Ph.D Thesis

Ph.D StudentKisilyov Pavel
SubjectApplication of Wavelet Representations in Inverse Problems
DepartmentDepartment of Electrical and Computer Engineering
Supervisors PROF. Arkadi Nemirovski


Wavelets have become a popular tool in various fields of signal and image processing. In this thesis we show that wavelets are as crucial in inverse problems, such as Blind Source Separation (BSS) and some tomographic applications. The first facet of this thesis is a multiscale approach to the problem of BSS. In the proposed approach we take advantage of the properties of multiscale transforms, such as wavelet and wavelet packets, and decompose signals according to sets of local features. The resulting partial representations depict various degrees of sparsity. We study how the separation error is affected by the sparsity, and by the misfit between the probabilistic model and the actual distribution of the wavelet coefficients. Based on this study, we define error estimator used in selection of the best subsets of coefficients, utilized in turn for further separation. Experiments with simulated and real signals and images demonstrate superiority of the proposed method over other known techniques. The second facet of this thesis is the application of local multiscale strategies in tomography, wherein the main concern is the reconstruction of medical images from their projection data. We define a new class of Local Radon Transform - the Radon-Wavelet representations, combining properties of the Radon Transform and the local features of the Wavelet Transform. These representations result in a new class of spatially localized wavelet functions. We also propose novel approaches for utilizing wavelet representations in the Maximum Likelihood reconstruction of images. In the first approach, we define various wavelet-type penalty functions. The second approach is based on transformation of the parameter space to the wavelet domain, and performing a reconstruction procedure in this transform domain. The proposed schemes allow reconstruction of images at desired resolution and with desired regularization, so that a trade-off between increasing resolution and noise suppression is achieved.