Wavelets and filterbanks

1
Wavelets and filterbanks
2
Outline

• Wavelets and Filterbanks
• Biorthogonal bases
• Separable bases (2D)
• Overcomplete bases

3
Wavelets and Filterbanks

• Wavelet side
• Scaling function
• Design (from multiresolution priors)
• Signal approximation
• Corresponding filtering operation
• Condition on the filter hn ? Conjugate Mirror
  Filter (CMF)
• Corresponding wavelet families

• Filterbank side
• Perfect reconstruction conditions (PR)
• Reversibility of the transform
• Equivalence with the conditions on the wavelet
  filters
• Special case CMFs ? Orhogonal wavelets
• General case ? Biorthogonal wavelets

4
Scaling function

• The approximation of f at scale 2j is defined as
  the orthogonal projection of PVjf on Vj
• To compute this projection we need an orthonormal
  basis for Vj
• Such a basis can be obtained by orthogonalizating
  the Rieszs basis ?(t-n)n?Z
• This results in an orthogonal basis for each
  subspace Vj obtained by translating and dilating
  a single function ? called the scaling function

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Scaling function

• Theorem 7.1
• Let Vjj?Z be a multiresolution approximation
  and ? be the scaling function, whose Fourier
  transform is
• Let us denote
• The family is an orthonormal
  basis of Vj for all j ?Z

6
Approximation

• Partition of unity
• The orthogonal projection of f over Vj is
  obtained with an expansion in the scaling
  orthonormal basis

projection as filtering
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Scaling equation

• A multiresolution decomposition is entirely
  characterized by the scaling function ? that
  generates an orthonormal basis for Vj
• The scaling function must satisfy some conditions
  in order that the space Vj satisfy all the
  conditions of a multiresolution approximation
• Any scaling function is specified by a discrete
  filter called a conjugate mirror filter
• Scaling equation

The scaling equation relates a dilation of ? by a
factor 2 to its integer translations The
sequence hn can be interpreted as a discrete
filter
Scaling equation
(1)
8
From the Scaling Equation to CMFs
Taking the F-transform of both sides of (1)
(which is a convolution in the signal domain)
Condition for ? to be the scaling function for a
multiresolution approximation ? conditions on h
can be derived
9
Conjugate Mirror Filters
Teorem 7.2 (MallatMeyer) Let ??L2(R) be an
integrable scaling function. The F-series of hn
satisfies Conversely, if h(?) is 2? periodic
and continuously differentiable in a neighborhood
of ?0, if it satisfies (2) and if Then,
is
the F-transform of a scaling function.
(2)
CMF
Im
?
??
Re
10
Corresponding orthogonal wavelet family

• Theorem 7.3 MallatMeyer
• Let ? be a scaling function and h the
  corresponding CMF. Let ? be such that
• with
• Let us denote
• For any scale, ?j,nj?Z is an orthonormal basis
  for Wj. For all j, it is an orthonormal basis for
  L2.
• Signal domain

(1)
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Corollaries

• Lemma 7.1. The family ?j,nn?Z is an orthonormal
  basis for Wj iif
• The proof of the theorem also shows that g is the
  Fourier series of
• which are the decomposition coefficients of
• taking the inverse transform of (1)

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Wavelet representation

• The orthogonal projection of a signal f into the
  detail space Wj is obtained by a partial
  expansion in its wavelet basis
• A signal expansion in a wavelet orthonormal basis
  can thus be viewed as an aggregation of details
  at scales 2j that go from 0 to infinity
• Intuition

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Wavelets and filterbanks

• The decomposition coefficients in a wavelet
  orthogonal basis are computed with a fast
  algorithm that cascades discrete convolutions
  with h and g, and subsamples the output
• Fast orthogonal WT

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Fast transform

• A fast WT decomposes successively each
  approximation PVjf into a coarser approximation
  PVj1f plus the detail coefficients carried by
  PWj1f.
• Since ?j,nn?Z is an orthonormal basis of Vj

15
Fast transform

• Theorem 7.6 (Mallat) At the decomposition
• At the reconstruction

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Proof decomposition
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Proof decomposition

• Coming back to the projection coefficients
• Similarly, one can prove the relations for both
  the details and the reconstruction formula

18
Proof Reconstruction
but
(see (3) and the analogous one for g)
thus
Taking the scalar product with f at both sides
CVD
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Summary
Analysis or decomposition
Synthesis or reconstruction
Teorem 7.2 (MallatMeyer) and Theorem 7.3
MallatMeyer
The fast orthogonal WT is implemented by a
filterbank that is completely specified by the
filter h, which is a CMF The filters are the same
for every j
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Perfect reconstruction FB

• Dual perspective given a filterbank consisting
  of 4 filters, we derive the perfect
  reconstruction conditions
• Goal determine the conditions on the filters
  ensuring that

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Perfect Reconstruction FB
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Perfect Reconstruction conditions

• Putting all together
• Theorem 7.7 (Vetterli) The FB performs an exact
  reconstruction for any input signal iif

1
0
(alias-free)
Matrix notations
(alias free)
23
PR filterbanks

• Theorem 7.8. Perfect reconstruction filters also
  satisfy
• Furthermore, if the filters have a finite
  impulse response there exists a in R and l in Z
  such that
• Conjugate Mirror Filters

a1, l0
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CMF
_
h
h
?2
?2
a0
a0

_
g
g
?2
?2
Taking as reference
(which amounts to choosing the analysis low-pass
filter) the following relations hold for an
orthogonal filter bank
synthesys low-pass (interpolation) filter
reverse the order of the coefficients
negate every other term
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CMF property
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Daubechies filters

• CMF, FIR, orthogonal, compactly supported, real
  causal filters hn
• Asymmetric
• other families build to be more symmetric are
  called symmlets which are almost linear phase
• Related theorems
• Proposition 7.2 Mallats book Vanishing
  moments.
• Let ? be a wavelet that generates an orthonormal
  basis. If ?(?) is p times continuously
  differentiable at ?0, the three following
  statements are equivalent
• The wavelet has p vanishing moments
• ?(?) and its first (p-1) derivatives are zero at
  ?0
• h(?) and its first (p-1) derivatives are zero at
  ??
• Theorem 7.4 Mallats book (Daubechies)
• A real conjugate mirror filter h, such that h(?)
  has p zeros at ??, has at least 2p non-zero
  coefficients. Daubechies filters have 2p
  non-zero coefficients

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Analysis filters (db3)
(The zero frequency is at the center of the
horizontal axis)
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Synthesis filters (db3)
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Daubechies
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Biorthogonal wavelet bases
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Biorthogonal bases

• Orthonormal basis
• enn?N basis of Hilbert space
• Ortogonality condition lt en, epgt0 ?n?p
• ?y ? H,
• There exists a sequence
• en21 ortho-normal basis

• Bi-orthogonal basis
• enn?N linearly independent
• ?y ? H, ?Agt0 and Bgt0
• Biorthogonality condition
• AB1 ? orthogonal basis

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Biorthogonal bases
Though, some other conditions must be imposed to
guarantee that ? and ? are FT of finite energy
functions. The theorem from Cohen, Daubechies and
Feaveau provide sufficient conditions
33
Biorthogonal filter banks

• A 2-channel multirate filter bank convolves a
  signal a0 with
• a low pass filter
• and a high pass filter
• and sub-samples the output by 2
• A reconstructed signal ã0 is obtained by
  filtering the zero-expanded signals with a dual
  low-pass and high pass filter
• Imposing the PR condition (output signalinput
  signal) one gets the relations that the different
  filters must satisfy (Theorem 7.7)

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PR in biorthogonal FB

• Theorem 7.7 Vetterli
• The filter bank performs an exact reconstruction
  for any input if and only if
• This can be written in matrix form
• When all the filters are FIR, the determinant
  can be evaluated, which yields simpler relations
  between the decomposition and the reconstruction
  filters.

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Finite Impulse Response (FIR)

• Theorem 7.8
• Perfect reconstruction filters satisfy
• (1)
• If the filters have a FIR than there exists a?R
  and l?Z such that
• (2)
• (1) comes from the fact that the system equation
  giving the synthesis filters provide a stable
  solution for FIR filters (determinant ?0)
• (2) a is a gain factor and l is a reverse shift
  usually a1, l0
• Signal domain

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Proof

• Given h and and setting a1 and l0 in (2)
  the remaining filters are given by the following
  relations
• The filters h and are related to the scaling
  functions ? and ? via the corresponding
  two-scale relations, as was the case for the
  orthogonal filters (see eq. 1).
• Switching to the z-domain
• Signal domain

(3)
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Summary of Biorthogonality relations

• An infinite cascade of PR filter banks
  yields two scaling functions and two
  wavelets whose Fourier transform satisfies

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Properties of biorthogonal filters

• Imposing the zero average condition to ? in
  equations (iii) and (iv)

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Construction of BWB

• Assuming that h and are FIR, one can prove
  that
• are the FT of distributions with compact
  support. Some other conditions must then be
  imposed to ensure that they are the FT of some
  finite energy functions (see Theorem 7.10 Cohen,
  Daubechies, Feauveau).

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Fast BWT

• Two different sets of basis functions are used
  for analysis and synthesis
• PR filterbank

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Be careful with notations!

• In the simplified notation where
• hn is the analysis low pass filter and gn is
  the analysis high pass filter, as it is the case
  in most of the literature
• the delay factor is not made explicit
• The relations among the filters modify as follows

a0
The high pass filters are obtained by the low
pass filters by negating the odd terms
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Orthogonal vs biorthogonal PRFB
Biorthogonal PRFB
Orthogonal PRFB
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Biorthogonal bases

• If the decomposition and reconstruction filters
  are different, the resulting bases is
  non-orthogonal
• The cascade of J levels is equivalent to a signal
  decomposition over a non-orthogonal bases
• The dual bases is needed for reconstruction

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Biorthogonal bases

• An infinite cascade of PR filter banks (h,g),
  (h,g) yields two scaling functions and two
  wavelets whose Fourier transform satisfy

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Example bior3.5
h -0.0138 0.0414 0.0525 -0.2679
-0.0718 0.9667 0.9667 -0.0718 -0.2679
0.0525 0.0414 -0.0138 g -0.1768 0.5303
-0.5303 0.1768 h 0.1768 0.5303
0.5303 0.1768 g -0.0138 -0.0414 0.0525
0.2679 -0.0718 -0.9667 0.9667 0.0718
-0.2679 -0.0525 0.0414 0.0138
Lo_D
Lo_R
Hi_D
Hi_R
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Example bior3.5
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Example bior3.5
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Biorthogonal bases
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Biorthogonal bases
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Summary

• PR filter banks decompose the signals in a basis
  of l2(?). This basis is orthogonal for Conjugate
  Mirror Filters (CMF).
• SmithBarnwell,1984 Necessary and sufficient
  condition for PR orthogonal FIR filter banks,
  called CMFs
• Imposing that the decomposition filter h is equal
  to the reconstruction filter h, eq. (1) becomes
• Correspondingly

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Separable bases
52
Separable bases

• To any wavelet orthonormal basis one can
  associate a separable wavelet orthonormal basis
  of L2(R2)
• Separable multiresolutions lead to another
  construction of separable wavelet bases whose
  elements are products of functions dilated at the
  same scale.
• Separable multiresolution
• is the space of finite energy functions f(x,y)
  that are linear expansions of separable functions

53
Separable bases

• It is possible to prove (Theorem A.3) that
• is an orthonormal basis of V2j.
• A 2D wavelet basis is constructed with separable
  products of a scaling function and a wavelet

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Separable bases

• Theorem 7.24
• Let ? be a scaling function and ? be the
  corresponding wavelet generating an orthonormal
  basis of L2(R). We define three wavelets
• and denote for 1ltklt3
• The wavelet family
• is an orthonormal basis of W2j and
• is an orthonormal basis of L2(R2)
• On the same line, one can define biorthogonal 2D
  bases.

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Fast 2D Wavelet Transform
Approximation at scale j
Details at scale j
Wavelet representation
Analysis
Synthesis
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Bi-dimensional wavelets
57
Fast DWT for images
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Fast DWT for images
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Example
V
60
Example

h
?2

h
?2

g
?2
H

h
?2


h
?2
h
?2


g
?2
g
?2

g
?2

h
?2

g
?2

g
?2
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Subband structure for images
cD1(h)
cD2(h)
cA2
cD2(d)
cD2(v)
cD1(v)
cD1(d)
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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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Overcomplete bases
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Translation Covariance
Translation covariance
If translation covariance does not hold
Signal
Translation
DWT
Wavelet coefficients
?
Signal
DWT
Translation
Wavelet coefficients
NOT good for signal analysis
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Translation covariance

• The signal descriptors should be covariant with
  translations
• Continuous WT and windowed FT are translation
  covariant. Uniformly sampling the translation
  parameter destroys covariance
• Translation invariant representations can be
  obtained by sampling the scale parameter s but
  not the translation parameter u

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Dyadic Wavelet Transform

• Sampling scheme
• Dyadic scale
• 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 a trous

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Algorithme a trous

• Similiar to a fast biorthogonal WT without
  subsampling
• Fast dyadic transform
• The samples of the discrete signal a0n are
  considered as averages of some function weighted
  by some scaling kernels ?(t-n)
• For any filter xn, we denote by xjn the
  filters obtained by inserting 2j-1 zeros between
  each sample of xn ? create holes (trous, in
  French)

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Algorithme a 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

71
Analysis
aj1
aj2
hj
hj1
aj
gj1
gj
dj2
dj1
2j-1trous
No subsampling!!
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Synthesis
aj1
aj
hj1
aj2
hj


gj1
dj2
gj
dj1
Overcomplete wavelet representation aJ,
dj1?j?J
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Algorithme a trous
d1f
g(z)
d2f
g(z2)
d3f
a1f
g(z4)
s(z)
h(z)
a2f
h(z2)
a3f
h(z4)
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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
• (3J1)NxNy
• Feature extraction

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Discrete WT vs Dyadic WT
DWT
Original
Dyadic WT
LL
HL
LH
HH
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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 Agt0 and B such that

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Oriented wavelets

• Then, there exists a reconstruction wavelet
  family such that
• Gabor wavelets
• Dyadic Frames of Directional Wavelets
  Vandergheynst 2000
• Curvelets DonohoCandes 1995
• Steerable pyramid Simoncelli-95
• Contourlets DoVetterli 2002

?y
?x
80
Dyadic Frames of Directional Wavelets
Pierre Vandergheynst (ITS-EPFL)
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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 2 pi /K
• Recipe
• Build a family of isotropic wavelets such that
  the Fourier transform of the mother wavelet
  expresses in polar coordinates is separable
• Split each isotropic wavelet in a set of oriented
  wavelets by an angular window
• Express the angular part T(?) as a sum of window
  functions centered at ?k

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Partitions of the F-domain
2 pi
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Building DDWF
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Dyadic Directional Wavelet Frames
4 orientations (K4)
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Dyadic Directional Wavelet Frames
Orientation
Scaling function
Scale

• Properties
• Overcomplete
• Translation covariance
• Rotation covariance for given angles

DDWF
Feature Images or Neural images
87
Contourlets

• Brief overview

Minh Do, CM University
Martin Vetterli, LCAV-EPFL
88
Contourlets

• Goal
• Design an efficient linear expansion for 2D
  signals, which are smooth away from
  discontinuities across smooth curves
• Efficiency means Sparseness
• DoVetterli Piecewise smooth images with smooth
  contours
• Inspired to curvelets DonohoCandes

89
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 -gt more
  concise (sparse) representation

c2-j/2
2-j
2-j
2-j
wavelets
curvelets
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Curvelets

• Double filterbank approach pyramidal directional
  filterbank

Laplacian pyramid
Directional filterbank
First, a standard multiscale decomposition is
computed, where the low-pass channel is
sub-sampled while the high-pass channel is not.
Then, a directional decomposition with a DFB is
applied to each high-pass channel. The number of
directions increases with frequency.
91
Laplacian Pyramid
residuals
prediction
Low-pass
Interpolation
?2
?2
-
residual
coarser version
92
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 plane. 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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Overcomplete bases

• Discrete Wavelet Transform
• Non-redundant
• Signal il subsampled
• Not translation invariant
• Total number of coefficients
• NxNy
• Compression

• Overcomplete representations
• DWF, DDWF
• Redundant (in general)
• Signal is not subsapled
• Filters are upsampled
• Translation invariant
• Total number of coefficients
• (MxJ1)NxNy
• Feature extraction

95
Summary of useful relations

• If f is real

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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 -gt isotropic wavelets
• Contours and geometry -gt Curvelets
• Adaptive basis
• More flexible tool for image representation
• Could be related to the RF of highly specialized
  neurons
• Next step Vision Statistics of the visual
  environment
• Inferring models for early vision processes
  (deriving RF shapes)
• Designing (generative) models for images
• Textures

97
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)