Existence theory for the EED inpainting problem

An existence theory is developed for an elliptic boundary value problem in image analysis known as edge-enhancing diffusion (EED) inpainting. The EED inpainting problem aims at restoration missing data in an image as the steady state of a nonlinear anisotropic diffusion process where the known data provide Dirichlet boundary conditions. The existence of a weak solution is established by applying the Leray–Schauder fixed point theorem, and it is shown that the set of all possible weak solutions is bounded. Moreover, it is demonstrated that under certain conditions the sequences resulting from iterative application of the operator from the existence theory contain convergent subsequences.