@article{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Regularization of Inverse Problems with Translation Invariant Frames}, series = {Proceedings of the 20th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM-2022), 2022, Heraklion (Crete, Greece)}, journal = {Proceedings of the 20th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM-2022), 2022, Heraklion (Crete, Greece)}, publisher = {AIP Publishing}, abstract = {In various fields of applications, inverse problems are characterized by their sensitivity to data perturbations which can cause severe reconstruction errors. Hence, regularization procedures are employed in order to ensure stability and reconstruction quality. To overcome limitations of classical approaches such as the filtered singular value decomposition (SVD), frame based diagonalization methods have been studied in the recent years, e.g., wavelet-vagulette (WVD) decomposition. While these methods can be well adapted to the problem at hand, it is well-known, that the lack of translation invariance in multiscale systems can cause specific artifacts in the recovered object. Thus, to overcome these drawbacks we use the translation invariant diagonal frame decomposition (TI-DFD) of linear operators. For illustration, we construct a TI-WVD for one-dimensional integration operator, and confirm our theoretical findings by numerical simulations.}, language = {en} } @unpublished{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Translation invariant diagonal frame decomposition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clearly demonstrate the benefits of our approach over non-translation invariant counterparts.}, language = {en} } @unpublished{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Data-proximal null-space networks for inverse problems}, abstract = {Inverse problems are inherently ill-posed and therefore require regularization techniques to achieve a stable solution. While traditional variational methods have wellestablished theoretical foundations, recent advances in machine learning based approaches have shown remarkable practical performance. However, the theoretical foundations of learning-based methods in the context of regularization are still underexplored. In this paper, we propose a general framework that addresses the current gap between learning-based methods and regularization strategies. In particular, our approach emphasizes the crucial role of data consistency in the solution of inverse problems and introduces the concept of data-proximal null-space networks as a key component for their solution. We provide a complete convergence analysis by extending the concept of regularizing null-space networks with data proximity in the visual part. We present numerical results for limited-view computed tomography to illustrate the validity of our framework.}, language = {en} } @article{BorgFrikelJorgensenetal., author = {Borg, Leise and Frikel, J{\"u}rgen and J{\o}rgensen, Jakob Sauer and Quinto, Eric Todd}, title = {Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data}, series = {SIAM Journal on Imaging Sciences}, volume = {11}, journal = {SIAM Journal on Imaging Sciences}, number = {4}, publisher = {SIAM PUBLICATIONS}, doi = {10.1137/18M1166833}, pages = {2786 -- 2814}, abstract = {This article provides a mathematical analysis of singular (nonsmooth) artifacts added to reconstructions by filtered backprojection (FBP) type algorithms for X-ray computed tomography (CT) with arbitrary incomplete data. We prove that these singular artifacts arise from points at the boundary of the data set. Our results show that, depending on the geometry of this boundary, two types of artifacts can arise: object-dependent and object-independent artifacts. Object-dependent artifacts are generated by singularities of the object being scanned, and these artifacts can extend along lines. They generalize the streak artifacts observed in limited-angle tomography. Object-independent artifacts, on the other hand, are essentially independent of the object and take one of two forms: streaks on lines if the boundary of the data set is not smooth at a point and curved artifacts if the boundary is smooth locally. We prove that these streak and curve artifacts are the only singular artifacts that can occur for FBP in the continuous case. In addition to the geometric description of artifacts, the article provides characterizations of their strength in Sobolev scale in certain cases. The results of this article apply to the well-known incomplete data problems, including limited-angle and regionof-interest tomography, as well as to unconventional X-ray CT imaging setups that arise in new practical applications. Reconstructions from simulated and real data are analyzed to illustrate our theorems, including the reconstruction that motivated this work a synchrotron data set in which artifacts appear on lines that have no relation to the object.}, language = {en} } @article{BorgJorgensenFrikeletal., author = {Borg, Leise and J{\o}rgensen, Jakob Sauer and Frikel, J{\"u}rgen and Sporring, Jon}, title = {Reduction of variable-truncation artifacts from beam occlusion during in situ x-ray tomography}, series = {Measurement Science and Technology}, volume = {28}, journal = {Measurement Science and Technology}, number = {12}, publisher = {IOP Publishing}, doi = {10.1088/1361-6501/aa8c27}, abstract = {Many in situ x-ray tomography studies require experimental rigs which may partially occlude the beam and cause parts of the projection data to be missing. In a study of fluid flow in porous chalk using a percolation cell with four metal bars drastic streak artifacts arise in the filtered backprojection (FBP) reconstruction at certain orientations. Projections with non-trivial variable truncation caused by the metal bars are the source of these variable-truncation artifacts. To understand the artifacts a mathematical model of variable-truncation data as a function of metal bar radius and distance to sample is derived and verified numerically and with experimental data. The model accurately describes the arising variable-truncation artifacts across simulated variations of the experimental setup. Three variable-truncation artifact-reduction methods are proposed, all aimed at addressing sinogram discontinuities that are shown to be the source of the streaks. The 'reduction to limited angle' (RLA) method simply keeps only non-truncated projections; the 'detector-directed smoothing' (DDS) method smooths the discontinuities; while the 'reflexive boundary condition' (RBC) method enforces a zero derivative at the discontinuities. Experimental results using both simulated and real data show that the proposed methods effectively reduce variable- truncation artifacts. The RBC method is found to provide the best artifact reduction and preservation of image features using both visual and quantitative assessment. The analysis and artifact-reduction methods are designed in context of FBP reconstruction motivated by computational efficiency practical for large, real synchrotron data. While a specific variable- truncation case is considered, the proposed methods can be applied to general data cut-offs arising in different in situ x-ray tomography experiments.}, language = {en} } @inproceedings{FrikelBorgJorgensenetal., author = {Frikel, J{\"u}rgen and Borg, Leise and J{\o}rgensen, Jakob Sauer and Quinto, Eric Todd}, title = {Singular artifacts in incomplete data x-ray tomography}, series = {Tomographic Inverse Problems: Theory and Applications; 27.01. - 02.02.2019}, booktitle = {Tomographic Inverse Problems: Theory and Applications; 27.01. - 02.02.2019}, editor = {Burger, Martin and Hahn, Bernadette and Quinto, Eric Todd}, issn = {1660-8941}, doi = {10.4171/OWR/2019/4}, pages = {295 -- 297}, language = {en} } @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} } @inproceedings{SeligBauerFrikeletal., author = {Selig, Tim and Bauer, Patrick and Frikel, J{\"u}rgen and M{\"a}rz, Thomas and Storath, Martin and Weinmann, Andreas}, title = {Two-stage Approach for Low-dose and Sparse-angle CT Reconstruction using Backprojection}, series = {Bildverarbeitung f{\"u}r die Medizin 2025 (BVM 2025): Proceedings, German Conference on Medical Image Computing, Regensburg March 09-11, 2025}, booktitle = {Bildverarbeitung f{\"u}r die Medizin 2025 (BVM 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 VS}, address = {Wiesbaden}, isbn = {978-3-658-47421-8}, doi = {10.1007/978-3-658-47422-5_67}, pages = {286 -- 291}, abstract = {This paper presents a novel two-stage approach for computed tomography (CT) reconstruction, focusing on sparse-angle and low-dose setups to minimize radiation exposure while maintaining high image quality. Two-stage approaches consist of an initial reconstruction followed by a neural network for image refinement. In the initial reconstruction, we apply the backprojection (BP) instead of the traditional filtered backprojection (FBP). This enhances computational speed and offers potential advantages for more complex geometries, such as fan-beam and cone-beam CT. Additionally, BP addresses noise and artifacts in sparse-angle CT by leveraging its inherent noise-smoothing effect, which reduces streaking artifacts common in FBP reconstructions. For the second stage, we fine-tune the DRUNet proposed by Zhang et al. to further improve reconstruction quality. We call our method BP-DRUNet and evaluate its performance on a synthetically generated ellipsoid dataset alongside thewell-established LoDoPaBCT dataset. Our results show that BP-DRUNet produces competetive results in terms of PSNR and SSIM metrics compared to the FBP-based counterpart, FBPDRUNet, and delivers visually competitive results across all tested angular setups.}, language = {en} } @unpublished{BorgFrikelJorgensenetal., author = {Borg, Leise and Frikel, J{\"u}rgen and J{\o}rgensen, Jakob Sauer and Quinto, Eric Todd}, title = {Theorems that Characterize Artifacts for Arbitrary Limited X-ray CT Data}, edition = {version 6}, abstract = {This article provides a mathematical classification of artifacts from arbitrary incom-plete X-ray tomography data when using the classical filtered backprojection algorithm. Usingmicrolocal analysis, we prove that all artifacts arise from points at the boundary of the data set.Our results show that, depending on the geometry of the data set boundary, two types of artifactscan arise: object-dependent and object-independent artifacts. The object-dependent artifacts aregenerated by singularities of the object being scanned and these artifacts can extend all along lines.This is a generalization of the streak artifacts observed in limited angle CT. The article also char-acterizes two new phenomena: the object-independent artifacts are caused only by the geometryof the data set boundary; they occur along lines if the boundary of the data set is not smooth andalong curves if the boundary of the data set is smooth. In addition to the geometric descriptionof artifacts, the article also provides characterizations of their strength in Sobolev scale in certaincases. Moreover, numerical reconstructions from simulated and real data are presented illustratingour theorems.This work is motivated by a reconstruction we present from a synchrotron data set in whichartifacts along lines appeared that were independent of the object.The results of this article apply to a wide range of well-known incomplete data problems, in-cluding limited angle CT and region of interest tomography, as well as to unconventional x-ray CTimaging setups. Some of those problems are explicitly addressed in this article, theoretically and numerically.}, language = {en} } @inproceedings{BorgJorgensenFrikeletal., author = {Borg, Leise and J{\o}rgensen, Jakob Sauer and Frikel, J{\"u}rgen and Quinto, Eric Todd and Sporring, Jon}, title = {Reducing artifacts from varying projection truncations}, series = {3rd International Conference on Tomography of Materials and Structures, Lund, Sweden, 26-30 June 2017, ICTMS2017-65-1}, booktitle = {3rd International Conference on Tomography of Materials and Structures, Lund, Sweden, 26-30 June 2017, ICTMS2017-65-1}, abstract = {We study samples with full and partial occlusion causing streak artifacts, and propose two mod-ifications of filtered backprojection for artifact removal. Data is obtained by the SPring-8 synchrotron using a monochromatic parallel-beam scan [1]. Thresholding in the sinogram segments the metal, resulting in edges on which we apply 1) a smooth transition, or 2) a Dirichlet boundary condition.}, language = {en} } @inproceedings{FrikelQuinto, author = {Frikel, J{\"u}rgen and Quinto, Eric Todd}, title = {Artifacts in limited view tomography}, series = {Oberwolfach Reports}, volume = {11}, booktitle = {Oberwolfach Reports}, number = {3}, editor = {Burger, Martin and Quinto, Eric Todd and Louis, Alfred K.}, doi = {10.4171/OWR/2014/37}, pages = {2047 -- 2114}, language = {en} } @inproceedings{BauerFrikel, author = {Bauer, Patrick and Frikel, J{\"u}rgen}, title = {BPConvNet: a deep learning based ρ-Filtered layergram reconstruction method for computed tomography}, series = {AIP Conference Proceedings}, volume = {3315}, booktitle = {AIP Conference Proceedings}, number = {1}, publisher = {AIP Publishing}, issn = {0094-243X}, doi = {10.1063/5.0286063}, abstract = {In this article, we address the reconstruction problem in computed tomography (CT) when dealing with sparse view data. Traditional approaches like filtered backprojection (FBP) often fail under these conditions, leading to streaking artifacts. We propose BPConvNet, a deep learning based version of the ρ-filtered layergram or backprojection filtration (BPF) technique (cf. [1]). Unlike FBP, the BPF method applies filtering (F) after backprojection (BP), hence the name. The proposed BPConvNet adapts the BPF workflow by substituting the filtering step with a residual convolutional neural network. Our numerical experiments demonstrate that BPConvNet is competitive to similar deep learning methods. Moreover, we explain that BPConvNet can be easily adapted to to different CT acquisition geometries, such as fan beam and 3D configurations}, language = {en} }