Tight frame methods for deconvolution

Abstract

This thesis devotes to analyzing deconvolution algorithms using framelet approaches, which has appeared in framelet based high resolution image reconstruction methods. We first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation. This formulation converts deconvolution to problems of filling missing framelet coefficients which satisfy certain minimization properties. The missing framelet coefficients are recovered iteratively together with a built-in denoising scheme which prevent noises in data from blowing up while iterating. This approach has been proven to be efficient in solving various problems in high resolution image reconstructions. However, analysis of convergence as well as stability of algorithms and the optimal properties of solutions were absent in those works. This thesis is to establish the theoretical foundation of this framelet approach. In particular, proofs of convergence, analysis of the stability of the algorithms and a study of the minimization property of solutions are given.