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

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

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.
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
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show all references

References:
[1]

Comput. Math. Appl., 56 (2008), 874-882. doi: 10.1016/j.camwa.2008.01.017.  Google Scholar

[2]

Ac. Press, New york, 1975.  Google Scholar

[3]

SIAM J. Numer. Anal, 29 (1992), 845-866. doi: 10.1137/0729052.  Google Scholar

[4]

Differential Integral Equations, 14 (2001), 321-360.  Google Scholar

[5]

Nonlinear Anal. Real World Appl., 18 (2014), 57-68. doi: 10.1016/j.nonrwa.2013.11.006.  Google Scholar

[6]

in: Proceeding of the 1997 IEEE International Conference on Image Processing, 3 (1997), 384-387. doi: 10.1109/ICIP.1997.632128.  Google Scholar

[7]

Ann.Inst. Fourier, 18 (1968), 115-175. doi: 10.5802/aif.280.  Google Scholar

[8]

Nonlinear Anal. Real World Appl., 11 (2010), 253-261. doi: 10.1016/j.nonrwa.2008.11.004.  Google Scholar

[9]

SIAM J. Numer. Anal., 29 (1992), 182-193. doi: 10.1137/0729012.  Google Scholar

[10]

Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar

[11]

2010 IEEE International Conference on Image Processing, (2010), 4137-4140. doi: 10.1109/ICIP.2010.5653199.  Google Scholar

[12]

Journal of Visual Communication and Image Representation, 18 (2007), 464-486. doi: 10.1016/j.jvcir.2006.12.004.  Google Scholar

[13]

Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376.  Google Scholar

[14]

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

[15]

Commun. Pure Appl. Anal., 6 (2007), 917-936. doi: 10.3934/cpaa.2007.6.917.  Google Scholar

[16]

J. London Math. Soc., (2) 72 (2005), 410-428. doi: 10.1112/S0024610705006630.  Google Scholar

[17]

Trans. Amer. Math. Soc., 190 (1974), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

[18]

Studia Math., 74 (1982), 17-24.  Google Scholar

[19]

Nonlinear Anal. Real World Appl., 12 (2011), 2904-2918. doi: 10.1016/j.nonrwa.2011.04.015.  Google Scholar

[20]

Math. Comput. Modelling, 53 (2011), 1336-1350. doi: 10.1016/j.mcm.2010.12.031.  Google Scholar

[21]

J. Math. Pures Appl., (9) 89 (2008), 174-197. doi: 10.1016/j.matpur.2007.10.006.  Google Scholar

[22]

2nd edn. SIAM, Philidelphia, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[23]

Nodhoff Groningen, 1969. Google Scholar

[24]

Academia, Prague, 1977.  Google Scholar

[25]

Ark. Mat., 25 (1987), 29-40. doi: 10.1007/BF02384435.  Google Scholar

[26]

Dunod; Gauthier-Villars, Paris 1969.  Google Scholar

[27]

The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures, AMS, 2001. doi: 10.1090/ulect/022.  Google Scholar

[28]

Multiscale Model. Simul., 1 (2003), 349-370. doi: 10.1137/S1540345902416247.  Google Scholar

[29]

IEEE, Trans, Pattern anal. Match.Intell, 12 (1990), 629-639. doi: 10.1109/34.56205.  Google Scholar

[30]

Physica D , 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[31]

J. Sci. Comput., 19 (2003), 553-572. doi: 10.1023/A:1025384832106.  Google Scholar

[32]

Teubner-Verlag, Stuttgart, Germany, 1998.  Google Scholar

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