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TV based image restoration with local constraints. (English) Zbl 1034.49036
Summary: The problem of recovering an image that has been blurred and corrupted with additive noise is ill-posed. Among the methods that have been proposed to solve this problem, one of the most successful ones is that of constrained Total Variation (TV) image restoration, proposed by L. Rudin, S. Osher, and E. Fatemi [Physica D 60, 259–268 (1992; Zbl 0780.49028)]. In its original formulation, to ensure the satisfaction of constraints, TV restoration requires the estimation of a global parameter $\lambda$ (a Lagrange multiplier). We observe that if $\lambda$ is global, the constraints of the method are also satisfied globally, but not locally. The effect is that the restoration is better achieved in some regions of the image than in others. To avoid this, we propose a variant of the TV restoration model including, instead of a single constraint $\lambda$, a set of constraints ${\lambda }_{i}$, each one corresponding to a region ${O}_{i}$ of the image. We discuss the existence and uniqueness of solutions of the proposed model and display some numerical experiments.
##### MSC:
 49N90 Applications of optimal control and differential games 68U10 Image processing (computing aspects) 49N10 Linear-quadratic optimal control problems 94A08 Image processing (compression, reconstruction, etc.)