Summary
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.
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References
Alikakos, N., Bates, P., Chen, X. (1994): Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Rational Mech. Anal. 128, 165–205
Barrett, J.W., Blowey, J.F. (1999): Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility. Math. Comp. 68, 487–517
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, 286–318
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, 233–280
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, 147–179
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, 287–301
Cahn, J.W., Hilliard, I.E. (1958): Free energy of a non-uniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267
Capuzzo Dolcetta, I., Ferretti, R. (2001): Optimal stopping time formulation of adaptive image filtering. Appl. Math. Optim. 43, 245–258
Capuzzo Dolcetta, I., Finzi Vita, S., March, R. (2002): Area-preserving curve-shortening flows: from phase separation to image processing. Interfaces Free Bound. 4, 325–343
Chen, X. (1993): The Hele-Shaw problem and area-preserving curve-shortening motions. Arch. Rational Mech. Anal. 123, 117–151
Chen, X. (1996): Global asymptotic limit of solutions of the Cahn-Hilliard equations. J. Differential Geom. 44, 262–311
Elliott, C.M., French, D.A., Milner, RA. (1989): A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54, 575–590
Elliott, C.M., Garcke, H. (1996): On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423
Marr, D., Hildreth, E. (1980): Theory of edge detection. Proc. Roy. Soc. London Ser. B 207, 187–217
Nochetto, R.H., Paolini, M., Verdi, C. (1996): A dynamic mesh algorithm for curvature dependent evolving interfaces. J. Comput. Phys. 123, 296–310
Pego, R.L. (1989): Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. London Ser A 422, 261–278
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© 2003 Springer-Verlag Italia
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Finzi Vita, S. (2003). A Cahn-Hilliard equation with non-homogeneous mobility and its application to image processing. In: Brezzi, F., Buffa, A., Corsaro, S., Murli, A. (eds) Numerical Mathematics and Advanced Applications. Springer, Milano. https://doi.org/10.1007/978-88-470-2089-4_48
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DOI: https://doi.org/10.1007/978-88-470-2089-4_48
Publisher Name: Springer, Milano
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