@article{EbnerFrikelLorenzetal., author = {Ebner, Andrea and Frikel, J{\"u}rgen and Lorenz, Dirk and Schwab, J. and Haltmeier, Markus}, title = {Regularization of inverse problems by filtered diagonal frame decomposition}, series = {Applied and Computational Harmonic Analysis}, volume = {62}, journal = {Applied and Computational Harmonic Analysis}, number = {January}, publisher = {Elsevier}, doi = {10.1016/j.acha.2022.08.005}, pages = {66 -- 83}, abstract = {Inverse problems are at the heart of many practical problems such as image reconstruction or nondestructive testing. A characteristic feature is their instability with respect to data perturbations. To stabilize the inversion process, regularization methods must be developed and applied. In this paper, we introduce the concept of filtered diagonal frame decomposition, which extends the classical filtered SVD to the case of frames. The use of frames as generalized singular systems allows a better match to a given class of potential solutions and is also beneficial for problems where the SVD is not analytically available. We show that filtered diagonal frame decompositions yield convergent regularization methods, derive convergence rates under source conditions and prove order optimality. Our analysis applies to bounded and unbounded forward operators. As a practical application of our tools, we study filtered diagonal frame decompositions for inverting the Radon transform as an unbounded operator on L2(R2).}, language = {en} } @inproceedings{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Data-proximal Neural Networks for Limited-view CT}, series = {Bildverarbeitung f{\"u}r die Medizin 2025 : Proceedings, German Conference on Medical Image Computing, Regensburg March 09-11, 2025}, booktitle = {Bildverarbeitung f{\"u}r die Medizin 2025 : Proceedings, German Conference on Medical Image Computing, Regensburg March 09-11, 2025}, editor = {Palm, Christoph and Breininger, Katharina and Deserno, Thomas M. and Handels, Heinz and Maier, Andreas and Maier-Hein, Klaus H. and Tolxdorff, Thomas M.}, publisher = {Springer Fachmedien Wiesbaden}, address = {Wiesbaden}, isbn = {978-3-658-47421-8}, doi = {10.1007/978-3-658-47422-5_41}, pages = {185 -- 190}, abstract = {Limited-angle computed tomography (CT) requires solving an inverse problem that is both ill-conditioned and underdetermined. In recent years, learned reconstruction methods have proven highly effective in addressing this challenge. Most of these methods follow a two-step process: first, an initial reconstruction method is applied to the data to generate an auxiliary reconstruction; second, a neural network is used to map the auxiliary reconstruction closer to the ground truth images. However, when applied to unseen data, there are no guarantees that the network's output will remain consistent with the available measurement data. To address this, we recently introduced a data-proximal network architecture. In this paper, we implement this approach for limited-angle CT and compare its performance with a standard residual network and a null space network.}, language = {en} }