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Titel
An optimal transport approach for solving dynamic inverse problems in spaces of measures
Verfasser/ VerfasserinBredies, Kristian ; Fanzon, Silvio
KörperschaftFonds zur Förderung der Wissenschaftlichen Forschung
Enthalten in
ESAIM: Mathematical Modelling and Numerical Analysis, 54 (2020), 6, S. 2351-2382
ErschienenEDP Sciences, 2020
Ausgabe
Accepted Version
VersionAccepted version
SpracheEnglisch
DokumenttypAufsatz in einer Zeitschrift
Schlagwörter (EN)dynamic inverse problems / optimal transport regularization / continuity equation / time dependent Bochner spaces / dynamic image reconstruction / dynamic MRI
ISSN0764-583X
URNurn:nbn:at:at-ubg:3-8776 
DOI10.1051/m2an/2020056 
Fördergeber
Fonds zur Förderung der Wissenschaftlichen Forschung
Projekt-/Reportnummer: PIR 27
Zugriffsbeschränkung
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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.

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