We develop a general mathematical framework for variational problems where the
unknown function takes 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.