Caputo fractional-time of a modified Cahn-Hilliard equation for the inpainting of binary images

Document Type : Research Article

Authors

1 Department of mathematics, Faculty of Sciences and Technologies Mohammedia, University Hassan II, Casablanca, Morocco.

2 LAMAI laboratory, university of Cadi Ayyad, Faculty of sciences and technology, Marrakesh, Morocco

Abstract

In this work, we present a new version of the Cahn-Hilliard equation to deal with binary image inpainting. The proposed model is unique due to its memory effect ability implemented by the time fractional derivative. Also, this model has a new diffusion term that gives a topological reconnection and a well sharpness of edges and corners. We give an existence result with some numerical tests implemented by the convexity splitting to show the efficiency of the proposed model.

Keywords

Main Subjects

References

[1] A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations, Differ. Equ. 46 (2010) 660–666.
[2] H. Attouch, G. Buttazzo, G. Michaille, Variational analysis in sobolev and bv spaces: applications to pdes and optimization, SIAM, 2014.
[3] A. Babaei, Solving a time-fractional inverse heat conduction problem with an unknown nonlinear boundary condition, J. Math. Model. 7 (2019) 85–106.
[4] B. Beisner, D. Haydon, K. Cuddington, Hysteresis, Encycl. Ecol. (2008) 1930-1935.
[5] A. Ben-Loghfyry, A. Hakim, Time-fractional diffusion equation for signal and image smoothing, Math. Model. Comput. 9 (2022) 351–364.
[6] A. Ben-loghfyry, A. Hakim, Reaction-diffusion equation based on fractional-time anisotropic diffusion for textured images recovery, Int. J. Comput. Math. 8 (2022) 177.
[7] A. Ben-loghfyry, A. Hakim, Robust time-fractional diffusion filtering for noise removal, Math. Methods Appl. Sci. 45 (2022) 9719–9735.
[8] A. Ben-Loghfyry, A. Hakim, A. Laghrib, A denoising model based on the fractional beltrami regularization and its numerical solution, J. Appl. Math. Comput. (2022) 1–33.
[9] M. Bertalmio, A.L. Bertozzi, G. Sapiro, Navier-stokes, fluid dynamics, and image and video inpainting, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001, vol. 1, IEEE, 2001, pp. I–I.
[10] M. Bertalmio, G. Sapiro, V. Caselles, C. Ballester, Image inpainting, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, (2000) 417–424.
[11] A. Bertozzi, S. Esedoglu, A. Gillette, Analysis of a two-scale cahn–hilliard model for binary image inpainting, Multiscale Model. Simul. 6 (2007) 913–936.
[12] A. Bertozzi, C.B. Schönlieb, Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci. 9 (2011) 413–457.
[13] A.L. Bertozzi, S. Esedoglu, A. Gillette, Inpainting of binary images using the cahn–hilliard equation, IEEE Trans. Image Process. 16 (2006) 285–291.
[14] J.F. Blowey, C.M. Elliott, The cahn–hilliard gradient theory for phase separation with non-smooth free energy part i: Mathematical analysis, Eur. J. Appl. Math. 2 (1991) 233–280.
[15] J.F. Blowey, C.M. Elliott, The cahn–hilliard gradient theory for phase separation with non-smooth free energy part ii: numerical analysis, Eur. J. Appl. Math. 3 (1992) 147–179.
[16] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, 2010.
[17] M. Burger, L. He, C.B. Schönlieb, Cahn–hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci. 2 (2009) 1129–1167.
[18] J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. i. interfacial free energy, Chem. Phys. 28 (1958) 258–267.
[19] M. Caputo, W. Plastino, Diffusion in porous layers with memory, Geophys. J. Int. 158 (2004) 385–396.
[20] L. Chen, F. Ghanbarnejad, D. Brockmann, Fundamental properties of cooperative contagion processes, New J. Phys. 19 (2017) 103041.
[21] L. Cherfils, H. Fakih, A. Miranville, A cahn–hilliard system with a fidelity term for color image inpainting, J. Math. Imaging Vis. 54 (2016) 117–131.
[22] L. Cherfils, H. Fakih, A. Miranville, A complex version of the cahn–hilliard equation for grayscale image inpainting, Multiscale Model. Simul. 15 (2017) 575–605.
[23] H. El-Saka, The fractional-order sir and sirs epidemic models with variable population size, Math. Sci. Lett. 2 (2013) 195.
[24] C.M. Elliott, The Cahn-Hilliard Model for the Kinetics of Phase Separation, Mathematical Models for Phase Change Problems, Springer, 1989.
[25] S. Esedoglu, J. Shen, Digital inpainting based on the mumford–shah–euler image model, Eur. J. Appl. Math. 13 (2002) 353–370.
[26] D.J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article 6 (1998).
[27] H. Garcke, K.F. Lam, V. Styles, Cahn–hilliard inpainting with the double obstacle potential, SIAM J. Imaging Sci. 11 (2018) 2064–2089.
[28] A. Hakim, A. Ben-Loghfyry, A total variable-order variation model for image denoising, AIMS Mathematics 4 (2019) 1320–1335.
[29] L. Li, J.G. Liu, Some compactness criteria for weak solutions of time fractional pdes, SIAM J. Math. Anal. 50 (2018) 3963–3995.
[30] J.T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1140–1153.
[31] E.M. Mendes, G.H. Salgado, L.A. Aguirre, Numerical solution of caputo fractional differential equations with infinity memory effect at initial condition, Commun. Nonlinear Sci. Numer. Simul. 69 (2019) 237–247.
[32] J. Mu, B. Ahmad, S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Comput. Math. with Appl. 73 (2017) 985–996.
[33] D.A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. with Appl. 56 (2008) 1138–1145.
[34] D.S. Oliveira, E.C. de Oliveira, On a caputo-type fractional derivative, Adv. Pure Appl. Math. 10 (2019) 81–91.
[35] A. Pimenov, T.C. Kelly, A. Korobeinikov, M.J. O'Callaghan, A.V. Pokrovskii, D. Rachinskii, Memory effects in population dynamics: spread of infectious disease as a case study, Math. Model. Nat. Phenom. 7 (2012) 204–226.
[36] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
[37] M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, G. Jafari, M. Ausloos, Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model, Phys. Rev. E 95 (2017) 022409.
[38] J.T. Schwartz, H. Karcher, Nonlinear Functional Analysis, CRC Press, 1969.
[39] J. Shen, T.F. Chan, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math. 62 (2002) 1019–1043.
[40] B.P. Vollmayr-Lee, A.D. Rutenberg, Fast and accurate coarsening simulation with an unconditionally stable time step, Phys. Rev. E. 68 (2003) 066703.
[41] T.K. Yamana, X. Qiu, E.A. Eltahir, Hysteresis in simulations of malaria transmission, Adv. Water Resour. 108 (2017) 416–422.
[42] Q. Zou, An image inpainting model based on the mixture of perona–malik equation and cahn–hilliard equation, J. Appl. Math. Comput. 66 (2021) 21–38.
Journal of Mathematical Modeling
Volume 11, Issue 2
July 2023
Pages 357-373

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