Solving QVIs for Image Restoration with Adaptive Constraint Sets

TL;DR

This paper advances the theory and algorithms for adaptive image restoration using quasi-variational inequalities, proving solution uniqueness independent of image size and providing a convergent numerical method validated by experiments.

Contribution

It extends theoretical results to larger classes of QVIs with guaranteed uniqueness and introduces a convergent numerical algorithm for adaptive image restoration.

Findings

Proved solution uniqueness independent of image size.

Developed a convergent numerical algorithm.

Experimental results support theoretical claims.

Abstract

We consider a class of quasi-variational inequalities (QVIs) for adaptive image restoration, where the adaptivity is described via solution-dependent constraint sets. In previous work we studied both theoretical and numerical issues. While we were able to show the existence of solutions for a relatively broad class of problems, we encountered problems concerning uniqueness of the solution as well as convergence of existing algorithms for solving QVIs. In particular, it seemed that with increasing image size the growing condition number of the involved differential operator poses severe problems. In the present paper we prove uniqueness for a larger class of problems and in particular independent of the image size. Moreover, we provide a numerical algorithm with proved convergence. Experimental results support our theoretical findings.