Image Processing with Complex Daubechies Wavelets.
Analyses based on Symmetric Daubechies Wavelets (SDW) lead to complex-valued multiresolution representations of real signals. After a recall of the construction of the SDW, we present some specific properties of these new types of Daubechies wavelets. We then discuss two applications in image processing: enhancement and restoration. In both cases, the efficiency of this multiscale representation relies on the information encoded in the phase of the complex wavelet coefficients.
INTRODUCTION
Many current investigations in mathematical imaging consist in finding the optimal representation to
perform specific enhancements by extracting the relevant information contained in an empirical signal. This
question of representation, present in many fields of applied mathematics and physics, is indeed the cornerstone
of the pionnering work of D. Marr in vision 1 . The "primal sketch" of an image he proposed was
based on the multiscale edge representation obtained through the action of some operators of different sizes.
More specifically, Marr and Hildreth 2 argued that the convolution of the image with the filter associated
with the Laplacian of the two dimensional Gaussian at different scales constitutes the most satisfactory
representation. Such a representation identifies edges with the zero-crossings of the filtered image. Physiological
experiments 1 gave credit to this model of vision generally known as the "Marr conjecture". In a more
computational approach to edge detection proposed by Canny 3 , edges are located at the local extrema of
the convolutions of the image with the directional first order derivatives of some smoothing function (e.g. a
Gaussian kernel). Unlike the zero-crossing technique, the Canny's edge detector characterizes the strength
of the discontinuities in the image intensity. The synthesis of those two approaches is due to Mallat and
Zhong 4;5 who demonstrated that zero-crossings and local extrema are unified in a single mathematical frame-
work: the wavelet theory of frames. They further established an iterative algorithm to restore the original
signal from these sparse representations. The aim of the present work is to show that similar ideas can
be developped with orthogonal multiresolution bases with compact support provided we use the symmetric
Daubechies wavelets 6;7 .
Being complex-valued, the symmetric Daubechies multiresolution analyses did not receive much attention
from the signal processing community since the resulting representation of a real field, such as an image, is
a redundant expansion with complex-valued coefficients. Needless to say, this is also true with the Fourier
transform; however, the reality condition is rather trivial and establishes a simple identity between the
Fourier modes. The present work investigates the same kind of relationship between the complex wavelet
coefficients: on the basis of comparison with the standard real Daubechies analyses, we show that the "natural
e-mail: lina@crm.umontreal.ca
redundancy" given by the complex Daubechies analyses of a real field provides a "dual representation"
combining zero-crossings and local extrema. In fact, this result relies on the existence of "hidden" differential
operators underlying the structure of some complex Symmetric Daubechies Wavelets (SDW). This result has
interesting potential applications in numerical simulations 8 and signal processing 9;10 .
Pursuing the analogy with Fourier representation, we can also wonder where the essential information
of a signal is located in the complex wavelet coefficients. In the context of Fourier analysis, this point has
been discussed by many authors who investigated the "importance of the phase in signals" 11;12;13 . In the
same way, most of the information is encoded in the phase of the wavelet coefficients and we describe a
"phase-only wavelet synthesis" algorithm that restore most of the signal from a "shrunk representation" as
suggested by many authors 14;15 .
The paper is organized as follows. Section 2 briefly describes the basics of the multiresolution analyses
and the SDW solutions following the standard factorisation approach. More details and results can be
found in Ref.[7,10]. In Section 3, the SDW are discussed in terms of parametrized filter banks. In Section
4, we describe the differential operators underlying some complex symmetric Daubechies multiresolution
bases. The consequences in two dimensions and application to edge enhancements are discussed in Section
5. Denoising and "phase reconstruction" are finally presented in Section 6. Conclusion will follow.
Notation: The derivatives and partial derivatives are denoted by a superscript as in Eq.(14) and by @ m
respectively. Complex conjugation of z is written z.
2. SYMMETRIC DAUBECHIES WAVELETS
A multiresolution analysis of L 2 (IR) is a sequence of closed subspaces V j ae L 2 (IR) such that
A scaling function ' 2 V 0 with unit integral exists such that f' 0;k is an orthonormal
basis of V 0 and, consequently, the set of functions
is an orthonormal basis of the space V j . Since a sequence of complex-valued coefficients a k
exists such that
a
Multiresolution aims to decompose L 2 (IR) as
where the spaces W j are defined as the orthogonal complement of V j in V . For a given
scale j, the space W j is generated by the set of orthonormal wavelets / j
associated with the
multiresolution analysis. Since / 2 W a sequence of complex-valued coefficients b k exists such that:
The choice b ensures that the set f/ j is an orthonormal basis of W j .
In general, a field with finite energy will be "empirically known" at some scale and approximated in some
c jmax
The discrete multiresolution analysis of f consists in the computation of the coefficients of the expansion
c j0
is a given low resolution scale. The coefficients in the previous expansion are computed through
the orthogonal projection of the field over the multiresolution basis:
Using the previous definition, we have the well-known fast wavelet decomposition algorithm composed with
the low-pass projection and the high-pass projection
- a k\Gamma2n c j
Conversely, any elements of and of W j \Gamma1 can combine to give a unique vector in V j ; this reconstruction
is expressed by the inverse fast wavelet transform:
The Symmetric Daubechies Wavelets are subject to the following constraints:
(i) Compactness of the support of ': We require that ' ( and consequently /) has a compact support inside
the interval [\GammaJ; J + 1] for some integer J that is, a k 6= 0 for
(ii) Orthogonality of the '(x \Gamma k): This condition defines in a large sense the Daubechies wavelets. Defining
the polynomial
an z n ; with F
where z is on the unit circle, the orthonormality of the set f' 0;k (x); k 2 ZZg can be stated through
the following identity
where the polynomial P (z) is defined as
(iii) Accuracy of the approximation (6): To maximize the regularity of the functions generated by the scaling
function ', we require the vanishing of the first J moments of the wavelet or, in terms of the polynomial
(11),
The usual Daubechies wavelets 16 are the real polynomial solutions of Eqs.(12) and (14). They differ from
each other through their degree of symmetry and, in general, the standard Daubechies wavelet (DAUBn) is
the "least asymmetric" solution being the length of the filters).
(iv) Symmetry: This condition on the filter,i.e a can be written as
As anticipated by Lawton 17 , only complex-valued solutions of ' and /, under the four above constraints, can
exist, and for J even, only. The first solutions (from were described in Ref.[7] by using the
parametrized solutions of Eqs. (12), (14) and (15). The solutions have also been investigated in the spirit of
the original Daubechies approach, i.e. by inspection of the roots of some so-called "valid polynomial" that
satisfies Eq.(12). Such a polynomial is defined by
where
Straightforward algebra shows that P J (z) does satisfy Eq.(12).
The 2J roots of p J (z) display obvious symmetries: the conjugate and the inverse of a root are also roots;
furthermore, no root is of unit modulus. If we denote by x k=1;2;:::;J the roots inside the unit circle (jx
and -
Y
Y
and the low-pass filter F (z) can be written as:
Y
The polynomial p(z) is defined through some particular roots of p J (z); R, R 0 are two arbitrary subsets of
Jg. The spectral factorization of P J (z), i.e. P J
and leads to the following constraint on R and
This selection of roots fulfills the conditions (i),(ii) and (iii). In her seminal work 16 , Daubechies investigates
all the real solutions for a k derived from the condition (20). This amounts to restrict the definition (20) and
to consider the rule
For instance, corresponds to the DAUBn solution with 2. The
symmetry condition (iv) defines an other subset of solutions of Eq.(20). It corresponds to the following rule
of selection for the roots
We first notice that "reality" (rule (21)) and symmetry (rule (22)) are incompatible: all the symmetric
Daubechies wavelets are complex valued. For any even value of J , Eq.(22) defines a subset of 2 J
solutions in
the original set of "orthogonal compactly supported regular wavelets" (2 J elements, complex or real). Notice
that a complex conjugate of a scaling function is also a scaling function.
Figure

1: The complex scaling function ' for J=2. Right: The complex wavelet / for J=2. (Imaginary part
in dashed line).
The complex scaling function and wavelet will be written as
where h, g, w and v are all real functions. In the present work, we will consider a particular family of
solutions, the so-called SDWJ Daubechies wavelets, that correspond to the following selection of roots:
that clearly satisfies the constraints (22). For 2, the unique solution (SDW2) is shown on Fig.1. The
two symmetric solutions for are shown on Fig.2 and Fig.3; the SDW4 solution corresponding to the
definition (24) is shown on Fig. 2. Finalyy, Figs. 4 display the four real functions h, g, w and v of SDW6.
The filter coefficients
a k for SDW2, SDW4 and SDW6 are:
6 0:000000 0:000000 0:003238 \Gamma 0:001300i
To conclude this section, we recall that the bi-dimensional multiresolution analysis is built from the tensor
product of two multiresolution spaces V i . We thus have one scaling function '(x)'(y) complemented with
three wavelets /(x)'(y); '(x)/(y) and /(x)/(y). We will write these complex-valued basis functions as
and expansions like (6) and (7) now generalize in two dimensions as

Figure

2: SDW4: The complex scaling function ' (left) and the complex wavelet / (Imaginary part in
dashed line).
-0.4
-0.4

Figure

3: Complex Daubechies scaling function ' (left) and wavelet / (right) for J=4. (Imaginary part in
dashed line).
m;n
c jmax
k1 ;k2
c j0
k1 ;k2 \Phi j0 ;k1 ;k2 (x; y) +X
k1 ;k2
where \Phi j;k1 ;k2 (x;
span the spaces
respectively. In the sequel, we denote by WN the wavelet transform
m;n g WN
\Gamma! fc jmax \GammaN
Figure

4:
-0.4
-0.4

Figure

5:
3. PARAMETRIZATION OF THE SYMMETRIC DAUBECHIES FILTER BANKS
A few years ago, D. Pollen studied the correspondance between the orthonormal multiresolution analysis
(with compact supported scaling function) and the element of the group of the two by two matrices of the
where z is still a complex variable on the unit circle and u(z), v(z) are polynomials in j
the unit vector
p' z
Pollen has shown that
solves the orthogonality condition expressed by Eqs.(12) and (13) if and only if G(z) is unitary with
i.e. G(z) element of the group SU I (2; j
Such element can be factorized in a unique way:
where G - (z) is an element of SU I (2; j
labelled with a complex parameter -:
For the expansion of F (z) defined as Eq.(33) leads to the following four parametrized scaling coefficient

a
For we recognize the solution already mentioned in the early works of Daubechies 16 . We further
notice that symmetry of the filter (37) is only possible for purely imaginary complex parameter -. In fact,
this observation can be generalized 7 : the scaling coefficients of any symmetric orthogonal multiresolution
analysis can be computed from the definition (34), the factorization (35) and a set of complex parameters
us emphasize that, at this point, no vanishing moments of the
wavelet have been requested. For a given value of J , the vanishing moments conditions expressed by Eq.(14)
define a finite set of parameters fr Jg. As studied in Ref.[7], this set which is empty when J is
odd, contains 2 J
solutions for J even. For instance, the solution displayed in the previous
section is given by -
q5 .
4. SOME PROPERTIES OF THE SYMMETRIC DAUBECHIES WAVELETS
All the Symmetric Daubechies Wavelets share the usual properties of the standard real Daubechies bases.
In this section, we show that SDW have an unexpected analytic structure that relates real and complex part
of the basis functions 10 .
Using the Fourier representation of real functions g(x), h(x), v(x) and w(x) that define '(x) and /(x), it
can be shown that, for the first values of J (say J ! 12), the complex scaling function and the wavelet of
the SDWJ multiresolution analysis can be written as
Those identities are verified on the frequency domain defined by the sampling rate of the analysed signal.
In other words, those two identites are verified in the interval [0; -] (with a sampling step renormalized to
unity) when they are written in the Fourier representation.
For J ? 10, higher derivative terms in g(x) become non negligible. Of course, the two real functions h(x)
and w(x) are the genuine scaling function and the true wavelet only in the cases of the real Daubechies
multiresolution analyses In the complex case, h(x) (the real part of the scaling function)
and w(x) (the real part of the wavelet) are still endowed with interesting properties. Indeed, h(x) is a good
interpolating function since we can easily verify that
Z
Z
Let us recall that, in the early age of the Daubechies wavelets, the introduction of vanishing moments for
the scaling function led to the construction of the well-known coiflets. Let us mention that w(x), that is
an "admissible wavelet" with J vanishing moments, can be well approximated with @ J+1
x h(x) up to some
multiplicative factor that can be computed. The parameters ff; fi and - in (38) can be also computed directly
from the filters coefficients a k by using the first non vanishing momentum of '(x) and /(x). Denoting by
Z
Z
we have
an and
and
l i\Gammaj
Straighforward integrations by part lead to
The numerical values found for are listed in the following table. They all agree with the
values estimated from the Fourier representation of the identities (38).
J=2 J=4 J=6
ff \Gamma0:164 \Gamma0:089 \Gamma0:076
In most applications, the signal to be analyzed is real valued. In the present work and Ref.[10], the complex
wavelet representation provides a redundant description of the signal. The previous result helps in interpreting
this redundancy since, using the Taylor expansion of a one dimensional field, we can estimate the real

Figure

5: Original picture: "lady256"
and imaginary parts of the coefficients c j
k in (7) as
with x
5. A SHARPENING ENHANCEMENT ALGORITHM
Let us now turn to the bi-dimensional case. The real and imaginary parts of the scaling function (25) are
where G(x; y) denotes the real smoothing kernel h(x)h(y). We see that in one hand the real part of the 2-d
scaling function is close (because ff 2 !! 1) to the smoothing kernel G(x; y) while, on the other hand, the
imaginary part is proportional to the Laplacian of G(x; y): \Psi(x; y) is thus the "Marr wavelet" associated
with \Theta(x; y) ' G(x; y).
The simultaneous presence of a smoothing kernel and its Laplacian in the complex scaling function can be
exploited to define some elementary operations on the wavelet coefficients. Since the real and imaginary
parts of the wavelet transform coefficients of some real image correspond to the convolution of the original
field with the real part and the imaginary part respectively of \Phi j;m;n (x; y), we then have access to the
(multiscaled) smoothed Laplacian of the image.
Let us consider an image, i.e a real matrix I m;n , and the bi-dimensional (complex valued) field f(x; y) defined
by the expansion (29) with c jmax
The field f(x; y) is a particular point in the space R jmax defined as the set of all fields constructed with
real scaling coefficients at scale j max . We denote by PR the orthogonal projector on this space; it is simply
defined by keeping the real part of the scaling coefficients only:
PR
m;n
m;n )\Phi jmax;m;n (x; y)
m;n
m;n \Phi jmax;m;n (x; y) (46)
Let us now define a smoothing operator Z: R jmax ! R jmax+1 by the inverse wavelet transform with the
real part of the SDW filters only. The image Zf is twice larger than the original image, without spurious
high-frequency components. Then, we can verify that the complex wavelet decomposition W 1 Zf gives, in
the imaginary part of the scaling coefficients, a good estimate of the Laplacian of I m;n . In other words,
is a good approximation of a real field in R jmax+1 .
For the sake of illustration, we consider the image displayed on Figure 5. Figure 6 shows the upper left
corner of the image resulting from the application of Z, and Figure 7 displays the imaginary part of the
scaling coefficients of W 1 Zf . As expected, the zero-crossings in this picture correspond to the edges in the
original image.
Let us now consider a N-level decomposition of Zf . The real and imaginary parts of the coefficients c l
k1 ;k2
can be written as (  denotes the usual convolution operator)
h l
k1
k1
l
k1
k1
To illustrate this result, we consider the sharpening operator, f ! ~
ae4f . It can be implemented at
the finest resolution scale through (we normalize the initial data such that j
c jmax
followed by the projection of this enhanced complex field on R
~
m;n
m;n \Phi jmax;m;n (x; y)
m;n
m;n \Phi jmax;m;n (x; y) (49)
This algorithm can easily be generalized to obtain a non-linear multiscale enhancement by considering a N
level decomposition of Zf and by applying the adapted enhancement
at any coarse scale with different sharpening parameter ae (ae ! ae j ). The coefficients c j
m;n are induced by the
synthesis process, whereas the coefficients g j
m;n are calculated during the decomposition process.
The

Figure

8 shows an example of such processing with SDW2 and In comparison with the original
image (Fig.4), the local contrast has been improved significantly. Other tests of this algorithm have been
performed on more challenging images, i.e. low-resolution and low-contrast mammographic images 9 .
Let us mention that other efficient multiscale sharpening transformations have been proposed in the recent
past 19;20 . The main difference here in the present work is the orthogonality property of the SDW transform.
We recall that the SDW bases are not derived from a representation that allows specific enhancements. On
the contrary, we have shown that the Laplacian was inherent to this particular orthogonal basis.
6. SHRINKAGE AND PHASE
Other important non-linear transformations in image processing are regression modeling and denoising.
Mostly advocated by Donoho and Johnstone 15 , the wavelet regression estimators, based on orthogonal multiresolution
bases and the shrinkage of the empirical wavelet coefficients, provide near-optimal estimates
of the true signal. Apart from the choice of the basis, the wavelet regression technique suggested by the
authors of the Ref.[15] relies on the definition of the thresholding rule (hard or soft) and the selection of
the appropriate threshold (optimal or universal). Based on the amplitude of the wavelet coefficients, this
regression estimator can be directly implemented on the complex coefficients.
Let us define first a general shrinkage operation. From the empirical expansion
m;n
I m;n \Phi jmax;m;n (x; y) (51)
we consider the field
~
m;n
~
I m;n \Phi jmax;m;n (x; y) (52)
with
~
where the shrinkage operator T ,
c jmax \GammaN
is defined by
This operator only modifies the amplitude of the wavelet coefficients; the phases are preserved.
Given the complex wavelet coefficients fc jmax \GammaN
associated with f 0 , we define
the space of all the fields f(x; y) such that
k1 ;k2
c jmax \GammaN
k1 ;k2 \Phi l;k 1 ;k2 (x; y) +X
k1 ;k2
with
We call this space the isophase space \Gamma jmax . Any shrinkage operator T maps f 0 towards some point in \Gamma jmax .
We denote by P \Gamma the orthogonal projector onto this space. This operator is simply given by the orthogonal
projections (i.e. the closest point) of each complex wavelet components f i
j;m;n on the half-line defined by the
phase Arg(d i
Notice that this projection can naturally "kill" a coefficient for which the closest point
on the half-line (defined by the phase) is the origin in the complex plane.
Of course, the initial empirical field f 0 and the "shrunk field" ~
(whatever the shrinkage operator) belong
to \Gamma jmax . The isophase space and the space R jmax are both convex spaces and the intersection
contains, at least, the field f 0 (see Fig. 9).
Let us now consider the sequence of real images I l
m;n defined by
m;n
I l
m;n \Phi jmax;m;n (x;
PR ~
Well-known theorems 21 prove the convergence of this sequence to a point in \Gamma jmax " R jmax , as l !1.
As a first example of shrinkage, we take the functions j in Eq.(55). This shrinkage is nothing
but the usual projection (denoted by P VN ) of f onto the approximation space V jmax \GammaN . A
typical example of such projection is shown on Figure 10. Starting from this "shrunk representation" of the
original image and using the isophase space associated with the original image (see Figure 5), the iterative
projections on the convex \Gamma jmax and R jmax lead to I
m;n (Fig.
m;n (Fig. 10). As well illustrated
in the present example, the "wavelet phase reconstruction algorithm" restores the edges and small features
blurred by the shrinkage. During the alternate projections, the percentage of phase effectively used to restore
the non-vanishing amplitudes of the wavelet coefficients, gradually increases from 50% (first iteration) to
75% (30 iterations) and then 90% (1000 iterations and more).
The choice of the SDW wavelet and the definition of the shrinkage (i.e. ~
apparently have no effect on the
limit; however, they obviously affect the speed of convergence.
The complete characterization of \Gamma jmax " R jmax is still an open problem. However, in one dimension, some
prior investigations tend to demonstrate the uniqueness of f 0 in this space. Let us sketch an heuristic proof.
We first observe that the complex conjugate of the wavelet, -
/, has a non-vanishing projection on V 0 and on
all the "details spaces" W j-0
The coefficients fi vanishing for
k can be acurately computed through
a cascade algorithm. Any function in \Gamma jmax " R jmax can be written as f 0
with
\Gamma1. Assuming that f 0 mostly a real-valued field for any small values of
\Gammaj;k , the reality condition leads to an over-determined system of constraints on the j's
since the expansion in Eq.(60) is only in terms of the orthogonal wavelets at a finite number of different
scales: the reality condition together with the expansion of -
severe constraints on ffi(x) and
the unique solution. More carefull analysis of this statement is presently under investigation.
An other typical example of shrinkage is the soft-thresholding function used in wavelet-regression:
where I(x) is the indicator function. The thresholds t i are the most important parameters of this regression
method. The universal thresholds, that lead to a noise-free estimate, are defined as t
being the size of the image and oe i the noise variance directly estimated from the modulus of the wavelet
coefficients d i
jmax \Gamma1;m;n . A simple estimation is given 15 by -
In most applications where noise is important (see Figure 13), the usual wavelet regression techniques tend
to underfit the true signal by shrinking too many coefficients 22 . The same observation is made with the SDW
when we consider the real image produced by the scaling coefficients of PR ~
Fig. 14). However, in many
experiments we noticed that local artefacts in the universal estimate were in correspondence with transients
in the imaginary part of the scaling coefficients of ~
f 0 . This observation leads us to consider the improvement
of the universal estimate by using the information contained in the phase of the wavelet coefficients and left
untouched by the shrinkage operation.

Figure

15 and Fig.16 display the resulting image after 30 and 1000 iterations of alternate projections re-
spectively. Here again, the restoration of the amplitudes of the wavelet coefficients is such that coherent
structures of the image are restored firstly. Noise is also induced during the projections but, quite remark-
ably, major part of the noise introduced with the phase (we use the isophase space of the noisy image in
is immediately pumped out with the elimination of imaginary part of the scaling coefficient (in PR ).
One-dimensional simulations have also shown that it was possible to improve the "wavelet phase-only re-
construction" algorithm by defining an isophase space from the statistical results of Donoho and Johnstone.
This amounts to interpret somewhat differently their results: the threshold being chosen (i.e defined from
the a priori knowledge of the noise content in the signal), the wavelet coefficients of modulus less than the
threshold are set to zero which means that the corresponding phases are undefined. In other words, no
relevant information for describing the true signal is encoded in those phases. However, the phase of the
remaining coefficients of modulus greater than the threshold are meaningful and we claim that those phases
and those phases only contain all the available informations for describing the true signal. This subset of
phases defines a new isophase space ~
simulations have shown that using this new
isophase space in the alternate projections (58) gives an effective shrinkage of the empirical wavelet coefficients
similar to a compromise between the "hard "and "soft" thresholdings. This "phase reconstruction
algorithm" improves significantly the initial estimate by adapting the thresholding rule to the coherence of
the signal.In this "holographic" interpretation of the shrinkage techniques, the wavelet coefficient amplitude
doest not carry pure information about the true signal but rather weights the information contained in the
phase of the coefficient.
7. CONCLUSION
After a review of the construction of the Symmetric Daubechies Wavelets, we have described a simple
multiresolution processing algorithm based on the inherent Laplacian kernel in this type of orthogonal bases.
Beside this operation which acts on the scaling coefficients only, we have discussed the importance of the
phase in the wavelet coefficients. Most information of a real signal is indeed encoded in this phase of the
wavelet coefficients. We note that, despite the orthogonality of the present wavelet bases, the redundancy of
the complex representation of real signals is utilized. In fact, both algorithms (enhancement and restoration)
emphasize the role of the phases in this type of complex multiresolution decompositions.
The phase representation of images is quite appealing. It is worth mentionning that experiments in physiology
also demonstrate the crucial role of both a redundant multiresolution signal processing 23;24 and the phase
information 25;26 in the Human Vision System. Quite remarkably, the symmetry condition, which is required
in image analysis for very practical reasons, leads to multiresolution representations that seem to be of main
relevance in Nature 27 . Far from being an immediate model for the visual perception, the complex symmetric
Daubechies wavelets suggest,at least, new directions of investigation in image processing like, for instance,
the use of complex convolutions in layered neural networks for pattern recognition.
8.

ACKNOWLEDGMENTS

The author wishes to thank L. Gagnon and M. Mayrand for their precious collaboration, and St'ephane
Mallat for discussions about the "phase reconstruction" algorithm. This work was supported in part by the
Natural Sciences and Engineering Research Council (NSERC) of Canada.
8.



--R


"Theory of edge detection"
"A computational approach to edge detection"
"Zero-crossings of a wavelet transform"
"Characterization of signals from multiscale edges"
"Applications of Complex Valued Wavelet Transforms to Subband Decomposition"
"Complex Daubechies Wavelets"
"Symmetric Daubechies wavelet and numerical solution of NLS equations"
"Complex linear phase filters for efficient image coding"
"From Daubechies to Marr"
"The importance of phase in signals"
"The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform"
"Signal restoration from phase by projections onto convex sets"
"Fast wavelet techniques for near-optimal image processing"
"Adapting to unknown smoothness via wavelet shrinkage"
"Orthonormal bases of compactly supported wavelets"
Ten lectures on Wavelets
"SU I (2; F [z; 1=z]) for F a subfield of C"
"Mammographic Feature Enhancement by Multiscale Analysis"
"Image restoration by the method of convex projections"
"Wavelet regression by cross-validation"
"Uncertainty relation for resolution in space, spacial frequency, and orientation optimized by two-dimentional visual cortical filters"
"Multifrequency channel decomposition of images and wavelet models"
"Phase relationships between adjacent simple cells in the visual cortex"
"Visual signal detection. I. Ability to use phase information"
"Feature detection in human vision: a phase-dependent energy model"
--TR
A computational approach to edge detection
Ten lectures on wavelets
Characterization of Signals from Multiscale Edges

--CTR
R. G. Stockwell, A basis for efficient representation of the S-transform, Digital Signal Processing, v.17 n.1, p.371-393, January, 2007
Diego Clonda , J.-M. Lina , Bernard Goulard, Complex Daubechies wavelets: properties and statistical image modelling, Signal Processing, v.84 n.1, p.1-23, January 2004
daubechies wavelets;complex signals;restoration;image processing