TY - JOUR A1 - Bringout, Gaël A1 - Erb, Wolfgang A1 - Frikel, Jürgen T1 - A new 3D model for Magnetic Particle Imaging using realistic magnetic field topologies for algebraic reconstruction JF - Inverse Problems N2 - We derive a new 3D model for magnetic particle imaging (MPI) that is able to incorporate realistic magnetic fields in the reconstruction process. In real MPI scanners, the generated magnetic fields have distortions that lead to deformed magnetic low-field volumes with the shapes of ellipsoids or bananas instead of ideal field-free points (FFP) or lines (FFL), respectively. Most of the common model-based reconstruction schemes in MPI use however the idealized assumption of an ideal FFP or FFL topology and, thus, generate artifacts in the reconstruction. Our model-based approach is able to deal with these distortions and can generally be applied to dynamic magnetic fields that are approximately parallel to their velocity field. We show how this new 3D model can be discretized and inverted algebraically in order to recover the magnetic particle concentration. To model and describe the magnetic fields, we use decompositions of the fields in spherical harmonics. We complement the description of the new model with several simulations and experiments, exploring the effects of magnetic fields distortion and reconstruction parameters on the reconstruction. Y1 - 2020 U6 - https://doi.org/10.1088/1361-6420/abb446 VL - 36 IS - 12 PB - IOP Publishing ER - TY - INPR A1 - Ebner, Andrea A1 - Frikel, Jürgen A1 - Lorenz, Dirk A1 - Schwab, Johannes A1 - Haltmeier, Markus T1 - Regularization of inverse problems by filtered diagonal frame decomposition N2 - The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singular value decomposition to the frame case. Frames as generalized singular system allows to better adapt to a given class of potential solutions. In this paper, we show that filtered diagonal frame decomposition yield a convergent regularization method. Moreover, we derive convergence rates under source type conditions and prove order optimality under the assumption that the considered frame is a Riesz-basis. Y1 - 2020 UR - https://arxiv.org/pdf/2008.06219.pdf UR - https://applied-math.uibk.ac.at/images/Preprints/2021/pp_76_2021.pdf ER - TY - CHAP A1 - Frikel, Jürgen A1 - Göppel, Simon A1 - Haltmeier, Markus ED - Palm, Christoph T1 - Combining Reconstruction and Edge Detection in Computed Tomography T2 - Bildverarbeitung für die Medizin 2021 : Proceedings, German Workshop on Medical Image Computing: Regensburg, March 7-9, 2021 N2 - We present two methods that combine image reconstruction and edge detection in computed tomography (CT) scans. Our first method is as an extension of the prominent filtered backprojection algorithm. In our second method we employ l1-regularization for stable calculation of the gradient. As opposed to the first method, we show that this approach is able to compensate for undersampled CT data. Y1 - 2021 SN - 978-3-658-33197-9 U6 - https://doi.org/10.1007/978-3-658-33198-6_37 SP - 153 EP - 157 PB - Springer Vieweg CY - Wiesbaden, Germany ER - TY - JOUR A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Feature Reconstruction from Incomplete Tomographic Data without Detour JF - Mathematics N2 - In this paper, we consider the problem of feature reconstruction from incomplete X-ray CT data. Such incomplete data problems occur when the number of measured X-rays is restricted either due to limit radiation exposure or due to practical constraints, making the detection of certain rays challenging. Since image reconstruction from incomplete data is a severely ill-posed (unstable) problem, the reconstructed images may suffer from characteristic artefacts or missing features, thus significantly complicating subsequent image processing tasks (e.g., edge detection or segmentation). In this paper, we introduce a framework for the robust reconstruction of convolutional image features directly from CT data without the need of computing a reconstructed image first. Within our framework, we use non-linear variational regularization methods that can be adapted to a variety of feature reconstruction tasks and to several limited data situations. The proposed variational regularization method minimizes an energy functional being the sum of a feature dependent datafitting term and an additional penalty accounting for specific properties of the features. In our numerical experiments, we consider instances of edge reconstructions from angular under-sampled data and show that our approach is able to reliably reconstruct feature maps in this case. KW - Computed tomography KW - Radon transform KW - reconstruction KW - limited data KW - sparse data KW - feature reconstruction, KW - edge detection Y1 - 2022 U6 - https://doi.org/10.3390/math10081318 VL - 10 IS - 8 PB - MDPI ER - TY - JOUR A1 - Frikel, Jürgen A1 - Quinto, Eric Todd T1 - Limited Data Problems for the Generalized Radon Transform in Rⁿ JF - SIAM JOURNAL ON MATHEMATICAL ANALYSIS N2 - We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in R-n. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in R-n and define modified versions of reconstruction operators that do not generate added artifacts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators and provide conditions for the filters under which the reconstruction operators are elliptic for the visible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in R-2 and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to analyze more general imaging situations. KW - ARTIFACTS KW - computed tomography KW - Fourier integral operators KW - Lambda tomography KW - limited angle tomography KW - LOCAL TOMOGRAPHY KW - microlocal analysis KW - Radon transforms Y1 - 2016 U6 - https://doi.org/10.1137/15M1045405 VL - 48 IS - 4 SP - 2301 EP - 2318 PB - SIAM ER - TY - JOUR A1 - Erb, Wolfgang A1 - Weinmann, Andreas A1 - Ahlborg, Mandy A1 - Brandt, Christina A1 - Bringout, Gael A1 - Buzug, Thorsten M. A1 - Frikel, Jürgen A1 - Kaethner, C. A1 - Knopp, Tobias A1 - März, T. A1 - Moddel, Martin A1 - Storath, Martin A1 - Weber, A. T1 - Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging JF - Inverse Problems N2 - Magnetic particle imaging (MPI) is a promising new in vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator. KW - FORMULATION KW - ill-posedness KW - INTERPOLATION KW - inverse problems KW - magnetic particle imaging KW - model-based reconstruction KW - MPI KW - NODES KW - reconstruction operator KW - RELAXATION KW - RESOLUTION Y1 - 2018 U6 - https://doi.org/10.1088/1361-6420/aab8d1 VL - 34 IS - 5 PB - IOP Publishing ER - TY - JOUR A1 - Agranovsky, Mark A1 - Boman, Jan A1 - Hasanov, Alemdar A1 - Felea, Raluca A1 - Frikel, Jürgen A1 - Krishnan, Venky A1 - Novikov, Roman A1 - Ramlau, Ronny A1 - Sebu, Cristiana T1 - Research biography of a distinguished expert in the field of inverse problems: Professor Eric Todd Quinto JF - Journal of Inverse and Ill-posed Problems N2 - This article gives a brief overview of the research in microlocal analysis, tomography, and integral geometry of Professor Eric Todd Quinto, Robinson Professor of Mathematics at Tufts University, along with the collaborators and colleagues who influenced his work. Y1 - 2022 U6 - https://doi.org/10.1515/jiip-2022-0031 SN - 1569-3945 VL - 30 IS - 4 SP - 613 EP - 617 PB - De Gruyter ER - TY - JOUR A1 - Barannyk, Lyudmyla L. A1 - Frikel, Jürgen A1 - Nguyen, Linh V. T1 - On artifacts in limited data spherical Radon transform: Curved observation surface JF - Inverse Problems N2 - We study the limited data problem of the spherical Radon transform in two and three-dimensional spaces with general acquisition surfaces. In such situations, it is known that the application of filtered-backprojection reconstruction formulas might generate added artifacts and degrade the quality of reconstructions. In this article, we explicitly analyze a family of such inversion formulas, depending on a smoothing function that vanishes to order k on the boundary of the acquisition surfaces. We show that the artifacts are k orders smoother than their generating singularity. Moreover, in two-dimensional space, if the generating singularity is conormal satisfying a generic condition then the artifacts are even k+1/2 orders smoother than the generating singularity. Our analysis for three-dimensional space contains an important idea of lifting up space. We also explore the theoretical findings in a series of numerical experiments. Our experiments show that a good choice of the smoothing function leads to a significant improvement of reconstruction quality. Y1 - 2016 U6 - https://doi.org/10.1088/0266-5611/32/1/015012 SN - 1361-6420 SN - 0266-5611 VL - 32 IS - 1 PB - Institute of Physics CY - Bristol [u.a.] ER - TY - JOUR A1 - Frikel, Jürgen T1 - Sparse regularization in limited angle tomography JF - Applied and Computational Harmonic Analysis N2 - We investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy, etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, e.g. filtered backprojection (FBP), do not perform well in such situations. To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this work, we propose the use of the sparse regularization technique in combination with curvelets. We argue that this technique gives rise to an edge-preserving reconstruction. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of the limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will show that the theoretical results directly translate into practice and that the proposed method outperforms classical reconstructions. KW - Curvelets KW - Dimensionality reduction KW - limited angle tomography KW - Radon transform KW - Sparse regularization Y1 - 2013 U6 - https://doi.org/10.1016/j.acha.2012.03.005 VL - 34 IS - 1 SP - 117 EP - 141 PB - Elsevier ER - TY - JOUR A1 - Frikel, Jürgen T1 - Short communication: Dimensionality reduction of curvelet sparse regularizations in limited angle tomography JF - Proceedings in applied mathematics and mechanics : PAMM N2 - We investigate the reconstruction problem for limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well in such situations. To stabilize the inversion we propose the use of a sparse regularization technique in combination with curvelets. We argue that this technique has the ability to preserve edges. As our main result, we present a characterization of the kernel of the limited angle Radon transform in terms of curvelets. Moreover, we characterize reconstructions which are obtained via curvelet sparse regularizations at a limited angular range. As a result, we show that the dimension of the limited angle problem can be significantly reduced in the curvelet domain. Y1 - 2011 U6 - https://doi.org/10.1002/pamm.201110412 SN - 1617-7061 VL - 11 IS - 1 SP - 847 EP - 848 PB - Wiley-VCH CY - Weinheim ER - TY - JOUR A1 - Frikel, Jürgen A1 - Quinto, Eric Todd T1 - Artifacts in Incomplete Data Tomography with Applications to Photoacoustic Tomography and Sonar JF - SIAM Journal on Applied Mathematics N2 - We develop a paradigm using microlocal analysis that allows one to characterize the visible and added singularities in a broad range of incomplete data tomography problems. We give precise characterizations for photoacoustic and thermoacoustic tomography and sonar, and provide artifact reduction strategies. In particular, our theorems show that it is better to arrange sonar detectors so that the boundary of the set of detectors does not have corners and is smooth. To illustrate our results, we provide reconstructions from synthetic spherical mean data as well as from experimental photoacoustic data. KW - computed tomography KW - Fourier integral operators KW - Lambda tomography KW - limited angle tomography KW - microlocal analysis KW - photoacoustic tomography KW - radon transforms KW - sonar KW - spherical mean KW - thermoacoustic tomography Y1 - 2015 U6 - https://doi.org/10.1137/140977709 SN - 1095-712X SN - 0036-1399 VL - 75 IS - 2 SP - 703 EP - 725 PB - Society for Industrial and Applied Mathematics CY - Philadelphia, Pa. ER - TY - JOUR A1 - Frikel, Jürgen A1 - Quinto, Eric Todd T1 - Characterization and reduction of artifacts in limited angle tomography JF - Inverse Problems N2 - We consider the reconstruction problem for limited angle tomography using filtered backprojection (FBP) and lambda tomography. We use microlocal analysis to explain why the well-known streak artifacts are present at the end of the limited angular range. We explain how to mitigate the streaks and prove that our modified FBP and lambda operators are standard pseudodifferential operators, and so they do not add artifacts. We provide reconstructions to illustrate our mathematical results. Y1 - 2013 U6 - https://doi.org/10.1088/0266-5611/29/12/125007 SN - 0266-5611 SN - 1361-6420 VL - 29 IS - 12 PB - IOP Publishing CY - Bristol ER - TY - INPR A1 - Frikel, Jürgen A1 - Quinto, Eric Todd T1 - A paradigm for the characterization of artifacts in tomography N2 - We present a paradigm for characterization of artifacts in limited data tomography problems. In particular, we use this paradigm to characterize artifacts that are generated in reconstructions from limited angle data with generalized Radon transforms and general filtered backprojection type operators. In order to find when visible singularities are imaged, we calculate the symbol of our reconstruction operator as a pseudodifferential operator. KW - computed tomography KW - Fourier integral operators KW - Lambda tomography KW - limited angle tomography KW - microlocal analysis KW - radon transforms Y1 - 2014 U6 - https://doi.org/10.48550/arXiv.1409.4103 ER - TY - JOUR A1 - Storath, Martin A1 - Weinmann, Andreas A1 - Frikel, Jürgen A1 - Unser, Michael T1 - Joint image reconstruction and segmentation using the Potts model JF - Inverse Problems N2 - We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford–Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider limited data situations. For instance, our method is able to recover all segments of the Shepp–Logan phantom from seven angular views only. We illustrate the practical applicability on a real positron emission tomography dataset. As further applications, we consider spherical Radon data as well as blurred data. Y1 - 2015 U6 - https://doi.org/10.1088/0266-5611/31/2/025003 SN - 0266-5611 SN - 1361-6420 VL - 31 IS - 2 PB - IOP Publishing CY - Bristol ER - TY - JOUR A1 - Wieczorek, Matthias A1 - Frikel, Jürgen A1 - Vogel, Jakob A1 - Eggl, Elena A1 - Kopp, Felix A1 - Noël, Peter B. A1 - Pfeiffer, Franz A1 - Demaret, Laurent A1 - Lasser, Tobias T1 - X-ray computed tomography using curvelet sparse regularization JF - Medical physics N2 - PURPOSE Reconstruction of x-ray computed tomography (CT) data remains a mathematically challenging problem in medical imaging. Complementing the standard analytical reconstruction methods, sparse regularization is growing in importance, as it allows inclusion of prior knowledge. The paper presents a method for sparse regularization based on the curvelet frame for the application to iterative reconstruction in x-ray computed tomography. METHODS In this work, the authors present an iterative reconstruction approach based on the alternating direction method of multipliers using curvelet sparse regularization. RESULTS Evaluation of the method is performed on a specifically crafted numerical phantom dataset to highlight the method's strengths. Additional evaluation is performed on two real datasets from commercial scanners with different noise characteristics, a clinical bone sample acquired in a micro-CT and a human abdomen scanned in a diagnostic CT. The results clearly illustrate that curvelet sparse regularization has characteristic strengths. In particular, it improves the restoration and resolution of highly directional, high contrast features with smooth contrast variations. The authors also compare this approach to the popular technique of total variation and to traditional filtered backprojection. CONCLUSIONS The authors conclude that curvelet sparse regularization is able to improve reconstruction quality by reducing noise while preserving highly directional features. Y1 - 2015 U6 - https://doi.org/10.1118/1.4914368 SN - 0094-2405 SN - 2473-4209 VL - 42 IS - 4 SP - 1555 EP - 1565 PB - American Association of Physicists in Medicine ER - TY - CHAP A1 - Wieczorek, Matthias A1 - Frikel, Jürgen A1 - Vogel, Jakob A1 - Pfeiffer, Franz A1 - Demaret, Laurent A1 - Lasser, Tobias T1 - Curvelet sparse regularization for differential phase-contrast X-ray imaging T2 - Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine : Proceedings ; June 16-21, 2013, Granlibakken Resort, Lake Tahoe, California N2 - Differential phase contrast imaging (DPCI) enables the visualization of soft tissue contrast using X-rays. In this work we introduce a reconstruction framework based on curvelet expansion and sparse regularization for DPCI. We will show that curvelets provide a suitable data representation for DPCI reconstruction that allows preservation of edges as well as an exact analytic representation of the system matrix. As a first evaluation, we show results using simulated phantom data Y1 - 2013 UR - https://www.researchgate.net/publication/255913330_Curvelet_sparse_regularization_for_differential_phase-contrast_X-ray_imaging SP - 489 EP - 492 ER - TY - CHAP A1 - Frikel, Jürgen T1 - A new framework for sparse regularization in limited angle x-ray tomography T2 - 2010 7th IEEE International Symposium on Biomedical Imaging: From Nano to Macro ; Rotterdam 14.04.2010 - 17.04.2010 N2 - We propose a new framework for limited angle tomographic reconstruction. Our approach is based on the observation that for a given acquisition geometry only a few (visible) structures of the object can be reconstructed reliably using a limited angle data set. By formulating this problem in the curvelet domain, we can characterize those curvelet coefficients which correspond to visible structures in the image domain. The integration of this information into the formulation of the reconstruction problem leads to a considerable dimensionality reduction and yields a speedup of the corresponding reconstruction algorithms. KW - computerised tomography KW - curvelet transforms KW - Curvelets KW - Dimensionality reduction KW - image reconstruction KW - limited angle tomography KW - medical image processing KW - sparse matrices KW - Sparse regularization KW - wavefront set Y1 - 2010 SN - 978-1-4244-4125-9 U6 - https://doi.org/10.1109/ISBI.2010.5490113 SN - 1945-7928 SP - 824 EP - 827 PB - IEEE CY - Piscataway, NJ ER - TY - RPRT A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Translation invariant diagonal frame decomposition of inverse problems and their regularization N2 - 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. KW - convergence rates KW - frames KW - Inverse problems KW - operator decomposition KW - regularization KW - translation invariance KW - vaguelettes KW - wavelets Y1 - 2022 U6 - https://doi.org/10.48550/arXiv.2208.08500 ER - TY - JOUR A1 - Frikel, Jürgen T1 - A measurement-based model for image reconstruction in MPI using a FFL JF - Proceedings of the 20th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM-2022), 2022, Heraklion (Crete, Greece) N2 - Most of the common model-based reconstruction schemes in magnetic particle imaging (MPI) use idealized assumptions, e.g., of an ideal field-free-line (FFL) topology. However, the magnetic fields that are generated in real MPI scanners have distortions and, therefore, model-based approaches often lead to inaccurate reconstructions and may contain artifacts. In order to improve the reconstruction quality in MPI, more realistic MPI models need to be derived. In the present work, we address this problem and present a hybrid model for MPI that allows us to incorporates real measurements of the applied magnetic fields. We will explain that the measurements, that are needed to setup a model for the magnetic fields, can be obtained in a novel calibration procedure that is independent of the resolution and which is much less time-consuming than the one employed in measurement-based MPI reconstructions.We will also present a discretization strategy for this model, that can be used in context of algebraic reconstructions. The presented approach was validated on simulated data in [1], however, its evaluation on real data is a topic for future research. Y1 - 2023 PB - AIP Publishing ER - TY - JOUR A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Regularization of Inverse Problems with Translation Invariant Frames JF - Proceedings of the 20th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM-2022), 2022, Heraklion (Crete, Greece) N2 - 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. Y1 - 2023 PB - AIP Publishing ER - TY - INPR A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Translation invariant diagonal frame decomposition for the Radon transform N2 - 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. KW - DFD KW - translation invariance KW - Radon transform KW - inverse problems KW - regularization Y1 - 2023 ER - TY - INPR A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Data-proximal null-space networks for inverse problems N2 - 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. KW - Regularization KW - null-space network KW - data-proximal network KW - convergence analysis KW - data consistency Y1 - 2023 ER - TY - JOUR A1 - Borg, Leise A1 - Frikel, Jürgen A1 - Jørgensen, Jakob Sauer A1 - Quinto, Eric Todd T1 - Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data JF - SIAM Journal on Imaging Sciences N2 - 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. KW - DATA TOMOGRAPHY KW - Fourier integral operators KW - GENERALIZED RADON-TRANSFORM KW - IMAGE-RECONSTRUCTION KW - incomplete data tomography KW - limited-angle tomography KW - METAL ARTIFACTS KW - microlocal analysis KW - PROJECTIONS KW - reconstruction artifact KW - region-of-interest tomography KW - SINGULARITIES KW - wavefront set KW - X-ray tomography Y1 - 2018 U6 - https://doi.org/10.1137/18M1166833 VL - 11 IS - 4 SP - 2786 EP - 2814 PB - SIAM PUBLICATIONS ER - TY - JOUR A1 - Borg, Leise A1 - Jørgensen, Jakob Sauer A1 - Frikel, Jürgen A1 - Sporring, Jon T1 - Reduction of variable-truncation artifacts from beam occlusion during in situ x-ray tomography JF - Measurement Science and Technology N2 - 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. KW - computed tomography KW - COMPUTED-TOMOGRAPHY KW - CT KW - filtered backprojection KW - IMAGE-RECONSTRUCTION KW - in situ measurements KW - sinogram discontinuities KW - variable-truncation artifacts Y1 - 2017 U6 - https://doi.org/10.1088/1361-6501/aa8c27 VL - 28 IS - 12 PB - IOP Publishing ER - TY - CHAP A1 - Frikel, Jürgen A1 - Borg, Leise A1 - Jørgensen, Jakob Sauer A1 - Quinto, Eric Todd ED - Burger, Martin ED - Hahn, Bernadette ED - Quinto, Eric Todd T1 - Singular artifacts in incomplete data x-ray tomography T2 - Tomographic Inverse Problems: Theory and Applications; 27.01. - 02.02.2019 Y1 - 2019 U6 - https://doi.org/10.4171/OWR/2019/4 SN - 1660-8941 SP - 295 EP - 297 ER - TY - JOUR A1 - Ebner, Andrea A1 - Frikel, Jürgen A1 - Lorenz, Dirk A1 - Schwab, J. A1 - Haltmeier, Markus T1 - Regularization of inverse problems by filtered diagonal frame decomposition JF - Applied and Computational Harmonic Analysis N2 - 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). Y1 - 2023 U6 - https://doi.org/10.1016/j.acha.2022.08.005 N1 - Preprint unter: https://arxiv.org/pdf/2008.06219.pdf OPUS-Einrag unter: https://opus4.kobv.de/opus4-oth-regensburg/admin/document/edit/id/2290 VL - 62 IS - January SP - 66 EP - 83 PB - Elsevier ER - TY - CHAP A1 - Göppel, Simon A1 - Frikel, Jürgen A1 - Haltmeier, Markus ED - Palm, Christoph ED - Breininger, Katharina ED - Deserno, Thomas M. ED - Handels, Heinz ED - Maier, Andreas ED - Maier-Hein, Klaus H. ED - Tolxdorff, Thomas M. T1 - Data-proximal Neural Networks for Limited-view CT T2 - Bildverarbeitung für die Medizin 2025 : Proceedings, German Conference on Medical Image Computing, Regensburg March 09-11, 2025 N2 - 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. Y1 - 2025 SN - 978-3-658-47421-8 U6 - https://doi.org/10.1007/978-3-658-47422-5_41 SP - 185 EP - 190 PB - Springer Fachmedien Wiesbaden CY - Wiesbaden ER - TY - CHAP A1 - Selig, Tim A1 - Bauer, Patrick A1 - Frikel, Jürgen A1 - März, Thomas A1 - Storath, Martin A1 - Weinmann, Andreas ED - Palm, Christoph ED - Breininger, Katharina ED - Deserno, Thomas M. ED - Handels, Heinz ED - Maier, Andreas ED - Maier-Hein, Klaus H. ED - Tolxdorff, Thomas M. T1 - Two-stage Approach for Low-dose and Sparse-angle CT Reconstruction using Backprojection T2 - Bildverarbeitung für die Medizin 2025 (BVM 2025): Proceedings, German Conference on Medical Image Computing, Regensburg March 09-11, 2025 N2 - 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. Y1 - 2025 SN - 978-3-658-47421-8 U6 - https://doi.org/10.1007/978-3-658-47422-5_67 SP - 286 EP - 291 PB - Springer VS CY - Wiesbaden ER - TY - INPR A1 - Borg, Leise A1 - Frikel, Jürgen A1 - Jørgensen, Jakob Sauer A1 - Quinto, Eric Todd T1 - Theorems that Characterize Artifacts for Arbitrary Limited X-ray CT Data N2 - 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. Y1 - 2017 UR - https://www.researchgate.net/publication/318360078_Theorems_that_Characterize_Artifacts_for_Arbitrary_Limited_X-ray_CT_Data ET - version 6 ER - TY - CHAP A1 - Borg, Leise A1 - Jørgensen, Jakob Sauer A1 - Frikel, Jürgen A1 - Quinto, Eric Todd A1 - Sporring, Jon T1 - Reducing artifacts from varying projection truncations T2 - 3rd International Conference on Tomography of Materials and Structures, Lund, Sweden, 26-30 June 2017, ICTMS2017-65-1 N2 - 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. KW - X-ray tomography KW - projection truncation KW - streak artifacts KW - boundary conditions Y1 - 2017 UR - https://meetingorganizer.copernicus.org/ICTMS2017/ICTMS2017-65-1.pdf ER - TY - CHAP A1 - Frikel, Jürgen A1 - Quinto, Eric Todd ED - Burger, Martin ED - Quinto, Eric Todd ED - Louis, Alfred K. T1 - Artifacts in limited view tomography T2 - Oberwolfach Reports Y1 - 2014 U6 - https://doi.org/10.4171/OWR/2014/37 VL - 11 IS - 3 SP - 2047 EP - 2114 ER - TY - CHAP A1 - Bauer, Patrick A1 - Frikel, Jürgen T1 - BPConvNet: a deep learning based ρ-Filtered layergram reconstruction method for computed tomography T2 - AIP Conference Proceedings N2 - 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 KW - Convolutional neural network KW - Deep learning KW - Learning and learning models KW - Computed tomography Y1 - 2025 U6 - https://doi.org/10.1063/5.0286063 SN - 0094-243X VL - 3315 IS - 1 PB - AIP Publishing ER -