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 -