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Image inpainting involves filling in damaged or missing regions of an imageby utilizing information from the surrounding areas. In this paper, weinvestigate a highly nonlinear partial differential equation inspired by themodified Cahn-Hilliard equation. Instead of using standard potentials thatdepend solely on pixel intensities, we consider morphological image enhancementfilters 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 FilterCahn-Hilliard Equation. This equation is nonlinear with respect to thesecond-order derivative, which poses significant mathematical challenges. Toaddress these, we make use of a specific approximation argument, establishingthe existence of a family of approximate solutions through the Leray-Schauderfixed point theorem and the Aubin-Lions lemma. In the limit, we obtain asolution strategy wherein we can prove the existence and uniqueness ofsolutions. Proving the latter involves the use of Young measures and Kruzhkoventropy type-admissibility conditions. Additionally, we use a numerical methodbased on the convexity splitting idea to approximate solutions of the nonlinearpartial differential equation and achieve fast inpainting results. Todemonstrate the effectiveness of our approach, we apply our method to standardbinary images and compare it with variations of the Cahn-Hilliard equationcommonly used in the field.

Navigating the Complex Landscape of Shock Filter Cahn-Hilliard Equation: From Regularized to Young Measure Solutions