Singular Regularization of Inverse Problems

This thesis comments on the use of Bregman distances in the context of singular regularization schemes for inverse problems. According to previous works the use of Bregman distances in combination with variational frameworks, based on singular regularization energies, leads to improved approximations of inverse problems solutions. The Bregman distance has become a powerful tool for the analysis of these frameworks, and has brought iterative algorithms to life that enhance the quality of solutions of existing frameworks significantly. However, most works have yet considered Bregman distances in the context of variational frameworks with quadratic fidelity only. One of the goals of this thesis is to extend analytical results to more general, nonlinear fidelity terms arising from applications as e.g. medical imaging. Moreover, the concept of Eigenfunctions of linear operators is transferred to nonlinear operators arising from the optimality conditions of the variational frameworks. From a computational perspective, a novel compressed sensing algorithm based on an inverse scale space formulation is introduced. Furthermore, important concepts related to Bregman distances are carried over to non-quadratic frameworks arising from the applications of dynamic Positron Emission Tomography and Bioluminescence Tomography.