Some variants of the Cahn-Hilliard equation for image inpainting
Hussein Fakih1,2,3, Noura Nasreddine4, Salwa Mansour1,2, Ragheb Mghames1,3
1Lebanese International University (LIU), Department of Mathematics and Physics, Nabatieh campus, Lebanon
2Lebanese International University (LIU), Department of Mathematics and Physics, Beirut campus
3Lebanese International University (LIU), Khawarizmi Laboratory for Mathematics and Applications, Bekaa campus
4The University of Texas at Dallas, School of Natural Sciences and Mathematics, Texas, Canada
Keywords: complex Cahn-Hilliard equation, Cahn-Hilliard systems, fidelity term, image inpainting, steady state, stability, numerical simulations
Abstract
In this article, we are interested in the complex version of Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation for grayscale image inpainting as well as the multi-component Cahn-Hilliard systems for image inpainting, that is an extension approach for color image inpainting. We have studied well-posedness of the steady state problem associated to the complex Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation as well as to Bertozzi-Esedoglu-Gillete-Cahn Hilliard systems. Then, Backward Euler discretization on time has been considered in both models mentioned above. We were able to prove the stability of the backward Euler scheme. Finally, we do some numerical simulations that confirm the theoretical results and show the efficiency of the scheme. The simulations were done using FreeFem++.
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