Wavelet Shrinkage in Signal and Image Processing : An Investigation of Relations and Equivalences

Abstract

This thesis is a contribution to the field "equivalences of different methods of mathematical image processing´´. During the last decade this field has become an independent field of mathematical image processing. The intention of this thesisis to present an extensive collection of equivalence results for special denoising methods: the wavelet shrinkage methods.Wavelet methods are applied in signal and image processing very successfully for almost fifteen years and it has been shown in several papers that wavelet shrinkage methods arise "naturally´´ in many different mathematical models for signal and image denoising. These results come from very different fields of mathematics: harmonic analysis, functional analysis, partial differential equations, or statistics. The aim of this thesis is to present all these equivalence results in a unifying framework.Besides these "classical´´ results some generalizations are presented, for example: Hard and soft wavelet shrinkagecan be treated in a common framework and it is possible to construct a natural interpolation between both of them; the abstract concept of ``shrinkage´´ also applies to other methods for denoising, for example for BV denoising or even for regularizations which involve Sobolev or Hölder spaces.