Measure-Valued Variational Models with Applications to Diffusion-Weighted Imaging
Provides a theoretical foundation for measure-valued variational models, with a specific application to diffusion-weighted imaging, but the numerical results are preliminary and not compared to SOTA.
The paper develops a mathematical framework for variational problems with measure-valued functions, proving existence of minimizers and applying it to restore ODF-valued images in Diffusion MRI with numerical feasibility demonstrated.
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.