Optimal Sparse Control Formulation for Reconstruction of Noise-Affected Images

Abstract

We discuss the optimal control formulation for enhancement and denoising of satellite multiband images and propose to take it in the form of an L1 control problem for a quasi-linear parabolic equation with a nonlocal p[u] Laplacian and with a cost functional of a tracking type. The main characteristic features of the considered parabolic problem is that the variable exponent p(t,x) and the diffusion anisotropic tensor D(t,x) are not predefined well a priori; instead, these characteristics nonlocally depend on the form of the solution of this problem (i.e., pu=p(t,x,u) and Du=D(t,x,u)). We prove the existence of optimal pairs with sparse L1 controls used for the indirect approach and a special family of approximation problems.