Cahn--Hilliard inpainting with the double obstacle potential

TL;DR

This paper analyzes a Cahn--Hilliard inpainting model with the double obstacle potential, providing the first mathematical results on solution existence and demonstrating its effectiveness for binary and grayscale image restoration.

Contribution

It offers the first analytical results for the double obstacle potential in Cahn--Hilliard inpainting, including existence of solutions, and validates the approach with numerical experiments.

Findings

Existence of global solutions for the time-dependent model.

Existence of stationary solutions without parameter constraints.

Effective inpainting results for binary and grayscale images.

Abstract

The inpainting of damaged images has a wide range of applications, and many different mathematical methods have been proposed to solve this problem. Inpainting with the help of Cahn--Hilliard models has been particularly successful, and it turns out that Cahn--Hilliard inpainting with the double obstacle potential can lead to better results compared to inpainting with a smooth double well potential. However, a mathematical analysis of this approach is missing so far. In this paper we give first analytical results for a Cahn--Hilliard double obstacle inpainting model regarding existence of global solutions to the time-dependent problem and stationary solutions to the time-independent problem without constraints on the parameters involved. With the help of numerical results we show the effectiveness of the approach for binary and grayscale images.