The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting
Moez Kallel, Maher Moakher, Anis Theljani · Inverse Problems and Imaging · 2015
Image inpainting or disocclusion, which refers to the process of restoringa damaged image with missing information, has many applications in differentfields. Different techniques can be applied to solve this problem. Inparticular, many variational models have appeared in the literature. Thesemodels give rise to partial differential equations for which Dirichletboundary conditions are usually used. The basicidea of the algorithms that have been proposed in the literature is tofill-in damaged regions with available information from theirsurroundings. The aim of this work is to treat the case wherethis information is not available in a part of the boundary ofthe damaged region. We formulate the image inpainting problem as a nonlinearCauchy problem. Then, we give a Nash-game formulation ofthis Cauchy problem and we present different numerical experiments usingthe finite-element method for solving the image inpainting problem.