Jürgen Frikel

Low-dose and sparse-angle computed tomography (CT) reduces radiation exposure but makes image reconstruction challenging due to noisy and limited projection data. Popular reconstruction methods are based on two-stage approaches, typically involving filtered backprojection (FBP) followed by a neural network to enhance the image. FBP, however, amplifies noise and struggles with irregular sampling. Therefore, we explore filter-free initial reconstructions, shifting the filtering step to the neural network. In particular, we investigate how two-stage methods can be adapted for cases where implementing explicit filters is difficult, such as with irregular sampling. Specifically, we propose backprojection (BP) or a small number of Landweber iterations as the initial reconstruction, followed by a fine-tuned DRUNet model, referred to as BP-DRUNet and Landweber-DRUNet, respectively. For evaluation, we consider both regular and irregular sampling conditions: For regular sampling, we compare BP-DRUNet with FBP-DRUNet (using FBP as the initial stage) in order to benchmark against standard two-stage approaches. BP-DRUNet performs comparably to FBP-DRUNet under regular sampling. In irregular sampling, Landweber-DRUNet improves reconstruction quality with more iterations, though at the cost of longer training and inference times. Experiments are carried out on synthetic and real CT datasets with parallel- and fan-beam acquisitions across different sparse-angle setups.

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) topo 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) topo

Inverse problems are inherently ill-posed and therefore require regularization techniques to achieve a stable solution. While traditional variational methods have well-established 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.

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 decomposition, which are well adapted to the underlying signal class and the forward model and furthermore 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 translation invariant diagonal frame decomposition (TI-DFD) of linear operators as a novel concept generalizing the singular value decomposition. We characterize ill-posedness via the TI-DFD and 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.

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 furthermore 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 translation invariant diagonal frame decomposition (TI-DFD) of linear operators as a novel concept generalizing the SVD. We characterize ill-posedness via the TI-DFD and 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.