Measure-Valued Variational Models with Applications to Diffusion-Weighted Imaging

TL;DR

This paper introduces a mathematical framework for variational problems with functions valued in probability measures, applying it to improve diffusion MRI imaging through a novel optimal transport-based approach.

Contribution

It develops a new total variation framework for measure-valued functions and applies it to diffusion MRI, demonstrating practical numerical feasibility.

Findings

Existence of minimizers in the proposed variational framework.

Application of the framework to diffusion MRI data.

Numerical feasibility demonstrated on multiple data sets.

Abstract

We develop a general mathematical framework for variational problems where the unknown function assumes values in the space of probability measures on some metric space. We study weak and strong topologies and define a total variation seminorm for functions taking values in a Banach space. The seminorm penalizes jumps and is rotationally invariant under certain conditions. We prove existence of a minimizer for a class of variational problems based on this formulation of total variation, and provide an example where uniqueness fails to hold. Employing the Kan\-torovich-Rubinstein transport norm from the theory of optimal transport, we propose a variational approach for the restoration of orientation distribution function (ODF)-valued images, as commonly used in Diffusion MRI. We demonstrate that the approach is numerically feasible on several data sets.