A Cahn-Hilliard equation with non-homogeneous mobility and its application to image processing

  • S. Finzi Vita
Conference paper


We propose a Cahn-Hilliard type model with a mobility coefficient depending on the position in the reference domain; such a model, which also seems to have a physical motivation in problems of phase separation theory, can be applied to the detection of contours in image processing, since it is able to generate noise filtering and smoothing effects in planar curve evolution with long-time accordance with the initial data. We discuss the theoretical validation of this approach, together with the results of numerical experiments.


Finite Element Approximation Mobility Coefficient Mobility Term Initial Front Double Obstacle 
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  1. [1]
    Alikakos, N., Bates, P., Chen, X. (1994): Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Rational Mech. Anal. 128, 165–205MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Barrett, J.W., Blowey, J.F. (1999): Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility. Math. Comp. 68, 487–517MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Barrett, J.W., Blowey, J.F., Garcke, H. (1999): Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37, 286–318MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Blowey, J.F., Elliott, C.M. (1991): The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis. European J. Appl. Math. 2, 233–280MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Blowey, J.F., Elliott, C.M. (1992): The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. European J. Appl. Math. 3, 147–179MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Cahn, J.W., Elliott, C.M., Novick-Cohen, A. (1996): The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. European J. Appl. Math. 7, 287–301MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Cahn, J.W., Hilliard, I.E. (1958): Free energy of a non-uniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267CrossRefGoogle Scholar
  8. [8]
    Capuzzo Dolcetta, I., Ferretti, R. (2001): Optimal stopping time formulation of adaptive image filtering. Appl. Math. Optim. 43, 245–258MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Capuzzo Dolcetta, I., Finzi Vita, S., March, R. (2002): Area-preserving curve-shortening flows: from phase separation to image processing. Interfaces Free Bound. 4, 325–343MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Chen, X. (1993): The Hele-Shaw problem and area-preserving curve-shortening motions. Arch. Rational Mech. Anal. 123, 117–151MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Chen, X. (1996): Global asymptotic limit of solutions of the Cahn-Hilliard equations. J. Differential Geom. 44, 262–311MathSciNetMATHGoogle Scholar
  12. [12]
    Elliott, C.M., French, D.A., Milner, RA. (1989): A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54, 575–590MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Elliott, C.M., Garcke, H. (1996): On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Marr, D., Hildreth, E. (1980): Theory of edge detection. Proc. Roy. Soc. London Ser. B 207, 187–217CrossRefGoogle Scholar
  15. [15]
    Nochetto, R.H., Paolini, M., Verdi, C. (1996): A dynamic mesh algorithm for curvature dependent evolving interfaces. J. Comput. Phys. 123, 296–310MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Pego, R.L. (1989): Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. London Ser A 422, 261–278MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • S. Finzi Vita
    • 1
  1. 1.Dipartimento di MatematicaUniversita di Roma “La Sapienza”RomaItaly

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