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Branislav Vidakovic

Room 306, Statistics
Georgia Institute of Technology - College of Engineering

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).

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