SIAM J. IMAGING SCIENCES c ⃝ xxxx Society for Industrial and Applied Mathematics Vol. xx, pp. x x–x Solving QVIs for Image Restoration with Adaptive Constraint Sets

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. Key words. Quasi-variational inequalities, denoising, deblurring, adaptive regularization, total variation regularization, non-convex AMS subject classifications. 49J40, 49J53, 49M30, 49N45, 52A20, 65K15 1. Introduction. Whe