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

Multiresolution analysis & wavelets (quick tutorial) Application : image modeling André Jalobeanu

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

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

Space / Frequency representation (wavelet basis functions) scale or spatial frequency space Compromise between spatial and frequential localization uncertainty principle …different wavelet shapes 4

1D wavelet basis • L 2 (R) Wavelet ψ : • || 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

2D tensor product wavelet basis details approximations φ 2 φ 2 ψ 2 φ 2 φ 1 φ 1 ψ 1 φ 1 φ 2 ψ 2 ψ 2 ψ 2 imag e φ 1 ψ 1 ψ 1 ψ 1 6

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 … a j+1 h  2 h  2 d 1 g  2 j+1 a j d 2 h  2 j+1 g  2 convolution decimation d 3 g  2 j+1 rows columns 7

Wavelet transform tree j=0 j=1 j=2 j=3 8

Wavelet packet transform tree j=0 j=1  decompose the detail subbands [Mallat] j=2 9

Wavelet packet basis approximations details Wavelet packets φ 2 φ 2 φ 1 φ 1 ψ 1 φ 1 imag e φ 1 ψ 1 ψ 1 ψ 1 10

Complex wavelet packets  Shift invariance  Directional selectivity Properties : Properties :  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

Quad-tree : 1 st level a 1A d 1 a 1 d 1 1A 1 a 1B d 1 1B a 0 d 2 d 3 a 1C d 1 1A 1A 1C d 2 d 3 a 1D d 1 (image) 1B 1B 1D d 2 d 3 d 2 d 3 1C 1C 1 1 d 2 d 3 1D 1D Parallel trees ABCD Non-decimated transform A C A C A A A A B B B B B D B D A A C C A A C C Perfect B B D D B B D D A A C C A C C A A C A C reconstruction : B B B D D B D D A A C C A A C C mean B D B D B B D B B D D D C C C C (A+B+C+D)/4 D D D D 12

Quad-tree : level j different length filters : h o , g o , h e , g e  shift < pixel a j+1,A h e  2 e a j+1,B h o  2 o a j+1,C h e  2 e h e  2 e a j+1,D h o  2 o d 1 h e g e  2 e  2 e j+1, d 1 h o g o  2 o  2 o a j,A j+1, d 1 h o g e  2 o  2 e A a j,B j+1, d 1 g o  2 o d 2 h e B  2 e a j,C j+1, j+1, d 2 g e h o C  2 e  2 o a j,A j+1, d 2 g e h e D  2 e  2 e A j+1, d 2 g o h o  2 o  2 o d 3 g e B  2 e j+1, g o j+1,  2 o d 3 g o C  2 o j+1, d 3 g e D  2 e A j+1, d 3 g o  2 o B j+1, C D 13

Frequency plane partition 14

Directional selectivity impulse responses – real part Complex wavelets Complex wavelet packets 15

Why use wavelets ? 16

Self-similarity of natural images : P1 (1) IMAGE Spectrum log w Energy w Power spectrum decay ? log r radial frequency r 17

Self-similarity of natural images (2) Vannes Vannes (1) IMAGE scale invariance or self-similarity Spectrum log w Energy w Power spectrum decay w = w 0 r -q log r radial frequency r 18

Non-stationarity of natural images : P2 textures Smooth areas Small features edges 19 2. Modélisation des images

Image modeling Fractional brownian motion (w 0 ,q) P1 Fractal model P2 Non-stationary multiplier function P1 P2 Wavelet transform  ~ independent oefficients (~K-L) Image Frequency space space Subband histogram Frequency plane partition P2 Heavy-tailed distribution 20

Inter-scale dependence Wavelet transform level 3 level 2 level 1 Inter-scale persistence of the details 21

Basis choice (1) Optimal representation of features by different wavelet shapes Haar Symmlet-8 Complex [Haar, 10] [Daubechies, 88] [Kingsbury, 98] Sparse representation : keep a small number of coefficients log approximation error Asymptote E~N -1/2 image Haar Symmlet-8 log coefficients number 22

Basis choice (2) : invariance properties Shift invariance ? Shifted image Haar Spline Symmlet 8 Complex Rotation invariance ? Haar Spline Symmlet 8 Complex 23

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