2D Wavelets - Università degli Studi di Verona · Advanced concepts • Wavelet packets •...
Transcript of 2D Wavelets - Università degli Studi di Verona · Advanced concepts • Wavelet packets •...
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2D Wavelets
Hints on advanced Concepts
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Advanced concepts
• Wavelet packets
• Laplacian pyramid
• Overcomplete bases – Discrete wavelet frames (DWF)
• Algorithme à trous – Discrete dyadic wavelet frames (DDWF)
• Overview on edge sensitive wavelets – Contourlets
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Wavelet packets
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Wavelet packets Both the approximation and the detail subbands are further decomposed
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Packet tree
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Wavelet Packets
time
frequ
ency
η
η /2
incr
easi
ng th
e sc
ale
η /4
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Laplacian Pyramid
Low-pass Interpolation ↓2 ↑2
coarser version
prediction
- residual
residuals
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Overcomplete bases
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Translation Covariance
Translation DWT
DWT Translation
Signal
Signal
Wavelet coefficients
Wavelet coefficients
{ }{ } { }{ }),(),( yxfTDWTyxfDWTT ≠
Translation covariance
If translation covariance does not hold:
{ }{ } { }{ }),(),( yxfTDWTyxfDWTT =
NOT good for signal analysis
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Rationale
• In pattern recognition it is important to build representations that are translation invariant. This means that when the pattern is translated the descriptors should be translated but not modified in the value.
• CWTs and windowed FT provide translation covariance, while sampling the translation parameter might destroy translation covariance unless some conditions are met.
• Intuition: either the sampling step is very small compared to the translation or the translation is a multiple of the sampling step. Adaptive sampling could also be a solution (tracking local maxima)
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Translation invariant representations
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Translation covariance
• The signal descriptors should be covariant with translations – Continuous WT and windowed FT are translation covariant.
– Wavelet frames (DWF) are constructed by sampling continuous transforms over uniform time grids.
– In general, the sampling grid removes the translation covariance because the translation factor τ is a priori not equal to the translation interval
),(1)(),(
)(1)(),(
)()(
suWfdtsut
stfsuWf
ufdtsut
stfsuWf
tftf
s
τψτ
ψψ
τ
τ
τ
−=⎟⎠
⎞⎜⎝
⎛ −−=
∗=⎟⎠
⎞⎜⎝
⎛ −=
−=
∫
∫
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Sampling and translation covariance
τ
If the translation does not correspond to a multiple of the sampling step, a different set of samples will be obtained by keeping the same sampling grid when the signal shifts
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Sampling and translation covariance
τ
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Translation invariant representation
• If the sampling interval is small enough than the samples of are approximately translated when f is shifted.
• Translation covariance holds if namely it is a multiple of the sampling interval
0ja u ( )ja
f tψ∗
0jku aτ =
Wftau(u,aj)
Wf(u,aj)
u
u τ 0
ja u
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Translation invariant representation
• Translation invariant representations can be obtained by sampling the scale parameter s but not the translation parameter u
• Uniformly sampling the translation parameter destroys covariance unless the translation is very small
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Dyadic Wavelet Transform
• Sampling scheme – Dyadic scales – Integer translations
– If the frequency axis is completely covered by dilated dyadic wavelets, then it defines a complete and stable representation
• The normalized dyadic wavelet transform operator has the same properties of a frame operator, thus both an analysis and a reconstruction wavelets can be identified
• Special case: algorithme à trous
)(22
1)()2,( 2 ufdtuttfuWf jjjj ψψ ∗=⎟
⎠
⎞⎜⎝
⎛ −= ∫
⎟⎠
⎞⎜⎝
⎛ −=−= jj
ttt jj22
1)()( 22 ψψψ
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Algorithme à trous
• Similiar to a fast biorthogonal WT without subsampling
• Fast dyadic transform – The samples of the discrete signal a0[n] are considered as averages of some function
weighted by some scaling kernels φ(t-n)
– For any filter x[n], we denote by xj[n] the filters obtained by inserting 2j-1 zeros between each sample of x[n] → create holes (trous, in French)
)2,()(),(][
)(),(][
)(),(][
2
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0
jnWfnttfnd
nttfna
nttfna
jj
jj
=−=
−=
−=
>
≥
ψ
ϕ
ϕ
grid integer the over 0j for computed are tscoefficien eletdyadic wav The
denote we0jany For
][][ nxnx jj −=
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Algorithme à trous
• Proposition
The dyadic wavelet representation of a0 is defined as the set of wavelet coefficients up to the scale 2J plus the remaining low-pass frequency information aJ
– Fast filterbank implementation
( )][~][~21][
][][
][][0
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1
1
ngdnhana
ngand
nhanaj
jjjjj
jjj
jjj
∗+∗=
∗=
∗=
≥
++
+
+
any For
{ }1
,J j j Ja d
≤ ≤
⎡ ⎤⎢ ⎥⎣ ⎦
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Analysis
~ hj ~ hj+1
~ gj ~ gj+1
aj aj+1
dj+1
aj+2
dj+2
2j-1”trous” No subsampling!!
2102
2101
210
000000][~00][~
][~
hhhnh
hhhnh
hhhnh
=
=
=
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Synthesis
hj+1
gj+1
aj+2
dj+2
+ hj
gj
aj+1
dj+1
+ aj
Overcomplete wavelet representation: [aJ, {dj}1≤j≤J]
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Algorithme a trous
g(z2)
h(z2)
a2f
d2f
g(z4)
h(z4)
a3f
d3f
g(z)
h(z) s(z) a1f
d1f
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DWT vs DWF
• DWT – Non-redundant – Signal il subsampled – Not translation invariant – Total number of coefficients:
NxNy
• Compression
• DWF – Redundant (in general) – Signal is not subsapled – Filters are upsampled – Translation invariant – Total number of coefficients:
(3J+1)NxNy
• Feature extraction
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Discrete WT vs Dyadic WT
Original
HL LH HH LL
DWT
Dyadic WT
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Example 1
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Example 2
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Rotation covariance
• Oriented wavelets
– In 2D, a dyadic WT is computed with several wavelets which have different spatial orientations
• We denote
• The WT in the direction k is defined as
– One can prove that this is a complete and stable representation if there exist A>0 and B>0 such that
Kkk yx ≤≤1)},({ψ
ψ2 jk (x, y) = 1
2 j ψk x2 j ,
y2 j
!
"#
$
%&
Wk f (u,v, 2 j ) = f (x, y),ψ2 jk (x −u, y− v) = f ∗ψ
2 jk (u,v)
{ } ( )2
2
1
ˆ( , ) 0,0 , 2 ,2K
j jx y x y
k jA Bω ω ψ ω ω
+∞
= =−∞
∀ ∈ − ≤ ≤∑ ∑R
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Oriented wavelets
– Then, there exists a reconstruction wavelet family such that
• Gabor wavelets
• In the Fourier plane the energy of the Gabor wavelet is mostly concentrated in
f (x, y) = 12 j
W k f (w, y,2 j )k=1
K
∑j=−∞
+∞
∑ ∗ ψ2 jk (x, y)
ψk x, y( ) = g(x, y)exp −iη(xcosαk + ysinαk ){ }g(x, y) = 1
2πexp −
x2 + y2
2
!
"##
$
%&&
'()
*)
+,)
-)→Gaussian → Gabor wavelets
ψ2 jk (ωx ,ω y ) = 2 j g(2 jωx −ηcosαk ,2
jω y −ηsinαk )
ωy
ωx
2− jηcosαk ,2− jηsinαk( ) in a neighborhood proportional to 2-j
j increases
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Gabor wavelets – dyadic scales
ψ2 jk (ωx ,ω y ) = 2 j g(2 jωx −ηcosαk ,2
jω y −ηsinαk )
( ),x yg ω ω
η2 j−1
kα
ωy
ωx
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Gabor wavelets
( ),x yg ω ωkα
( )cos , sinx yg k kω η α ω η α− −
ωy
ωx
η2 j
ψk x, y( ) = g(x, y)exp −iη(xcosαk + ysinαk ){ }g(x, y) = 1
2πexp −
x2 + y2
2
!
"##
$
%&&
'()
*)
+,)
-)→Gaussian → Gabor wavelets
ψ2 jk (ωx ,ω y ) = 2 j g(2 jωx −ηcosαk ,2
jω y −ηsinαk )
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Gabor wavelets – dyadic scales
• Other directional wavelet families – Dyadic Frames of Directional Wavelets [Vandergheynst 2000] – Curvelets [Donoho&Candes 1995] – Steerable pyramids [Simoncelli-95] – Contourlets [Do&Vetterli 2002]
( ) ( )
( )
2 2
1
cos sin2 2j k kj j
k ktg
η ηρ α α
θ α−
= + =
=
jρ
αk
2-j
ωy
ωx
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Dyadic Frames of Directional Wavelets
Pierre Vandergheynst (LTS-EPFL) http://people.epfl.ch/pierre.vandergheynst
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Dyadic Directional WF
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Dyadic Directional Wavelet Frames
• Directional selectivity at any desired angle at any scale – Not only horizontal, vertical and diagonal as for DWT and DWF – Rotation covariance for multiples of 2π/K
• Recipe – Build a family of isotropic wavelets such that the Fourier transform of the mother wavelet
expressed in polar coordinates is separable
– Split each isotropic wavelet in a set of oriented wavelets by an angular window • Express the angular part Θ(φ) as a sum of window functions centered at θk
Ψ^ω,φ( ) = Γ ω( )Θ φ( )
ω = ωx2 +ωy
2( )
ωy
ωx
ω
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Partitions of the F-domain
2π ϕ
( )θ ϕ
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Dyadic Directional Wavelet Frames
4 orientations (K=4)
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Dyadic Directional Wavelet Frames
DDWF
Feature Images or Neural images
Orientation
Scale
Scaling function
Properties • Overcomplete
• Translation covariance
• Rotation covariance for given angles
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Contourlets Brief overview
Minh Do, CM University Martin Vetterli, LCAV-EPFL
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Contourlets
• Goal – Design an efficient linear expansion for 2D signals, which are smooth away from
discontinuities across smooth curves – Efficiency means Sparseness
– [Do&Vetterli] Piecewise smooth images with smooth contours – Inspired to curvelets [Donoho&Candes]
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Curvelets
• Basic idea – Curvelets can be interpreted as a grouping of nearby wavelet basis functions into linear
structures so that they can capture the smooth discontinuity curve more efficiently
– More efficient in capturing the geometry -> more concise (sparse) representation
2-j
2-j
2-j
c2-j/2
wavelets curvelets
Parabolic scaling
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M. Do and M. Vetterli, The Contourlet Transform: An Efficient Directional Multiresolution Image Representation, IEEE-TIP
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Curvelets
Embedded grids of approximations in spatial domain. Upper line represents the coarser scale and the lower line the finer scale. Two directions (almost horizontal and almost vertical) are considered. Each subspace is spanned by a shift of a curvelet prototype function. The sampling interval matches with the support of the prototype function, for example width w and length l, so that the shifts would tile the R2 plan. The functions are designed to obey the key anisotropy scaling relation:
width ∝ length2
Close resemblance with complex cells’ (orientation selective RF)!
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Better approximation properties for geometries: for a given rate, a better representation of edges is reached Application: coding
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Summary of useful relations
• If f is real
( )
*
* ( )
* ( ) 1
ˆ ˆ( ) ( )ˆ ˆ( ) ( )ˆ ˆ ˆ( ) ( ) ( )
ˆ ˆˆ( ) ( ) ( ) [ ] ( 1) [ ]ˆ ˆˆ( ) ( ) ( ) [ ] ( 1) [1 ]
j
j
j
j n
j j j n
f f ef f ef f e f
t f f e t n f ng e f e f e g n f n
ω
ω π
ω
ω π
ω ω ω π
ωω πω ω
ω ω πω ω π
+
−
− + −
− − − + −
=
+ =
− = =
= + = → = − −
= + = → = − −
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Conclusions
• Multiresolution representations are the fixed point of vision sciences and signal processing
• Different types of wavelet families are suitable to model different image features – Smooth functions -> isotropic wavelets – Contours and geometry -> Curvelets
• Adaptive basis – More flexible tool for image representation – Could be related to the RF of highly specialized neurons
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References
• A Wavelet tour of Signal Processing, S. Mallat, Academic Press
• Papers – A theory for multiresolution signal decomposition, the wavelet representation, S. Mallat,
IEEE Trans. on PAMI, 1989 – Dyadic Directional Wavelet Transforms: Design and Algorithms, P. Vandergheynst and
J.F. Gobbers, IEEE Trans. on IP, 2002 – Contourlets, M. Do and M. Vetterli (Chapter)