Convex Variational Image Restoration with Histogram Priors
This work addresses image restoration for applications like denoising and inpainting, presenting an incremental improvement by integrating histogram priors into existing variational methods.
The paper tackles image restoration by combining spatial smoothness priors with a novel histogram-based prior using Wasserstein distance, resulting in a convex variational model that is mathematically analyzed and experimentally validated for denoising.
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.