  Image inpainting involves filling in damaged or missing regions of an image
by utilizing information from the surrounding areas. In this paper, we
investigate a highly nonlinear partial differential equation inspired by the
modified Cahn-Hilliard equation. Instead of using standard potentials that
depend solely on pixel intensities, we consider morphological image enhancement
filters that are based on a variant of the shock filter: : \begin{align*}
\partial_t u &= \Delta \left(-\nu \arctan(\Delta u)|\nabla u| - \mu \Delta u
\right)+ \lambda(u_0 - u). \end{align*} This is referred to as the Shock Filter
Cahn-Hilliard Equation. This equation is nonlinear with respect to the
second-order derivative, which poses significant mathematical challenges. To
address these, we make use of a specific approximation argument, establishing
the existence of a family of approximate solutions through the Leray-Schauder
fixed point theorem and the Aubin-Lions lemma. In the limit, we obtain a
solution strategy wherein we can prove the existence and uniqueness of
solutions. Proving the latter involves the use of Young measures and Kruzhkov
entropy type-admissibility conditions. Additionally, we use a numerical method
based on the convexity splitting idea to approximate solutions of the nonlinear
partial differential equation and achieve fast inpainting results. To
demonstrate the effectiveness of our approach, we apply our method to standard
binary images and compare it with variations of the Cahn-Hilliard equation
commonly used in the field.
