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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), 913-936. 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), 285-291. 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), 1129-1167. doi: 10.1137/080728548.  Google Scholar

[5]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. 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), 258-267. 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, Czech Technical University in Prague, (2004), 4-7. 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), 2013-2026. 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), 1511-1533. 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), 237-249. 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), 325-343. 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, John Wiley, New York, 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), 713-718. 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), 11-31. 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), 575-590. 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), 051129. 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), 0319021, 8pp. 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), 1301-1321. 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, 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.  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), 245-297. 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), 931-980. doi: 10.1103/RevModPhys.69.931.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

S. Tremaine, On the origin of irregular structure in Saturn's rings, Astron. J., 125 (2003), 894-901. 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), 913-936. 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), 285-291. 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), 1129-1167. doi: 10.1137/080728548.  Google Scholar

[5]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. 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), 258-267. 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, Czech Technical University in Prague, (2004), 4-7. 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), 2013-2026. 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), 1511-1533. 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), 237-249. 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), 325-343. 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, John Wiley, New York, 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), 713-718. 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), 11-31. 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), 575-590. 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), 051129. 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), 0319021, 8pp. 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), 1301-1321. 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, 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.  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), 245-297. 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), 931-980. doi: 10.1103/RevModPhys.69.931.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

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

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