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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
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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
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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• …