Measure-Valued Variational Models with Applications to Diffusion-Weighted Imaging
arXiv:1710.00798 · doi:10.1007/s10851-018-0827-8
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.
Accepted by Journal of Mathematical Imaging and Vision (SSVM 2017 special issue)