@unpublished{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Translation invariant diagonal frame decomposition for the Radon transform}, abstract = {In this article, we address the challenge of solving the ill-posed reconstruction problem in computed tomography using a translation invariant diagonal frame decomposition (TIDFD). First, we review the concept of a TI-DFD for general linear operators and the corresponding filter-based regularization concept. We then introduce the TI-DFD for the Radon transform on L 2 (R 2) and provide an exemplary construction using the TI wavelet transform. Presented numerical results clearly demonstrate the benefits of our approach over non-translation invariant counterparts.}, language = {en} } @techreport{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Translation invariant diagonal frame decomposition of inverse problems and their regularization}, doi = {10.48550/arXiv.2208.08500}, abstract = {Solving inverse problems is central to a variety of important applications, such as biomedical image reconstruction and non-destructive testing. These problems are characterized by the sensitivity of direct solution methods with respect to data perturbations. To stabilize the reconstruction process, regularization methods have to be employed. Well-known regularization methods are based on frame expansions, such as the wavelet-vaguelette (WVD) decomposition, which are well adapted to the underlying signal class and the forward model and allow efficient implementation. However, it is well known that the lack of translational invariance of wavelets and related systems leads to specific artifacts in the reconstruction. To overcome this problem, in this paper we introduce and analyze the concept of translation invariant diagonal frame decomposition (TI-DFD) of linear operators. We prove that a TI-DFD combined with a regularizing filter leads to a convergent regularization method with optimal convergence rates. As illustrative example, we construct a wavelet-based TI-DFD for one-dimensional integration, where we also investigate our approach numerically. The results indicate that filtered TI-DFDs eliminate the typical wavelet artifacts when using standard wavelets and provide a fast, accurate, and stable solution scheme for inverse problems.}, language = {en} }