Calculus of Variations and Geometric Measure Theory
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K. Bredies - S. Fanzon

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

created by fanzon on 01 Mar 2019
modified on 01 Aug 2020


Published Paper

Inserted: 1 mar 2019
Last Updated: 1 aug 2020

Journal: ESAIM: Mathematical Modelling and Numerical Analysis
Year: 2020

ArXiv: 1901.10162 PDF


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 bases on dynamic 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), modeling relevant examples of time varying acquisition strategies, as well as patient motion and presence of contrast agents.


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