**Solving Quasi-Variational Inequalities for Adaptive Total Variation Regularization**

Dr. Frank Lenzen, University of Heidelberg** **

**Abstract**

Due to its property to encourage piecewise constant solutions, Total Variation (TV) plays an important role as regularizer for various image restoration tasks.

To even further improve the restoration quality of TV regularization, various adaptive methods have been proposed in literature. Among these, we find the class of direction-dependent TV methods, which we refer to as anisotropic TV methods.

The main focus of the adaptive methods are e.g. the preservation of contrast, edges, corners, slopes and fine image structures.

To steer the adaptivity, additional information is required. We therefore can distinguish between data-driven approaches, where this information is taken from the input image or additional data, and solution-driven approaches, where the adaptivity is depending on the solution of the problem.

We also remark that, while the classical TV regularization is convex, some of the proposed approaches in literature yield non-convex problems. While in view of the restoration quality these non-convex methods show advantages, their theoretical and numerical treatment comes with some difficulties.

In our talk we discuss a method which implements solution driven adaptivity in terms of a fixed point problem, where the unknown on the one hand is the sought solution to the problem and on the other hand steers the adaptivity.

Reformulating the fixed-point problem as a Quasi-Variational Inequality Problem (QVIP) enables us to provide theoretical results for existence and uniqueness of the fixed point. Deriving a solution of the QVIP amounts to solving a sequence of convex sub-problems.

Therefore, our strategy combines advantages of convex and non-convex approaches, since on the one hand we can built on the convexity of the sub-problems, on the other hand achieve a performance similar to non-convex regularizers.

As applications, we exemplarily consider image denoising, (non-blind) deblurring and inpainting.