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A Cahn–Hilliard System with a Fidelity Term for Color Image Inpainting

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Abstract

In this paper, we propose a model for multi-color image inpainting composed of n colors. In particular, as in the binary model, i.e., the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation (Bertozzi et al. in IEEE Trans Image Proc 16:285–291, 2007, Multiscale Model Simul 6:913–936, 2007), we add a fidelity term to the corresponding Cahn–Hilliard system. We are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of the dynamical system associated with the problem. The main difficulty here is that we no longer have the conservation of mass, i.e., of the spatial average of the order parameter c, as in the Cahn–Hilliard system. Instead, we prove that the spatial average of c is dissipative. We finally give numerical simulations which confirm and extend previous ones on the efficiency of the binary model.

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Acknowledgments

The authors wish to thank the referees for their careful reading of the article and useful comments.

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Authors and Affiliations

  1. Laboratoire Mathématiques, Image et Applications, Université de la Rochelle, Avenue Michel Crépeau, 17042, La Rochelle Cedex, France

    Laurence Cherfils

  2. Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Université de Poitiers, Boulevard Marie et Pierre Curie - Téléport 2, 86962, Chasseneuil Futuroscope Cedex, France

    Hussein Fakih & Alain Miranville

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  1. Laurence Cherfils
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  2. Hussein Fakih
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  3. Alain Miranville
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Correspondence to Alain Miranville.

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Cherfils, L., Fakih, H. & Miranville, A. A Cahn–Hilliard System with a Fidelity Term for Color Image Inpainting. J Math Imaging Vis 54, 117–131 (2016). https://doi.org/10.1007/s10851-015-0593-9

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  • DOI: https://doi.org/10.1007/s10851-015-0593-9

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