Journal of Mathematical Imaging and Vision

, Volume 54, Issue 1, pp 117–131 | Cite as

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

  • Laurence Cherfils
  • Hussein Fakih
  • Alain Miranville


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.


Cahn–Hilliard system Fidelity term Color image inpainting Well-posedness Exponential attractor Simulations 

Mathematics Subject Classification

35B40 35K55 68M07 80A23 



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


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Laurence Cherfils
    • 1
  • Hussein Fakih
    • 2
  • Alain Miranville
    • 2
  1. 1.Laboratoire Mathématiques, Image et ApplicationsUniversité de la RochelleLa Rochelle CedexFrance
  2. 2.Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MIUniversité de PoitiersChasseneuil Futuroscope CedexFrance

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