Volume 54, Number 6, November-December 2020
|Page(s)||2351 - 2382|
|Published online||16 November 2020|
An optimal transport approach for solving dynamic inverse problems in spaces of measures
University of Graz, Institute of Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria
* Corresponding author: email@example.com
Accepted: 31 July 2020
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.
Mathematics Subject Classification: 65J20 / 49J20 / 35F05 / 46G12 / 92C55
Key words: Dynamic inverse problems / optimal transport regularization / continuity equation / time dependent Bochner spaces / dynamic image reconstruction / dynamic MRI
© EDP Sciences, SMAI 2020
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