Multiresolution analysis & wavelets (quick tutorial) - - PowerPoint PPT Presentation

Multiresolution analysis & wavelets (quick tutorial)

Application : image modeling

André Jalobeanu

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Multiresolution analysis

Set of closed nested subspaces

• f

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

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Multiresolution decomposition

Set of approximations and details Basis of Wj

k at scale j ; l = spatial index

k = subband index (orientation, etc.)

= wavelets

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Space / Frequency representation

(wavelet basis functions)

space scale

• r

spatial frequency

Compromise between spatial and frequential localization uncertainty principle …different wavelet shapes

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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

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2D tensor product wavelet basis

φ1φ1 φ1ψ1 ψ1φ1 ψ1ψ1

φ2φ2 ψ2φ2 ψ2ψ2 φ2ψ2

imag e approximations details

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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 d1

j+1

d2

j+1

d3

j+1

columns

…

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Wavelet transform tree

j=0 j=1 j=2 j=3

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Wavelet packet transform tree

j=0 j=1  decompose the detail subbands

[Mallat]

j=2

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Wavelet packet basis

φ1φ1 φ1ψ1 ψ1φ1 ψ1ψ1

φ2φ2

imag e approximations details

Wavelet packets

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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 :

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Quad-tree : 1st level

a0

(image) a1A d2

1A

d1

1A

d3

1A

a1B d2

1B

d1

1B

d3

1B

a1C d2

1C

d1

1C

d3

1C

a1D d2

1D

d1

1D

d3

1D

a1 d2

1

d1

1

d3

1

Non-decimated transform Parallel trees ABCD

A B C D A B C D A B C D A B C D A A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C C D D D D D D D D D D D D D D D D

Perfect reconstruction : mean (A+B+C+D)/4

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2e 2e

he ge aj,A he ge

2e 2e

he ge

2e 2e

aj+1,A d1

j+1, A

d2

j+1, A

d3

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 d1

j+1, B

d2

j+1, B

d3

j+1, B

2o 2o

ho go aj,C he ge

2e 2e

he ge

2e 2e

aj+1,C d1

j+1, C

d2

j+1, C

d3

j+1, C

2o 2o

ho go aj,A ho go

2o 2o

ho go

2o 2o

aj+1,D d1

j+1, D

d2

j+1, D

d3

j+1, D

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Frequency plane partition

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Directional selectivity

impulse responses – real part

Complex wavelets Complex wavelet packets

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Why use wavelets ?

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Self-similarity of natural images : P1 (1)

IMAGE

Spectrum radial frequency r Energy w log w log r

Power spectrum decay ?

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Self-similarity of natural images (2)

scale invariance

• r self-similarity

IMAGE

Vannes (1) Vannes

Spectrum radial frequency r Energy w log w log r

w = w0 r -q

Power spectrum decay

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Non-stationarity of natural images : P2

• 2. Modélisation des images

textures edges Smooth areas Small features

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Image modeling

Wavelet transform  ~ independent oefficients (~K-L)

Frequency plane partition

Heavy-tailed distribution P2 P1 Fractional brownian motion (w0,q) Fractal model P2 Non-stationary multiplier function

Subband histogram

P1 P2

Frequency space Image space

21 level 3 level 2 level 1

Inter-scale dependence

Inter-scale persistence of the details

Wavelet transform

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Haar [Haar, 10] Symmlet-8 [Daubechies, 88] Complex [Kingsbury, 98]

Basis choice (1)

log approximation error 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

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Haar Spline Symmlet 8 Complex Haar Spline Symmlet 8 Complex

Basis choice (2) : invariance properties

Shifted image

Shift invariance ? Rotation invariance ?

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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
• …