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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 & 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. doi: 10.1137/060660631. Google Scholar

[2]

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

[3]

C. Braverman, Photoshop Retouching Handbook,, IDG Books Worldwide, (1998). Google Scholar

[4]

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

[5]

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

[6]

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

[7]

V. Chalupecki, Numerical studies of Cahn-Hilliard equations and applications in image processing,, Proceedings of Gzech-Japanese Seminar in Applied Mathematics, (2004), 4. Google Scholar

[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. doi: 10.3934/dcdsb.2014.19.2013. Google Scholar

[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. doi: 10.3934/dcds.2010.27.1511. Google Scholar

[10]

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

[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. doi: 10.4171/IFB/64. Google Scholar

[12]

A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Expenential Attractors for Dissipative Evolution Equations,, Vol. 37, (1994). Google Scholar

[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. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar

[14]

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

[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. doi: 10.1007/BF01396363. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

D. King, The Commissar Vanishes,, Henry Holt and Company, (1997). Google Scholar

[20]

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

[21]

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

[22]

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

[23]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations, 4 (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar

[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. doi: 10.1080/03605308908820597. Google Scholar

[25]

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

[26]

B. Saoud, Attracteurs Pour Des Systèmes Dissipatifs Non Autonomes,, PhD thesis, (2011). Google Scholar

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[28]

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

[29]

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

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. doi: 10.1137/060660631. Google Scholar

[2]

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

[3]

C. Braverman, Photoshop Retouching Handbook,, IDG Books Worldwide, (1998). Google Scholar

[4]

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

[5]

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

[6]

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

[7]

V. Chalupecki, Numerical studies of Cahn-Hilliard equations and applications in image processing,, Proceedings of Gzech-Japanese Seminar in Applied Mathematics, (2004), 4. Google Scholar

[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. doi: 10.3934/dcdsb.2014.19.2013. Google Scholar

[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. doi: 10.3934/dcds.2010.27.1511. Google Scholar

[10]

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

[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. doi: 10.4171/IFB/64. Google Scholar

[12]

A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Expenential Attractors for Dissipative Evolution Equations,, Vol. 37, (1994). Google Scholar

[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. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar

[14]

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

[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. doi: 10.1007/BF01396363. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

D. King, The Commissar Vanishes,, Henry Holt and Company, (1997). Google Scholar

[20]

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

[21]

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

[22]

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

[23]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations, 4 (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar

[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. doi: 10.1080/03605308908820597. Google Scholar

[25]

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

[26]

B. Saoud, Attracteurs Pour Des Systèmes Dissipatifs Non Autonomes,, PhD thesis, (2011). Google Scholar

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[28]

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

[29]

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

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