  We present a novel variational approach to image restoration (e.g.,
denoising, inpainting, labeling) that enables to complement established
variational approaches with a histogram-based prior enforcing closeness of the
solution to some given empirical measure. By minimizing a single objective
function, the approach utilizes simultaneously two quite different sources of
information for restoration: spatial context in terms of some smoothness prior
and non-spatial statistics in terms of the novel prior utilizing the
Wasserstein distance between probability measures. We study the combination of
the functional lifting technique with two different relaxations of the
histogram prior and derive a jointly convex variational approach. Mathematical
equivalence of both relaxations is established and cases where optimality holds
are discussed. Additionally, we present an efficient algorithmic scheme for the
numerical treatment of the presented model. Experiments using the basic
total-variation based denoising approach as a case study demonstrate our novel
regularization approach.
