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February  2015, 9(1): 105-125. doi: 10.3934/ipi.2015.9.105

Finite-dimensional attractors for the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation in image inpainting

1. 

Université de La Rochelle, Laboratoire Mathématiques, Image et Applications, Avenue Michel Crépeau, F-17042 La Rochelle Cedex

2. 

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

3. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex

Received  August 2013 Revised  May 2014 Published  January 2015

In this article, we are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of a generalization of the Cahn--Hilliard equation with a fidelity term (integrated over $\Omega\backslash D$ instead of the entire domain $\Omega$, $D \subset \subset \Omega$). Such a model has, in particular, applications in image inpainting. The difficulty here is that we no longer have the conservation of mass, i.e. of the spatial average of the order parameter $u$, as in the Cahn--Hilliard equation. Instead, we prove that the spatial average of $u$ is dissipative. We finally give some numerical simulations which confirm previous ones on the efficiency of the model.
Citation: Laurence Cherfils, Hussein Fakih, Alain Miranville. Finite-dimensional attractors for the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation in image inpainting. Inverse Problems and Imaging, 2015, 9 (1) : 105-125. doi: 10.3934/ipi.2015.9.105
References:
[1]

A. Bertozzi, S. Esedoglu, and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936. doi: 10.1137/060660631.

[2]

A. Bertozzi, S. Esedoglu, and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Proc., 16 (2007), 285-291. doi: 10.1109/TIP.2006.887728.

[3]

C. Braverman, Photoshop Retouching Handbook, IDG Books Worldwide, 1998.

[4]

M. Burger, L. He, and C. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imag. Sci., 2 (2009), 1129-1167. doi: 10.1137/080728548.

[5]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[7]

V. Chalupecki, Numerical studies of Cahn-Hilliard equations and applications in image processing, Proceedings of Gzech-Japanese Seminar in Applied Mathematics, Czech Technical University in Prague, (2004), 4-7.

[8]

L. Cherfils, A. Miranville, and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Cont. Dyn. Systems B, 19 (2014), 2013-2026. doi: 10.3934/dcdsb.2014.19.2013.

[9]

L. Cherfils, M. Petcu, and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Cont. Dyn. Systems, 27 (2010), 1511-1533. doi: 10.3934/dcds.2010.27.1511.

[10]

D. Cohen and J. M. Murray, A generalized diffusion model for growth and dispersion in a population, J. Math. Biol., 12 (1981), 237-249. doi: 10.1007/BF00276132.

[11]

I. C. Dolcetta, S. F. Vita, and R. March, Area-preserving curve-shortening flows: From phase separation to image processing, Interfaces Free Bound., 4 (2002), 325-343. doi: 10.4171/IFB/64.

[12]

A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Expenential Attractors for Dissipative Evolution Equations, Vol. 37, John Wiley, New York, 1994.

[13]

M. Efendiev, A. Miranville, and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^{3}$, C.R. Acad. Sci. Paris Série I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[14]

M. Efendiev, A. Miranville, and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nach., 272 (2004), 11-31. doi: 10.1002/mana.200310186.

[15]

C. M. Elliott, D. A. French, and F. A. Milner, A second order splitting method for the Cahn-Hilliard equation, Numer. Math., 54 (1989), 575-590. doi: 10.1007/BF01396363.

[16]

G. Emile-Male, The Restorer's Handbook of Easel Painting,, Van Nostrand Reinold., (). 

[17]

, FreeFem++,, Freely available at , (). 

[18]

E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological application, Phys. Rev. E, 77 (2008), 051129. doi: 10.1103/PhysRevE.77.051129.

[19]

D. King, The Commissar Vanishes, Henry Holt and Company, 1997.

[20]

I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E, 74 (2006), 0319021, 8pp. doi: 10.1103/PhysRevE.74.031902.

[21]

A. C. Kokaram, Motion Picture Restoration: Digital Algorithms for Artefact Suppression in Degraded Motion Picture Film and Video, Springer-Verlag, 1998.

[22]

A. Miranville, Asymptotic behavior of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal., 92 (2013), 1301-1321. doi: 10.1080/00036811.2012.671301.

[23]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations, Evolutionary Partial Differential Equations (eds. C. M. Dafermos and M. Pokorny), Elsevier, Amsterdam, 4 (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[24]

B. Nicolaenko, B. Scheurer, and R. Temam, Some global dynamical properties of a class of pattern formation equations, Commun. Diff. Eqns., 14 (1989), 245-297. doi: 10.1080/03605308908820597.

[25]

A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980. doi: 10.1103/RevModPhys.69.931.

[26]

B. Saoud, Attracteurs Pour Des Systèmes Dissipatifs Non Autonomes, PhD thesis, Université de Poitiers, 2011.

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[28]

U. Thiele and E. Knobloch, Thin liquid films on a slightly inclined heated plate, Phys. D, 190 (2004), 213-248. doi: 10.1016/j.physd.2003.09.048.

[29]

S. Tremaine, On the origin of irregular structure in Saturn's rings, Astron. J., 125 (2003), 894-901. doi: 10.1086/345963.

show all references

References:
[1]

A. Bertozzi, S. Esedoglu, and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936. doi: 10.1137/060660631.

[2]

A. Bertozzi, S. Esedoglu, and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Proc., 16 (2007), 285-291. doi: 10.1109/TIP.2006.887728.

[3]

C. Braverman, Photoshop Retouching Handbook, IDG Books Worldwide, 1998.

[4]

M. Burger, L. He, and C. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imag. Sci., 2 (2009), 1129-1167. doi: 10.1137/080728548.

[5]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[7]

V. Chalupecki, Numerical studies of Cahn-Hilliard equations and applications in image processing, Proceedings of Gzech-Japanese Seminar in Applied Mathematics, Czech Technical University in Prague, (2004), 4-7.

[8]

L. Cherfils, A. Miranville, and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Cont. Dyn. Systems B, 19 (2014), 2013-2026. doi: 10.3934/dcdsb.2014.19.2013.

[9]

L. Cherfils, M. Petcu, and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Cont. Dyn. Systems, 27 (2010), 1511-1533. doi: 10.3934/dcds.2010.27.1511.

[10]

D. Cohen and J. M. Murray, A generalized diffusion model for growth and dispersion in a population, J. Math. Biol., 12 (1981), 237-249. doi: 10.1007/BF00276132.

[11]

I. C. Dolcetta, S. F. Vita, and R. March, Area-preserving curve-shortening flows: From phase separation to image processing, Interfaces Free Bound., 4 (2002), 325-343. doi: 10.4171/IFB/64.

[12]

A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Expenential Attractors for Dissipative Evolution Equations, Vol. 37, John Wiley, New York, 1994.

[13]

M. Efendiev, A. Miranville, and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^{3}$, C.R. Acad. Sci. Paris Série I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[14]

M. Efendiev, A. Miranville, and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nach., 272 (2004), 11-31. doi: 10.1002/mana.200310186.

[15]

C. M. Elliott, D. A. French, and F. A. Milner, A second order splitting method for the Cahn-Hilliard equation, Numer. Math., 54 (1989), 575-590. doi: 10.1007/BF01396363.

[16]

G. Emile-Male, The Restorer's Handbook of Easel Painting,, Van Nostrand Reinold., (). 

[17]

, FreeFem++,, Freely available at , (). 

[18]

E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological application, Phys. Rev. E, 77 (2008), 051129. doi: 10.1103/PhysRevE.77.051129.

[19]

D. King, The Commissar Vanishes, Henry Holt and Company, 1997.

[20]

I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E, 74 (2006), 0319021, 8pp. doi: 10.1103/PhysRevE.74.031902.

[21]

A. C. Kokaram, Motion Picture Restoration: Digital Algorithms for Artefact Suppression in Degraded Motion Picture Film and Video, Springer-Verlag, 1998.

[22]

A. Miranville, Asymptotic behavior of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal., 92 (2013), 1301-1321. doi: 10.1080/00036811.2012.671301.

[23]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations, Evolutionary Partial Differential Equations (eds. C. M. Dafermos and M. Pokorny), Elsevier, Amsterdam, 4 (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[24]

B. Nicolaenko, B. Scheurer, and R. Temam, Some global dynamical properties of a class of pattern formation equations, Commun. Diff. Eqns., 14 (1989), 245-297. doi: 10.1080/03605308908820597.

[25]

A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980. doi: 10.1103/RevModPhys.69.931.

[26]

B. Saoud, Attracteurs Pour Des Systèmes Dissipatifs Non Autonomes, PhD thesis, Université de Poitiers, 2011.

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[28]

U. Thiele and E. Knobloch, Thin liquid films on a slightly inclined heated plate, Phys. D, 190 (2004), 213-248. doi: 10.1016/j.physd.2003.09.048.

[29]

S. Tremaine, On the origin of irregular structure in Saturn's rings, Astron. J., 125 (2003), 894-901. doi: 10.1086/345963.

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