An optimal transport approach for solving dynamic inverse problems in spaces of measures

Author:

Bredies KristianORCID,Fanzon SilvioORCID

Abstract

In this paper we propose and study a novel optimal transport based regularization of linear dynamic inverse problems. The considered inverse problems aim at recovering a measure valued curve and are dynamic in the sense that (i) the measured data takes values in a time dependent family of Hilbert spaces, and (ii) the forward operators are time dependent and map, for each time, Radon measures into the corresponding data space. The variational regularization we propose is based on dynamic (un-)balanced optimal transport which means that the measure valued curves to recover (i) satisfy the continuity equation, i.e., the Radon measure at time t is advected by a velocity field v and varies with a growth rate g, and (ii) are penalized with the kinetic energy induced by v and a growth energy induced by g. We establish a functional-analytic framework for these regularized inverse problems, prove that minimizers exist and are unique in some cases, and study regularization properties. This framework is applied to dynamic image reconstruction in undersampled magnetic resonance imaging (MRI), modelling relevant examples of time varying acquisition strategies, as well as patient motion and presence of contrast agents.

Funder

Austrian Science Fund

Christian Doppler Forschungsgesellschaft

Publisher

EDP Sciences

Subject

Applied Mathematics,Modelling and Simulation,Numerical Analysis,Analysis,Computational Mathematics

Reference67 articles.

1. $rm K$-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation

2. A uniqueness result for the continuity equation in two dimensions

3. Aliprantis C.D. and Border K., Infinite Dimensional Analysis. Springer-Verlag, Berlin, Heidelberg (2006).

4. An abstract framework for parabolic PDEs on evolving spaces

5. Ambrosio L., Fusco N. and Pallara D., Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications. Clarendon Press (2000).

Cited by 9 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3