@phdthesis{Frikel, author = {Frikel, J{\"u}rgen}, title = {Reconstructions in limited angle x-ray tomography: Characterization of classical reconstructions and adapted curvelet sparse regularization}, publisher = {Technische Universit{\"a}t M{\"u}nchen}, address = {M{\"u}nchen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:91-diss-20130328-1115037-0-3}, abstract = {This thesis is devoted to the problem of tomographic reconstruction at limited angular range. In the first part, we prove a characterization of filtered backprojection reconstructions from limited angle data. Moreover, we develop a strategy for artifact reduction and stabilization. In the second part, we introduce a new edge-preserving reconstruction algorithm for limited angle tomography and analyze this algorithm mathematically. Some numerical experiments are also presented.}, language = {en} } @unpublished{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {A complementary ℓ1 -TV reconstruction algorithm for limited data CT}, abstract = {In a variety of tomographic applications, data cannot be fully acquired, leading to severely underdetermined image reconstruction. Conventional methods result in reconstructions with significant artifacts. In order to remove these artifacts, regularization methods have to be applied that incorporate additional information. An important example is TV reconstruction which is well known to efficiently compensate for missing data and well reduces reconstruction artifacts. At the same time, however, tomographic data is also contaminated by noise, which poses an additional challenge. The use of a single regularizer within a variational regularization framework must therefore account for both the missing data and the noise. However, a single regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction over different scales, in which case ℓ1 curvelet regularization methods work well. To address this issue, in this paper we introduce a novel variational regularization framework that combines the advantages of two different regularizers. The basic idea of our framework is to perform reconstruction in two stages, where the first stage mainly aims at accurate reconstruction in the presence of noise, and the second stage aims at artifact reduction. Both reconstruction stages are connected by a data proximity condition. The proposed method is implemented and tested for limited-view CT using a combined curvelet-TV approach. We define and implement a curvelet transform adapted to the limited view problem and demonstrate the advantages of our approach in a series of numerical experiments in this context.}, language = {en} } @misc{FrikelGoeppelHaltmeier, author = {Frikel, J{\"u}rgen and G{\"o}ppel, Simon and Haltmeier, Markus}, title = {Combining reconstruction and edge detection in CT}, series = {BVM Workshop 2023, 02.-04. Juli 2023, Braunschweig}, journal = {BVM Workshop 2023, 02.-04. Juli 2023, Braunschweig}, abstract = {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.}, language = {en} } @misc{Frikel, author = {Frikel, J{\"u}rgen}, title = {Adapted curvelet sparse regularization in limited angle tomography}, series = {Summer School: New Trends and Directions in Harmonic Analysis, Fractional Operator Theory, and Image Analysis, Inzell, Germany, September 17 - 21, 2012}, journal = {Summer School: New Trends and Directions in Harmonic Analysis, Fractional Operator Theory, and Image Analysis, Inzell, Germany, September 17 - 21, 2012}, language = {en} } @incollection{FrikelHaltmeier, author = {Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Sparse Regularization of Inverse Problems by Operator-Adapted Frame Thresholding}, series = {Mathematics of Wave Phenomena}, booktitle = {Mathematics of Wave Phenomena}, editor = {D{\"o}rfler, Willy and Hochbruck, Marlis and Hundertmark, Dirk and Reichel, Wolfgang and Rieder, Andreas}, edition = {1st ed.}, publisher = {Birkh{\"a}user; Springer International Publishing}, address = {Cham}, isbn = {978-3-030-47173-6}, doi = {10.1007/978-3-030-47174-3_10}, pages = {163 -- 178}, abstract = {We analyze sparse frame based regularization of inverse problems by means of a diagonal frame decomposition (DFD) for the forward operator, which generalizes the SVD. The DFD allows to define a non-iterative (direct) operator-adapted frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse ℓ1-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the three versions of sparse regularization, namely synthesis and analysis variants of ℓ1-regularization as well as the DFD thresholding are equivalent. However, in the redundant case, those three approaches are pairwise different.}, language = {en} } @article{FrikelHaltmeier, author = {Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Efficient regularization with wavelet sparsity constraints in photoacoustic tomography}, series = {Inverse Problems}, volume = {34}, journal = {Inverse Problems}, number = {2}, doi = {10.1088/1361-6420/aaa0ac}, pages = {1 -- 28}, abstract = {In this paper, we consider the reconstruction problem of photoacoustic tomography (PAT) with a flat observation surface. We develop a direct reconstruction method that employs regularization with wavelet sparsity constraints. To that end, we derive a wavelet-vaguelette decomposition (WVD) for the PAT forward operator and a corresponding explicit reconstruction formula in the case of exact data. In the case of noisy data, we combine the WVD reconstruction formula with soft-thresholding, which yields a spatially adaptive estimation method. We demonstrate that our method is statistically optimal for white random noise if the unknown function is assumed to lie in any Besov-ball. We present generalizations of this approach and, in particular, we discuss the combination of PAT-vaguelette soft-thresholding with a total variation (TV) prior. We also provide an efficient implementation of the PAT-vaguelette transform that leads to fast image reconstruction algorithms supported by numerical results.}, language = {en} } @article{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Data-Proximal Complementary ℓ1-TV Reconstruction for Limited Data Computed Tomography}, series = {Mathematics}, volume = {12}, journal = {Mathematics}, number = {10}, publisher = {MDPI}, issn = {2227-7390}, doi = {10.3390/math12101606}, abstract = {In a number of tomographic applications, data cannot be fully acquired, resulting in severely underdetermined image reconstruction. Conventional methods in such cases lead to reconstructions with significant artifacts. To overcome these artifacts, regularization methods are applied that incorporate additional information. An important example is TV reconstruction, which is known to be efficient in compensating for missing data and reducing reconstruction artifacts. On the other hand, tomographic data are also contaminated by noise, which poses an additional challenge. The use of a single regularizer must therefore account for both the missing data and the noise. A particular regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction over multiple scales, in which case ℓ1 curvelet regularization methods are well suited. To address this issue, in this paper, we present a novel variational regularization framework that combines the advantages of different regularizers. The basic idea of our framework is to perform reconstruction in two stages. The first stage is mainly aimed at accurate reconstruction in the presence of noise, and the second stage is aimed at artifact reduction. Both reconstruction stages are connected by a data proximity condition. The proposed method is implemented and tested for limited-view CT using a combined curvelet-TV approach. We define and implement a curvelet transform adapted to the limited-view problem and illustrate the advantages of our approach in numerical experiments.}, language = {en} } @article{BringoutErbFrikel, author = {Bringout, Ga{\"e}l and Erb, Wolfgang and Frikel, J{\"u}rgen}, title = {A new 3D model for Magnetic Particle Imaging using realistic magnetic field topologies for algebraic reconstruction}, series = {Inverse Problems}, volume = {36}, journal = {Inverse Problems}, number = {12}, publisher = {IOP Publishing}, doi = {10.1088/1361-6420/abb446}, abstract = {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.}, language = {en} } @unpublished{EbnerFrikelLorenzetal., author = {Ebner, Andrea and Frikel, J{\"u}rgen and Lorenz, Dirk and Schwab, Johannes and Haltmeier, Markus}, title = {Regularization of inverse problems by filtered diagonal frame decomposition}, abstract = {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.}, language = {en} } @inproceedings{FrikelGoeppelHaltmeier, author = {Frikel, J{\"u}rgen and G{\"o}ppel, Simon and Haltmeier, Markus}, title = {Combining Reconstruction and Edge Detection in Computed Tomography}, series = {Bildverarbeitung f{\"u}r die Medizin 2021 : Proceedings, German Workshop on Medical Image Computing: Regensburg, March 7-9, 2021}, booktitle = {Bildverarbeitung f{\"u}r die Medizin 2021 : Proceedings, German Workshop on Medical Image Computing: Regensburg, March 7-9, 2021}, editor = {Palm, Christoph}, publisher = {Springer Vieweg}, address = {Wiesbaden, Germany}, isbn = {978-3-658-33197-9}, doi = {10.1007/978-3-658-33198-6_37}, pages = {153 -- 157}, abstract = {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.}, language = {en} } @article{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Feature Reconstruction from Incomplete Tomographic Data without Detour}, series = {Mathematics}, volume = {10}, journal = {Mathematics}, number = {8}, publisher = {MDPI}, doi = {10.3390/math10081318}, abstract = {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.}, language = {en} } @article{FrikelQuinto, author = {Frikel, J{\"u}rgen and Quinto, Eric Todd}, title = {Limited Data Problems for the Generalized Radon Transform in Rⁿ}, series = {SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, volume = {48}, journal = {SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, number = {4}, publisher = {SIAM}, doi = {10.1137/15M1045405}, pages = {2301 -- 2318}, abstract = {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.}, language = {en} } @article{ErbWeinmannAhlborgetal., author = {Erb, Wolfgang and Weinmann, Andreas and Ahlborg, Mandy and Brandt, Christina and Bringout, Gael and Buzug, Thorsten M. and Frikel, J{\"u}rgen and Kaethner, C. and Knopp, Tobias and M{\"a}rz, T. and Moddel, Martin and Storath, Martin and Weber, A.}, title = {Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging}, series = {Inverse Problems}, volume = {34}, journal = {Inverse Problems}, number = {5}, publisher = {IOP Publishing}, doi = {10.1088/1361-6420/aab8d1}, abstract = {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.}, language = {en} } @article{AgranovskyBomanHasanovetal., author = {Agranovsky, Mark and Boman, Jan and Hasanov, Alemdar and Felea, Raluca and Frikel, J{\"u}rgen and Krishnan, Venky and Novikov, Roman and Ramlau, Ronny and Sebu, Cristiana}, title = {Research biography of a distinguished expert in the field of inverse problems: Professor Eric Todd Quinto}, series = {Journal of Inverse and Ill-posed Problems}, volume = {30}, journal = {Journal of Inverse and Ill-posed Problems}, number = {4}, publisher = {De Gruyter}, issn = {1569-3945}, doi = {10.1515/jiip-2022-0031}, pages = {613 -- 617}, abstract = {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.}, language = {en} } @article{BarannykFrikelNguyen, author = {Barannyk, Lyudmyla L. and Frikel, J{\"u}rgen and Nguyen, Linh V.}, title = {On artifacts in limited data spherical Radon transform: Curved observation surface}, series = {Inverse Problems}, volume = {32}, journal = {Inverse Problems}, number = {1}, publisher = {Institute of Physics}, address = {Bristol [u.a.]}, issn = {1361-6420}, doi = {10.1088/0266-5611/32/1/015012}, abstract = {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.}, language = {de} } @article{Frikel, author = {Frikel, J{\"u}rgen}, title = {Sparse regularization in limited angle tomography}, series = {Applied and Computational Harmonic Analysis}, volume = {34}, journal = {Applied and Computational Harmonic Analysis}, number = {1}, publisher = {Elsevier}, doi = {10.1016/j.acha.2012.03.005}, pages = {117 -- 141}, abstract = {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.}, language = {en} } @article{Frikel, author = {Frikel, J{\"u}rgen}, title = {Short communication: Dimensionality reduction of curvelet sparse regularizations in limited angle tomography}, series = {Proceedings in applied mathematics and mechanics : PAMM}, volume = {11}, journal = {Proceedings in applied mathematics and mechanics : PAMM}, number = {1}, publisher = {Wiley-VCH}, address = {Weinheim}, issn = {1617-7061}, doi = {10.1002/pamm.201110412}, pages = {847 -- 848}, abstract = {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.}, language = {en} } @article{FrikelQuinto, author = {Frikel, J{\"u}rgen and Quinto, Eric Todd}, title = {Artifacts in Incomplete Data Tomography with Applications to Photoacoustic Tomography and Sonar}, series = {SIAM Journal on Applied Mathematics}, volume = {75}, journal = {SIAM Journal on Applied Mathematics}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia, Pa.}, issn = {1095-712X}, doi = {10.1137/140977709}, pages = {703 -- 725}, abstract = {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.}, language = {en} } @article{FrikelQuinto, author = {Frikel, J{\"u}rgen and Quinto, Eric Todd}, title = {Characterization and reduction of artifacts in limited angle tomography}, series = {Inverse Problems}, volume = {29}, journal = {Inverse Problems}, number = {12}, publisher = {IOP Publishing}, address = {Bristol}, issn = {0266-5611}, doi = {10.1088/0266-5611/29/12/125007}, abstract = {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.}, language = {en} } @unpublished{FrikelQuinto, author = {Frikel, J{\"u}rgen and Quinto, Eric Todd}, title = {A paradigm for the characterization of artifacts in tomography}, doi = {10.48550/arXiv.1409.4103}, abstract = {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.}, language = {en} }