  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.
