Convex Variational Image Restoration with Histogram Priors

TL;DR

This paper introduces a convex variational method for image restoration that incorporates a histogram prior using Wasserstein distance, combining spatial smoothness with non-spatial statistical information.

Contribution

It develops a novel convex variational framework integrating histogram priors via Wasserstein distance, with two relaxations and an efficient algorithm for image restoration tasks.

Findings

Effective in denoising with histogram priors

Mathematically equivalent relaxations established

Demonstrated improved restoration results

Abstract

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.