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| Abstract: The Cahn--Hilliard equation provides a simple and fast tool for binary image inpainting. By now two generalizations to gray value images exist: Bitwise binary inpainting and TV-H$^{-1}$ inpainting. This paper outlines a model based on the vector-valued Cahn--Hilliard equation. Additionally we generalize our approach to a fractional-in-space version. Fourier spectral methods provide efficient solvers since they yield a fully diagonal scheme. Furthermore their application to three spatial dimensions is straightforward. Numerical examples show the superiority of the fractional approach over the classical one. It improves the peak signal-to-noise ratio and structural similarity index. Likewise the experiments confirm that the proposed model competes with previous inpainting methods such as the total variation inpainting approach and its fourth-order variant. |
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