Abstract: Wavelet shrinkage methods that use complex-valued wavelets provide additional
insights to shrinkage process compared to standardly used real-valued
wavelets. Typically, a location-type statistical model with an additive noise is
posed on the observed wavelet coefficients and the true signal/image part is estimated
as the location parameter. Under such approach the wavelet shrinkage
becomes equivalent to a location estimation in the wavelet domain. The most
popular type of models imposed on the wavelet coefficients are Bayesian. This
popularity is well justified: Bayes rules are typically well behaved shrinkage
rules, prior information about the signal can be incorporated in the shrinkage
procedure, and adaptivity of Bayes rules can be achieved by data-driven
selection of model hyperparameters.
Several papers considering
Bayesian wavelet shrinkage with complex wavelets are available. For
example, Lina (University of Montreal) and collaborators focus on image denoising, in which the phase
of the observed wavelet coefficients is preserved, but the modulus of the coefficients
is shrunk by a Bayes rule. The procedure introduced in Barber and
Nason in 2004 modifies both the phase and modulus of wavelet coefficients by a
bivariate shrinkage rule.
We propose a Bayesian model in the domain of a complex scale-mixing discrete
unitary, compactly supported wavelets that generalizes the method in Barber
and Nason to 2-D signals. In estimating the signal part the model it is
allowed to modify both phase and modulus. The choice of wavelet transform
is motivated by the symmetry / antisymmetry of decomposing wavelets, which
is possible only in the complex domain under condition of orthogonality (unitarity)
and compact support. Symmetry is considered a desirable property of
wavelets, especially when dealing with images.
The 2-D discrete scale mixing wavelet transform is computed by left- and right-multiplying
the image by a wavelet matrix W and its Hermitian transpose
W', respectively. Mallat's algorithm to perform this task is not used, but it is
implicit in the construction of matrix W.
The resulting shrinkage procedures cSM-EB and cMOSM-EB are based on
empirical Bayes approach and utilize non-zero covariances between real and
imaginary parts of the wavelet coefficients. We discuss the possibility of phase-preserving
shrinkage in this framework.
Overall, the methods we propose are
calculationally efficient and provide excellent denoising capabilities when contrasted
to comparable and standardly used wavelet-based techniques.
In the spirit of reproducible research a suite of MATLAB demo files for implementing
cSM-EB and cMOSM-EB shrinkage is compiled and posted at
This work is joint with my former student Norbert Remenyi, and Professors
Orietta Nicolis and Guy Nason. The paper on which this talk is based appeared in
IEEE Trans Image Processing in Fall 2014 (DOI: 10.1109/TIP.2014.2362058).