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Some class of parabolic systems applied to image processing
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 |
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] | |
[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] | |
[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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