@unpublished{EbnerFrikelLorenzetal., author = {Ebner, Andrea and Frikel, J{\"u}rgen and Lorenz, Dirk and Schwab, Johannes and Haltmeier, Markus}, title = {Regularization of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} } @techreport{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Translation invariant diagonal frame decomposition of inverse problems and their regularization}, doi = {10.48550/arXiv.2208.08500}, abstract = {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.}, 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} } @article{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Regularization of Inverse Problems with Translation Invariant Frames}, series = {Proceedings of the 20th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM-2022), 2022, Heraklion (Crete, Greece)}, journal = {Proceedings of the 20th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM-2022), 2022, Heraklion (Crete, Greece)}, publisher = {AIP Publishing}, abstract = {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.}, 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} } @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{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{EbnerFrikelLorenzetal., author = {Ebner, Andrea and Frikel, J{\"u}rgen and Lorenz, D. and Schwab, J. and Haltmeier, Markus}, title = {Regularization of inverse problems by filtered diagonal frame decomposition}, series = {Applied and Computational Harmonic Analysis}, volume = {62}, journal = {Applied and Computational Harmonic Analysis}, number = {January}, publisher = {Elsevier}, doi = {10.1016/j.acha.2022.08.005}, pages = {66 -- 83}, abstract = {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).}, language = {en} } @unpublished{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Data-proximal null-space networks for inverse problems}, abstract = {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.}, language = {en} } @unpublished{GoeppelFrikelHaltmeier, author = {G{\"o}ppel, Simon and Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Translation invariant diagonal frame decomposition for the Radon transform}, abstract = {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.}, 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} } @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 = {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} }