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

## TL;DR

This paper introduces a novel regularization method for dynamic inverse problems using optimal transport, enabling the recovery of measure-valued curves in time-dependent data spaces, with applications in dynamic MRI.

## Contribution

It develops a functional-analytic framework for optimal transport-based regularization of dynamic inverse problems, proving existence, uniqueness, and regularization properties of solutions.

## Key findings

- Established existence and uniqueness of minimizers in certain cases.
- Applied the framework to dynamic MRI reconstruction with promising results.
- Modeled time-varying acquisition, motion, and contrast agent effects.

## 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.