Multiresolution analysis & wavelets (quick tutorial)
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1 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 a j at scale j : projection of f on V j Basis of V j at scale j ; l = spatial index 2
3 Multiresolution decomposition Set of approximations and details k = subband index (orientation, etc.) Basis of W j k at scale j ; l = spatial index = wavelets 3
4 Space / Frequency representation (wavelet basis functions) scale or spatial frequency space Compromise between spatial and frequential localization uncertainty principle different wavelet shapes 4
5 1D wavelet basis Wavelet ψ : L 2 (R) ψ 2 = 1 zero mean Dilations / shifts : Basis of L 2 (R) Scale function φ Multiresolution analysis [Mallat] Basis of V j : approximation at res. 2 -j 5
6 2D tensor product wavelet basis approximations details imag e φ 2 φ 2 ψ 2 φ 2 φ 2 ψ 2 φ 1 φ 1 φ 1 ψ 1 ψ 2 ψ 2 ψ 1 φ 1 ψ 1 ψ 1 6
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,d 1,d 2,d 3 ) at scale 2 -j (a,d 1,d 2,d 3 ) at scale 2 -j-1 h 2 a j+1 h 2 g 2 d 1 j+1 a j g 2 h 2 d 2 j+1 convolution rows decimation g columns 2 d 3 j+1 7
8 Wavelet transform tree j=0 j=1 j=2 j=3 8
9 Wavelet packet transform tree j=0 j=1 decompose the detail subbands [Mallat] j=2 9
10 Wavelet packet basis approximations details Wavelet packets imag e φ 2 φ 2 φ 1 φ 1 φ 1 ψ 1 ψ 1 φ 1 ψ 1 ψ 1 10
11 Complex wavelet packets Properties : Shift invariance Directional selectivity Perfect reconstruction Fast algorithm O(N) quad-tree (4 parallel wavelet trees) [Kingsbury 98] filters shifted by ½ and ¼ pixel between trees combination of trees complex coefficients biorthogonal wavelets filter bank implementation 11
12 Quad-tree : 1 st level a 1 d 1 1 a 1A d 1 1A a 1B d 1 1B a 0 (image) d 2 1A da 3 1C 1A d 1 d 2 1B d 3 1C a 1D 1B d 1 1D d 2 1 d 3 1 d 2 1C d 3 1C d 2 1D d 3 1D Non-decimated transform Parallel trees ABCD Perfect reconstruction : mean (A+B+C+D)/4 A B C D A B C D A B C D A B C D A A B A A A A A A A A A A B C B B B A B A A A B B B C D B B B C C C B C B B B C C C D C C C D D D D C C C C D D D D D D D D D D 12
13 Quad-tree : level j different length filters : h o, g o, h e, g e shift < pixel a j,a a j,b a j,c a j,a h e h e h o h o g e g e g o g o 2 e 2 e 2 o 2 o 2 e 2 e 2 o 2 o h e h o h e g e h o g o g e h e g o h o h e g e h o g o g e g o 2 e 2 o 2 e 2 e 2 o 2 o 2 e 2 e 2 o 2 o 2 e 2 e 2 o 2 o 2 e 2 o a j+1,a a j+1,b a j+1,c d 1 a j+1, j+1,d d 1 j+1, A d 1 d 2 j+1, B d 1 j+1, d 2 j+1, C j+1, A d 2 D d 3 j+1, B d 2 j+1, d 3 j+1, C j+1, A d 3 D j+1, B d 3 j+1, C D 13
14 Frequency plane partition 14
15 Directional selectivity impulse responses real part Complex wavelets Complex wavelet packets 15
16 Why use wavelets? 16
17 Self-similarity of natural images : P1 (1) IMAGE Spectrum Energy w radial frequency r log w Power spectrum decay? log r 17
18 Self-similarity of natural images (2) IMAGE Vannes Vannes (1) scale invariance or self-similarity Spectrum log w Energy w Power spectrum decay w = w 0 r -q log r radial frequency r 18
19 2. Modélisation des images Non-stationarity of natural images : P2 textures Smooth areas Small features edges 19
20 Image modeling P1 P2 Fractional brownian motion (w 0,q) Fractal model Non-stationary multiplier function P1 Frequency space P2 Image space Wavelet transform ~ independent oefficients (~K-L) Subband histogram Frequency plane partition P2 Heavy-tailed distribution 20
21 Inter-scale dependence Wavelet transform level 3 level 2 level 1 Inter-scale persistence of the details 21
22 Basis choice (1) Optimal representation of features by different wavelet shapes Haar [Haar, 10] Symmlet-8 [Daubechies, 88] Complex [Kingsbury, 98] Sparse representation : keep a small number of coefficients image log approximation error log coefficients number Asymptote E~N -1/2 Haar Symmlet-8 22
23 Basis choice (2) : invariance properties Shift invariance? Shifted image Haar Spline Symmlet 8 Complex Rotation invariance? Haar Spline Symmlet 8 Complex 23
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 24
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