The aim of this paper is to provide a solid mathematical discussion of the inverse problem in Magnetorelaxometry Imaging (MRXI), a currently developed technique for quantitative biomedical imaging using magnetic nanoparticles. We provide a detailed discussion of the mathematical modeling of the forward problems including possible ways to activate and measure, leading to a severely ill-posed linear inverse problem. Moreover, we formulate an idealized version of the inverse problem for infinitesimal small activation coils, which allows for a more detailed analysis of uniqueness issues.
We propose a variational regularization approach to compute stable approximations of the solution and discuss its discretization and numerical solution. Results on synthetic are presented and improvements to methods used previously in practice are demonstrated. Finally we give an outlook to further questions and in particular experimental design.
Douglas-Rachford Algorithm for Magnetorelaxometry Imaging using Random and Deterministic Activations
University of Innsbruck（因斯布鲁克大学）；Private University for Health Sciences, Medical Informatics and Technology (UMIT)（健康科学、医学信息学和技术私立大学（UMIT））；Technische Universität Ilmenau（伊尔梅瑙工业大学）
Inverse transport problems in quantitative PAT for molecular imaging
荧光光声层析成像（fluorescence photoacoustic tomography，fPAT）是一种将荧光成像与光声层析成像相结合的分子成像技术，可获得异质介质内部荧光分布的高分辨率成像。这项工作的目的是研究反问题的定量步骤fPAT，我们打算重建的辐射输运方程的耦合系统使用内部数据从超声测量恢复的物理系数。我们得到的唯一性和稳定性的反问题的结果，并开发了一些有效的算法，图像重建。基于合成数据的数值模拟验证了理论分析。本文的结果补充了[Ren-Zhao，SIAM J. Imag.科学，[2013]在同一个问题上，但在扩散制度。
A direct imaging method for inverse scattering by unbounded rough surfaces
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers
Zhejiang University（浙江大学）；Taiyuan University of Technology（太原理工大学）；Purdue University（普渡大学）；Institute for Mathematics and its Applications, Minneapolis, MN 55455, USA（明尼苏达大学数学及其应用研究所）
A direct imaging method for inverse elastic scattering by unbounded rigid rough surfaces
Beijing Computational Science Research Center（北京计算科学研究中心）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；LSEC, NCMIS（计算数学与科学工程计算研究所，国家数学与交叉科学中心）
Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；NCMIS（数学与应用数学国家重点实验室）；LSEC（计算数学与科学工程计算研究所）
A Direct Imaging Method for Half-Space Inverse Elastic Scattering Problems
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Science, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Inverse diffraction for the Atmospheric Imaging Assembly in the Solar Dynamics Observatory
Università di Genova（热那亚大学）；Catholic University of America（美国天主教大学）；NASA Goddard Space Flight Center（美国国家航空航天局戈达德太空飞行中心）；Ecole Polytechnique（巴黎综合理工学院）；CNR - SPIN Genova（意大利国家研究委员会-SPIN热那亚）
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）
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
NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems
Shanghai Jiao Tong University（上海交通大学）；University of Birmingham（伯明翰大学）；Indian Institute of Technology (IIT) Kharagpur（印度理工学院卡拉格布尔分校）；University of Cambridge（剑桥大学）
