  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.
