Multiresolution analysis & wavelets(quick tutorial)

Application : image modeling

André Jalobeanu

2

Multiresolution analysis

Set of closed nested subspaces of

j = scale, resolution = 2-j (dyadic wavelets)

Approximation aj at scale j : projection of f on Vj

Basis of Vj at scale j ; l = spatial index

3

Multiresolution decomposition

Set of approximations and details

Basis of Wjk at scale j ; l = spatial index

k = subband index (orientation, etc.)

= wavelets

4

Space / Frequency representation(wavelet basis functions)

space

scaleor

spatialfrequency

Compromise between spatial and frequential localizationuncertainty principle …different wavelet shapes

5

1D wavelet basis

Dilations / shifts :

Wavelet ψ : • L2(R)• || ψ ||2 = 1• zero mean

Basis of L2(R)

Scale function φ

Multiresolution analysis [Mallat]

Basis of Vj : approximation at res. 2-j

6

2D tensor product wavelet basis

φ1φ1

φ1ψ1

ψ1φ1

ψ1ψ1

φ2φ2 ψ2φ2

ψ2ψ2φ2ψ2

image

approximationsdetails

7

2D Wavelet transform using filter banks

In practice : discrete wavelet transform [Mallat,Vetterli]φ et ψ completely defined by the discrete filters h and g

(a,d1,d2,d3) at scale 2-j (a,d1,d2,d3) at scale 2-j-1

2

2

decimation

h

g

aj

convolution

rows

h

g

2

2

h

g

2

2

aj+1

d1j+1

d2j+1

d3j+1

columns

…

8

Wavelet transform tree

j=0

j=1

j=2

j=3

9

Wavelet packet transform tree

j=0

j=1

decomposethe detail subbands[Mallat]

j=2

10

Wavelet packet basis

φ1φ1

φ1ψ1

ψ1φ1

ψ1ψ1

φ2φ2

image

approximations details

Wavelet packets

11

Complex wavelet packets

Shift invariance Directional selectivity Perfect reconstruction Fast algorithm O(N)

Properties :

• quad-tree (4 parallel wavelet trees) [Kingsbury 98]

• filters shifted by ½ and ¼ pixel between trees• combination of trees complex coefficients

• filter bank implementation • biorthogonal wavelets

Properties :

12

Quad-tree : 1st level

a0(image)

a1A

d21A

d11A

d31A

a1B

d21B

d11B

d31B

a1C

d21C

d11C

d31C

a1D

d21D

d11D

d31D

a1

d21

d11

d31

Non-decimated transform Parallel trees ABCD

AB

CD

AB

CD

AB

CD

AB

CD

AA A

A

AA A

A

AA A

A

AA A

ABB B

B BB B

B

BB B

B BB B

BCC C

CCC C

C

CC C

CCC C

CDD D

D

DD D

D

DD D

D

DD D

D

Perfect reconstruction :

mean (A+B+C+D)/4

13

2e

2e

he

ge

aj,A

he

ge

2e

2e

he

ge

2e

2e

aj+1,A

d1j+1,

Ad2

j+1,

Ad3

j+1,

A

Quad-tree : level j

different length filters : ho, go, he, ge shift < pixel

2e

2e

he

ge

aj,B

ho

go

2o

2o

ho

go

2o

2o

aj+1,B

d1j+1,

Bd2

j+1,

Bd3

j+1,

B

2o

2o

ho

go

aj,C

he

ge

2e

2e

he

ge

2e

2e

aj+1,C

d1j+1,

Cd2

j+1,

Cd3

j+1,

C

2o

2o

ho

go

aj,A

ho

go

2o

2o

ho

go

2o

2o

aj+1,D

d1j+1,

Dd2

j+1,

Dd3

j+1,

D

14

Frequency plane partition

15

Directional selectivityimpulse responses – real part

Complex wavelets

Complex wavelet packets

16

Why use wavelets ?

17

Self-similarity of natural images : P1 (1)

IMAGE

Spectrum

radial frequency r

Energy w log w

log r

Power spectrum decay ?

18

Self-similarity of natural images (2)

scale invariance

or self-similarity

IMAGEVannes (1)Vannes

Spectrum

radial frequency r

Energy wlog w

log rw = w0 r -qPower spectrum decay

19

Non-stationarity of natural images : P2

2. Modélisation des images

textures

edges

Smooth areas

Small features

20

Image modeling

Wavelet transform ~ independent oefficients (~K-L)

Frequency planepartition

Heavy-tailed distributionP2

P1 Fractional brownian motion (w0,q)Fractal model

P2 Non-stationary multiplier function

Subband histogram

P1 P2Frequency

spaceImagespace

21

level 3 level 2 level 1

Inter-scale dependence

Inter-scale persistence of the details

Wavelettransform

22

Haar [Haar, 10]

Symmlet-8[Daubechies, 88]

Complex[Kingsbury, 98]

Basis choice (1)

log

appr

oxim

atio

n er

ror

log coefficients number

image

Sparse representation : keep a small number of coefficients

Optimal representation of features by different wavelet shapes

Asymptote E~N-1/2

Haar

Symmlet-8

23

Haar Spline Symmlet 8 Complex

Haar Spline Symmlet 8 Complex

Basis choice (2) : invariance properties

Shifted image

Shift invariance ?

Rotation invariance ?

24

Wavelet zoo

• Orthogonal wavelets• Biorthogonal wavelets• Non-decimated (redundant) decompositions• Pyramidal representations (Burt-Adelson, etc.)• Wavelets-vaguelettes (deconvolution)• Non-linear multiscale transforms (lifting, non-linear prediction)• Curvelet transform (better represents curves)• Complex wavelets• Non-separable wavelets• Wavelets on manifolds• …