A Complex Version of the Cahn-Hilliard Equation for Grayscale Image Inpainting

Our aim in this article is to propose a generalization of the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation, introduced for binary image inpainting, for grayscale image inpainting. In particular, we consider the solution to the corresponding Cahn--Hilliard inpainting model as a complex valued function. We are interested in the study of the well-posedness and of the asymptotic behavior, in terms of finite-dimensional attractors, of the associated dynamical system. We have to face two major difficulties here. The first one comes from the fact that we no longer have the conservation of mass, i.e., of the spatial average of the order parameter $u$, contrary to the classical Cahn--Hilliard equation. The second one is due to the estimates on the nonlinear terms, combined with the fact that the order parameter $u$ is complex valued. We finally give numerical simulations which confirm and extend previous ones on the efficiency of the binary model.