Learned Finite Element-based Regularization of the Inverse Problem in Electrocardiographic Imaging
Institute for Applied Mathematics, University of Bonn, Germany（波恩大学应用数学研究所）；Department of Mathematics and Scientific Computing, University of Graz, Austria（格拉茨大学数学与科学计算系）；Institute of Artificial Intelligence, Medical University of Vienna, Austria（维也纳医学大学人工智能研究所）；Heart Center Bonn, Department of Internal Medicine II, University Hospital Bonn, Germany（波恩心脏中心，波恩大学医院第二内科部）；Department of Mathematics, University of Trento, Italy（特伦托大学数学系）
Manuel Haas, Thomas Grandits, Thomas Pinetz, Thomas Beiert, Simone Pezzuto, Alexander Effland
Electrocardiographic imaging (ECGI) seeks to reconstruct cardiac electrical activity from body-surface potentials noninvasively. However, the associated inverse problem is severely ill-posed and requires robust regularization. While classical approaches primarily employ spatial smoothing, the temporal structure of cardiac dynamics remains underexploited despite its physiological relevance. We introduce a space-time regularization framework that couples spatial regularization with a learned temporal Fields-of-Experts (FoE) prior to capture complex spatiotemporal activation patterns. We derive a finite element discretization on unstructured cardiac surface meshes, prove Mosco-convergence, and develop a scalable optimization algorithm capable of handling the FoE term. Numerical experiments on synthetic epicardial data demonstrate improved denoising and inverse reconstructions compared to handcrafted spatiotemporal methods, yielding solutions that are both robust to noise and physiologically plausible.
Finite element-based space-time total variation-type regularization of the inverse problem in electrocardiographic imaging
Institute of Applied Mathematics, University of Bonn, Germany（波恩大学应用数学研究所）；Euler Institute, Università della svizzera italiana, Switzerland（瑞士意大利欧拉研究所；格拉茨大学数学与计算科学系）；Department of Mathematics and Scientific Computing, University of Graz, Austria（波恩心脏中心，第二内科部，波恩大学医院）；Heart Center Bonn, Department of Internal Medicine II, University Hospital Bonn, Germany（特伦托大学数学系）；Department of Mathematics, University of Trento, Italy
A remark on an error analysis for classical and learned Tikhonov regularization schemes
Center for Industrial Mathematics, University of Bremen（不来梅大学工业数学中心）；LMAM, School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Faculty of Mathematics, Chemnitz University of Technology（开姆尼茨工业大学数学学院）
Total Generalized Variation for Inverse Problems Involving Piecewise Linear Finite Elements
Institute for Applied Mathematics, University of Bonn（波恩大学应用数学研究所）；Heart Center Bonn, Department of Internal Medicine II, University Hospital Bonn（波恩大学医院第二内科心脏中心）；Department of Mathematics, University of Trento（特伦托大学数学系）
