Cahn--Hilliard inpainting with the double obstacle potential
This work addresses image restoration for applications like art conservation or digital photography, but it is incremental as it builds on existing Cahn-Hilliard inpainting methods.
The paper tackled the problem of image inpainting using a Cahn-Hilliard model with a double obstacle potential, providing first analytical results for global and stationary solutions without parameter constraints, and demonstrated effectiveness through numerical tests on binary and grayscale images.
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.