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August  2016, 21(6): 1671-1687. doi: 10.3934/dcdsb.2016017

Some class of parabolic systems applied to image processing

Lekbir Afraites 1, , Abdelghafour Atlas 2, , Fahd Karami 3, and  Driss Meskine 3,

1. 

Faculté des Sciences et Techniques, Université Sultan Moulay Slimane, B.P. 523 Beni-Mellal, Morocco
2. 

Ecole Nationale des Sciences Appliquées de Sa , Université Cadi Ayyad, Route Sidi Bouzid B.P. 63, Safi, Morocco
3. 

Ecole Supérieure de Technologie d'Essaouira, Université Cadi Ayyad, B.P. 383 Essaouira El Jadida, Essaouira, Morocco, Morocco

Received  January 2015 Revised  May 2016 Published  June 2016

In this paper, we are interested in the mathematical and numerical study of a variational model derived as Reaction-Diffusion System for image denoising. We use a nonlinear regularization of total variation (TV) operator's, combined with a decomposition approach of $H^{-1}$ norm suggested by Guo and al. ([19],[20]). Based on Galerkin's method, we prove the existence and uniqueness of the solution on Orlicz space for the proposed model. At last, compared experimental results distinctly demonstrate the superiority of our model, in term of removing noise while preserving the edges and reducing staircase effect.
Keywords: Galerkin method, staircase effect., Image denoising, Orlicz space, reaction-diffusion system.
Mathematics Subject Classification: 35K57, 94A08, 46E3.
Citation: Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017
References:
[1]

R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882. doi: 10.1016/j.camwa.2008.01.017.  Google Scholar

[2]

R. Adams, Sobolev Spaces, Ac. Press, New york, 1975.  Google Scholar

[3]

L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal, 29 (1992), 845-866. doi: 10.1137/0729052.  Google Scholar

[4]

F. Andreu, C. Ballester, V. Caselles and J. L. Mazòn, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.  Google Scholar

[5]

A. Atlas, F. Karami and D. Meskine, The Perona-Malik inequality and application to image denoising, Nonlinear Anal. Real World Appl., 18 (2014), 57-68. doi: 10.1016/j.nonrwa.2013.11.006.  Google Scholar

[6]

P. Blomgren, P. Mulet, T. Chan and C. Wong, Total variation image restoration: numerical methods and extensions, in: Proceeding of the 1997 IEEE International Conference on Image Processing, 3 (1997), 384-387. doi: 10.1109/ICIP.1997.632128.  Google Scholar

[7]

H. Brezis, Equations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann.Inst. Fourier, 18 (1968), 115-175. doi: 10.5802/aif.280.  Google Scholar

[8]

Y. Cao, Yin, J. Liu, Qiang and M. Li, A class of nonlinear parabolic- hyperbolic equations applied to image restoration, Nonlinear Anal. Real World Appl., 11 (2010), 253-261. doi: 10.1016/j.nonrwa.2008.11.004.  Google Scholar

[9]

F. Catté, P. L. Lions, J. M. Morel and T. Call, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. doi: 10.1137/0729012.  Google Scholar

[10]

A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar

[11]

T. F. Chan, S. Esedoglu and F. E. Park, A fourth order dual method for staircase reduction in texture extraction and image restoration problems, 2010 IEEE International Conference on Image Processing, (2010), 4137-4140. doi: 10.1109/ICIP.2010.5653199.  Google Scholar

[12]

T. F. Chan, S. Esedoglu and F. E. Park, Image decomposition combining staircase reduction and texture extraction, Journal of Visual Communication and Image Representation, 18 (2007), 464-486. doi: 10.1016/j.jvcir.2006.12.004.  Google Scholar

[13]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376.  Google Scholar

[14]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[15]

C. M. Eliot and S. A. Smitheman, Analysis of the TV regularization and $H^{-1}$ fidelity model for decomposing an image into cartoon plus texture, Commun. Pure Appl. Anal., 6 (2007), 917-936. doi: 10.3934/cpaa.2007.6.917.  Google Scholar

[16]

A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., (2) 72 (2005), 410-428. doi: 10.1112/S0024610705006630.  Google Scholar

[17]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidely (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

[18]

J. P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math., 74 (1982), 17-24.  Google Scholar

[19]

Z. Guo, Q. Liu, J. Sun and B. Wu, Reaction-diffusion systems with $p(x)$-growth for image denoising, Nonlinear Anal. Real World Appl., 12 (2011), 2904-2918. doi: 10.1016/j.nonrwa.2011.04.015.  Google Scholar

[20]

Z. Guo, J. Yin and Q. Liu, On a reaction-diffusion system applied to image decomposition and restoration, Math. Comput. Modelling, 53 (2011), 1336-1350. doi: 10.1016/j.mcm.2010.12.031.  Google Scholar

[21]

P. Harjulehto, P.A. Hasto, V. Latvala, Minimizers of the variable exponent, non-uniformly convex Dirichlet energy, J. Math. Pures Appl., (9) 89 (2008), 174-197. doi: 10.1016/j.matpur.2007.10.006.  Google Scholar

[22]

P. Hartman, Ordinary Differential Equations, 2nd edn. SIAM, Philidelphia, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[23]

M. Krasnoselśkii and Ya. Rutickii, Convex Functions and Orlicz Spaces, Nodhoff Groningen, 1969. Google Scholar

[24]

A. Kufner, O. John and S. Fucík, Function Spaces, Academia, Prague, 1977.  Google Scholar

[25]

R. Landes and V. Mustonen, A strongly nonlinear parabolic initial-boundary value problem, Ark. Mat., 25 (1987), 29-40. doi: 10.1007/BF02384435.  Google Scholar

[26]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris 1969.  Google Scholar

[27]

Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures, AMS, 2001. doi: 10.1090/ulect/022.  Google Scholar

[28]

S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370. doi: 10.1137/S1540345902416247.  Google Scholar

[29]

P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE, Trans, Pattern anal. Match.Intell, 12 (1990), 629-639. doi: 10.1109/34.56205.  Google Scholar

[30]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D , 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[31]

L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553-572. doi: 10.1023/A:1025384832106.  Google Scholar

[32]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, Germany, 1998.  Google Scholar

show all references

References:
[1]

R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882. doi: 10.1016/j.camwa.2008.01.017.  Google Scholar

[2]

R. Adams, Sobolev Spaces, Ac. Press, New york, 1975.  Google Scholar

[3]

L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal, 29 (1992), 845-866. doi: 10.1137/0729052.  Google Scholar

[4]

F. Andreu, C. Ballester, V. Caselles and J. L. Mazòn, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.  Google Scholar

[5]

A. Atlas, F. Karami and D. Meskine, The Perona-Malik inequality and application to image denoising, Nonlinear Anal. Real World Appl., 18 (2014), 57-68. doi: 10.1016/j.nonrwa.2013.11.006.  Google Scholar

[6]

P. Blomgren, P. Mulet, T. Chan and C. Wong, Total variation image restoration: numerical methods and extensions, in: Proceeding of the 1997 IEEE International Conference on Image Processing, 3 (1997), 384-387. doi: 10.1109/ICIP.1997.632128.  Google Scholar

[7]

H. Brezis, Equations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann.Inst. Fourier, 18 (1968), 115-175. doi: 10.5802/aif.280.  Google Scholar

[8]

Y. Cao, Yin, J. Liu, Qiang and M. Li, A class of nonlinear parabolic- hyperbolic equations applied to image restoration, Nonlinear Anal. Real World Appl., 11 (2010), 253-261. doi: 10.1016/j.nonrwa.2008.11.004.  Google Scholar

[9]

F. Catté, P. L. Lions, J. M. Morel and T. Call, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. doi: 10.1137/0729012.  Google Scholar

[10]

A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar

[11]

T. F. Chan, S. Esedoglu and F. E. Park, A fourth order dual method for staircase reduction in texture extraction and image restoration problems, 2010 IEEE International Conference on Image Processing, (2010), 4137-4140. doi: 10.1109/ICIP.2010.5653199.  Google Scholar

[12]

T. F. Chan, S. Esedoglu and F. E. Park, Image decomposition combining staircase reduction and texture extraction, Journal of Visual Communication and Image Representation, 18 (2007), 464-486. doi: 10.1016/j.jvcir.2006.12.004.  Google Scholar

[13]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376.  Google Scholar

[14]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[15]

C. M. Eliot and S. A. Smitheman, Analysis of the TV regularization and $H^{-1}$ fidelity model for decomposing an image into cartoon plus texture, Commun. Pure Appl. Anal., 6 (2007), 917-936. doi: 10.3934/cpaa.2007.6.917.  Google Scholar

[16]

A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., (2) 72 (2005), 410-428. doi: 10.1112/S0024610705006630.  Google Scholar

[17]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidely (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

[18]

J. P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math., 74 (1982), 17-24.  Google Scholar

[19]

Z. Guo, Q. Liu, J. Sun and B. Wu, Reaction-diffusion systems with $p(x)$-growth for image denoising, Nonlinear Anal. Real World Appl., 12 (2011), 2904-2918. doi: 10.1016/j.nonrwa.2011.04.015.  Google Scholar

[20]

Z. Guo, J. Yin and Q. Liu, On a reaction-diffusion system applied to image decomposition and restoration, Math. Comput. Modelling, 53 (2011), 1336-1350. doi: 10.1016/j.mcm.2010.12.031.  Google Scholar

[21]

P. Harjulehto, P.A. Hasto, V. Latvala, Minimizers of the variable exponent, non-uniformly convex Dirichlet energy, J. Math. Pures Appl., (9) 89 (2008), 174-197. doi: 10.1016/j.matpur.2007.10.006.  Google Scholar

[22]

P. Hartman, Ordinary Differential Equations, 2nd edn. SIAM, Philidelphia, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[23]

M. Krasnoselśkii and Ya. Rutickii, Convex Functions and Orlicz Spaces, Nodhoff Groningen, 1969. Google Scholar

[24]

A. Kufner, O. John and S. Fucík, Function Spaces, Academia, Prague, 1977.  Google Scholar

[25]

R. Landes and V. Mustonen, A strongly nonlinear parabolic initial-boundary value problem, Ark. Mat., 25 (1987), 29-40. doi: 10.1007/BF02384435.  Google Scholar

[26]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris 1969.  Google Scholar

[27]

Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures, AMS, 2001. doi: 10.1090/ulect/022.  Google Scholar

[28]

S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370. doi: 10.1137/S1540345902416247.  Google Scholar

[29]

P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE, Trans, Pattern anal. Match.Intell, 12 (1990), 629-639. doi: 10.1109/34.56205.  Google Scholar

[30]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D , 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[31]

L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553-572. doi: 10.1023/A:1025384832106.  Google Scholar

[32]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, Germany, 1998.  Google Scholar

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