Shock filter as the classifier for image inpainting problem using the Cahn-Hilliard equation

We consider a problem of digital image inpainting using the modified Cahn-Hilliard equation, where the standard double well potential is replaced by the shock filter. Using fixed point arguments and Aubin-Lions lemma we prove the existence and uniqueness of the solution. In addition, we introduce a numerical method based on the convexity splitting idea to approximate the solutions of the considered problem. We apply this method to several binary images and demonstrate that this approach naturally extends image features and preserves their edges.