Solving Quasi-Variational Inequalities for Image Restoration with Adaptive Constraint Sets

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.The work of J.L. was supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST), EPSRC first grant No. EP/J009539/1, and Royal Society International Exchange Award No. IE110314.This is the author accepted manuscript. The final version is available from SIAM via http://dx.doi.org/10.1137/13093834