Thursday, February 12 2015, 3:30pm Room 306, Statistics Vidakovic 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 http://gtwavelet.bme.gatech.edu/. 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).
