A class of quasi‐variational inequalities for adaptive image denoising and decomposition

We introduce a class of adaptive non‐smooth convex variational problems for image denoising in terms of a common data fitting term and a support functional as regularizer. Adaptivity is modeled by a set‐valued mapping with closed, compact and convex values, that defines and steers the regularizer depending on the variational solution. This extension gives rise to a class of quasi‐variational inequalities. We provide sufficient conditions for the existence of fixed points as solutions, and an algorithm based on solving a sequence of variational problems. Denoising experiments with spatial and spatio‐temporal image data and an adaptive total variation regularizer illustrate our approach.