Some class of parabolic systems applied to image processing

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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

Abstract
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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 and 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.
[2]	R. Adams, Sobolev Spaces, Ac. Press, New york, 1975.
[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.
[4]	F. Andreu, C. Ballester, V. Caselles and J. L. Mazòn, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[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.
[16]	A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., (2) 72 (2005), 410-428. doi: 10.1112/S0024610705006630.
[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.
[18]	J. P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math., 74 (1982), 17-24.
[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.
[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.
[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.
[22]	P. Hartman, Ordinary Differential Equations, 2nd edn. SIAM, Philidelphia, 2002. doi: 10.1137/1.9780898719222.
[23]	M. Krasnoselśkii and Ya. Rutickii, Convex Functions and Orlicz Spaces, Nodhoff Groningen, 1969.
[24]	A. Kufner, O. John and S. Fucík, Function Spaces, Academia, Prague, 1977.
[25]	R. Landes and V. Mustonen, A strongly nonlinear parabolic initial-boundary value problem, Ark. Mat., 25 (1987), 29-40. doi: 10.1007/BF02384435.
[26]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris 1969.
[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.
[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.
[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.
[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.
[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.
[32]	J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, Germany, 1998.

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.
[2]	R. Adams, Sobolev Spaces, Ac. Press, New york, 1975.
[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.
[4]	F. Andreu, C. Ballester, V. Caselles and J. L. Mazòn, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[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.
[16]	A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., (2) 72 (2005), 410-428. doi: 10.1112/S0024610705006630.
[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.
[18]	J. P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math., 74 (1982), 17-24.
[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.
[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.
[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.
[22]	P. Hartman, Ordinary Differential Equations, 2nd edn. SIAM, Philidelphia, 2002. doi: 10.1137/1.9780898719222.
[23]	M. Krasnoselśkii and Ya. Rutickii, Convex Functions and Orlicz Spaces, Nodhoff Groningen, 1969.
[24]	A. Kufner, O. John and S. Fucík, Function Spaces, Academia, Prague, 1977.
[25]	R. Landes and V. Mustonen, A strongly nonlinear parabolic initial-boundary value problem, Ark. Mat., 25 (1987), 29-40. doi: 10.1007/BF02384435.
[26]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris 1969.
[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.
[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.
[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.
[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.
[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.
[32]	J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, Germany, 1998.

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