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 - 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 - 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 - 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 - 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 - 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 -