TY - JOUR A1 - Erb, Wolfgang A1 - Weinmann, Andreas A1 - Ahlborg, Mandy A1 - Brandt, Christina A1 - Bringout, Gael A1 - Buzug, Thorsten M. A1 - Frikel, Jürgen A1 - Kaethner, C. A1 - Knopp, Tobias A1 - März, T. A1 - Moddel, Martin A1 - Storath, Martin A1 - Weber, A. T1 - Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging JF - Inverse Problems N2 - 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. KW - FORMULATION KW - ill-posedness KW - INTERPOLATION KW - inverse problems KW - magnetic particle imaging KW - model-based reconstruction KW - MPI KW - NODES KW - reconstruction operator KW - RELAXATION KW - RESOLUTION Y1 - 2018 U6 - https://doi.org/10.1088/1361-6420/aab8d1 VL - 34 IS - 5 PB - IOP Publishing 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 - CHAP A1 - Selig, Tim A1 - Bauer, Patrick A1 - Frikel, Jürgen A1 - März, Thomas A1 - Storath, Martin A1 - Weinmann, Andreas ED - Palm, Christoph ED - Breininger, Katharina ED - Deserno, Thomas M. ED - Handels, Heinz ED - Maier, Andreas ED - Maier-Hein, Klaus H. ED - Tolxdorff, Thomas M. T1 - Two-stage Approach for Low-dose and Sparse-angle CT Reconstruction using Backprojection T2 - Bildverarbeitung für die Medizin 2025 (BVM 2025): Proceedings, German Conference on Medical Image Computing, Regensburg March 09-11, 2025 N2 - This paper presents a novel two-stage approach for computed tomography (CT) reconstruction, focusing on sparse-angle and low-dose setups to minimize radiation exposure while maintaining high image quality. Two-stage approaches consist of an initial reconstruction followed by a neural network for image refinement. In the initial reconstruction, we apply the backprojection (BP) instead of the traditional filtered backprojection (FBP). This enhances computational speed and offers potential advantages for more complex geometries, such as fan-beam and cone-beam CT. Additionally, BP addresses noise and artifacts in sparse-angle CT by leveraging its inherent noise-smoothing effect, which reduces streaking artifacts common in FBP reconstructions. For the second stage, we fine-tune the DRUNet proposed by Zhang et al. to further improve reconstruction quality. We call our method BP-DRUNet and evaluate its performance on a synthetically generated ellipsoid dataset alongside thewell-established LoDoPaBCT dataset. Our results show that BP-DRUNet produces competetive results in terms of PSNR and SSIM metrics compared to the FBP-based counterpart, FBPDRUNet, and delivers visually competitive results across all tested angular setups. Y1 - 2025 SN - 978-3-658-47421-8 U6 - https://doi.org/10.1007/978-3-658-47422-5_67 SP - 286 EP - 291 PB - Springer VS CY - Wiesbaden ER -