Analysis of a Two-Scale Cahn–Hilliard Model for Binary Image Inpainting

Image inpainting is the process of filling in missing parts of damaged images based on information gleaned from surrounding areas. We consider a model for inpainting binary images using a modified Cahn–Hilliard equation. We prove for the steady state problem that the isophote directions are matched at the boundary of inpainting regions. Our model has two scales, the diffuse interface scale, $\varepsilon$, on which it can accomplish topological transitions, and the feature scale of the image. We show via simulations that a dynamic two-step method involving the diffuse interface scale allows us to connect regions across larger inpainting domains. For the model problem of stripe inpainting, we show that this issue is related to a bifurcation structure with respect to the scale $\varepsilon$.