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 - Borg, Leise A1 - Frikel, Jürgen A1 - Jorgensen, 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 - INPR A1 - Borg, Leise A1 - Frikel, Jürgen A1 - Jørgensen, Jakob 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 - JOUR A1 - Borg, Leise A1 - Jorgensen, Jakob S. 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 - Borg, Leise A1 - Sauer Jørgensen, Jakob A1 - Frikel, Jürgen A1 - Todd Quinto, Eric 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 - 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 - JOUR A1 - Ebner, Andrea A1 - Frikel, Jürgen A1 - Lorenz, D. 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 - 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 - 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 -