ(2jt− n)|s(t)〉,

as function of discrete variables j et n, is the wavelet transform of the signal.



1. compact support : Haar≡ Daubechies 1, Shannon (FT of Haar),Daubechies 2, Daubechies 3, ....

2. noncompact support

3. biorthogonal etc.


Daubechies Wavelet Ordre 2


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Temperature Vostok sur profondeur 3311m : DWT, 6 niveaux, Daubechies 2

L =

0 co

efap

pr


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BASICS OF IMAGE PROCESSING

• A digital image a[m,n] described in a 2D discrete space is derived from ananalog image a(x, y) in a 2D continuous space through a sampling processthat is frequently referred to as digitization.

• The 2D continuous image a(x, y) is divided into N rows and M columns.The intersection of a row and a column is termed a pixel. The value assignedto the integer coordinates [m,n] with m = 0, 1, 2, . . . ,M−1 and n =0, 1, 2, . . . , N − 1 is a[m,n]. In fact, in most cases a(x, y)–which wemight consider to be the physical signal that impinges on the face of a 2Dsensor–is actually a function of many variables including depth (z), color(λ), and time (t). Unless otherwise stated, we will consider the case of 2D,static images.


Digitization of a continuous image

• The pixel at coordinates [m = 10, n = 3] has the integer brightness value110.

• This image has been divided into N = 16 rows and M = 16 columns.

• The value assigned to every pixel is the average brightness in the pixelrounded to the nearest integer value.

• The process of representing the amplitude of the 2D signal at a given coor-dinate as an integer value with L different gray levels is usually referred toas amplitude quantization or simply quantization.


There are standard values for the various parameters encountered in digitalimage processing. These values can be caused by video standards, by

algorithmic requirements, or by the desire to keep digital circuitry simple.Commonly encountered values :

Rows Columns Gray LevelsParameter N M Λ

Typical values 256, 512, 256, 512, 2, 64,525, 625, 768, 1024, 256, 1024,

1024, 1035 1320 16384


Quite frequently we see cases of M = N = 2K where K = 8, 9, 10. Thiscan be motivated by digital circuitry or by the use of certain algorithms such as

the fast Fourier transform.The number of distinct gray levels is usually a power of 2, that is, Λ = 2Bwhere B is the number of bits in the binary representation of the brightness

levels. When B > 1 we speak of a gray-level image; when B = 1 we speak ofa binary image. In a binary image there are just two gray levels which can be

referred to, for example, as “black” and “white” or “0” and “1”.


HISTOGRAMS

More and more cameras let you view histograms on the camera’s monitor. Thehistogram, like those found in most serious photo-editing programs such asPhotoshop and Picture Window, let you evaluate the distribution of tones.

Since most image corrections can be diagnosed by looking at a histogram, ithelps to look at it while still in a position to reshoot the image. Each pixel in animage can be set to any of 256 levels of brightness from pure black (0) to purewhite (255). A histogram is a graph that shows how the 256 possible levels of

brightness are distributed in the image.


How to read a Histogram


How to read a Histogram

• The horizontal axis represents the range of brightness from 0 (shadows) onthe left to 255 (highlights) on the right. Think of it as a line with 256 spaceson which to stack pixels of the same brightness. Since these are the onlyvalues that can be captured by the camera, the horizontal line also representsthe camera’s maximum potential dynamic range.

• The vertical axis represents the number of pixels that have each one of the256 brightness values. The higher the line coming up from the horizontalaxis, the more pixels there are at that level of brightness.

• To read the histogram, you look at the distribution of pixels. An image thatuses the entire dynamic range of the camera will have a reasonable numberof pixels at every level of brightness. An image that has low contrast willhave the pixels clumped together and have a narrower dynamic range.


Example

This high-key fog scene has most of its values towards the highlight end of thescale. The distinct vertical line to the left of middle gray shows how many

pixels there are in the uniformly gray frame border. You can see that there areno really dark values in the image. In fact, the image uses only a little more

than half the camera’s dynamic range.


CONTINUOUS WAVELET TRANSFORM OF IMAGES

A wavelet or probe ψ(x), x = (x1, x2) ∈ R2 is chosen

• well localized

• admissible, which means

cψ = cste∫∫R2

|ψ(k)|2 d2k

‖k‖2<∞⇔ ψ(0) = 0


Image analysis is carried out by affine transport of the probe ψ in the euclideanplane :

• Dilation/contraction (a > 0)

• Translations (b = (bx, by) ∈ R2)

• Rotations (θ ∈ [0, 2π])

ψ(x)→ ψb,θ,a(x) ≡ 1aψ(R−1(θ)

(x−ba

))Wavelet transform of a 2D signal :

L2(R2) 3 s(x)→ S(b, θ, a) =

∫∫R2

ψb,θ,a(x)s(x) d2x

=

∫∫R2

ψb,θ,a(k)s(k) d2k

s(k) =1

2π

∫∫R2

e−ik·xs(x) d2x, ψb,θ,a(k) = ae−ib·kψ(aR−1(θ)k)


RECONSTRUCTION

s(x) =1

cψ

∫∫R2

d2b

∞∫0

da

a3

2π∫0

dθS(b, θ, a)ψb,θ,a(x)

ENERGY CONSERVATION

∫∫R2

|s(x)|2d2x =1

cψ

∫∫R2

d2b

∞∫0

da

a3

2π∫0

dθ|S(b, θ, a)|2

RESOLUTION OF THE UNITY

I =1

cψ

∫∫R2

d2b

∞∫0

da

a3

2π∫0

dθ|ψb,θ,a〉〈ψb,θ,a|


EXAMPLES OF 2D WAVELETS

1. Isotropic

• Mexican Hat (Marr) :ψH(x) = −∆e−‖x‖

2/2 = (2− ‖x‖2)e−‖x‖2/2

• Difference wavelets (h(x) regular > 0):ψ(x) = α−2h(α−2x)− h(x), 0 < α < 1

2. Orientational

• Morlet wavelet : ψM(t) = eik0·xe−‖Ax‖2

2 + small correction terms s. t. ψH(0) = 0

A = diag[ 1√ε1, 1√

ε2] (anisotropy matrix)

• Cauchy wavelet :ψ has support in strictly convex cone


Anisotropic mexican hat


INTERPRETATIONIf the wavelet ψ is well localized in x and in k at once, then the wavelet

analysis acts with constant relative bandwidth : ∆‖k‖‖k‖ =cste⇒ efficiency at

large frequencies or at small scales⇒ detector of discontinuities in images

• e.g. point singularities ( contour vertices)

• e.g. orientational features (borders, edges, segments, mikado)

ConsequentlyThe 2D CWT acts as a

orientational mathematical microscope :• ψ ≡ optics

• b ≡ position

• 1/a ≡ global zoom

• θ ≡ orientation parameter


VISUALIZATION

6 possible choices of two-dimensional sections of the CWT S(b, θ, a) in spaceof parameters (bx1

, bx2, θ, a),

e.g. :

• representation position : (θ, a) is fixed

• representation direction-scale : (bx1, bx2

) is fixed


DISCRETE WAVELET TRANSFORM OF IMAGES

One here comes back to dyadic multiresolution: a wavelet orthonormal basis inL2(R2) is built up from (tensor) products involving

• a scale function ϕ associated to a multiresolution Vjj∈Z of L2(R)

• a wavelet ψ whose the dilated-translated 2j/2ψ(2jt− n) form an orthonor-mal basis of L2(R) =

⊕jWj

For this purpose, one defines three wavelets :

ψ1(x1, x2) = ϕ(x1)ψ(x2) (horizontal)ψ2(x1, x2) = ψ(x1)ϕ(x2) (vertical),ψ3(x1, x2) = ψ(x1)ψ(x2) (diagonal),

and one puts, for 1 ≤ k ≤ 3,

ψkj,n(x) = 2jψk(2jx1 − n1, 2jx2 − n2)


Then,

• ψ1j,n, ψ

2j,n, ψ

3j,n form an orthonormal basis of the subspace of details

W 2j = (Vj ⊗Wj)⊕ (Wj ⊗ Vj)⊕ (Wj ⊗Wj)

at scale j

• L2(R2) =⊕

jW2j

• The whole image s(x) decomposes ass(x) =

∑k,j,n

dkj,n2jψkj,n(x)

• The coefficient

dkj,n =

∞∫−∞

2jψk(2jx1 − n1, 2

jx2 − n2)s(x) d2x = 〈ψkj,n|s(t)〉,

as function of the three discrete variables k, j and n, is the discrete wavelettransform of the image.



1. compact support : Haar ≡ Daubechies 1, Shannon (TF de Haar),Daubechies 2,Daubechies 3, ....

2. noncompact support

3. biorthogonal etc.

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