American Institute of Mathematical Sciences

Advanced
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]

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

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

[1]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005
[2]

Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024
[3]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[4]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[5]

Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021081
[6]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004
[7]

Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021085
[8]

Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249
[9]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067
[10]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400
[11]

Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021077
[12]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[13]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[14]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030
[15]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011
[16]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003
[17]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217
[18]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001
[19]

Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020
[20]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3211-3240. doi: 10.3934/dcds.2020403

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (130)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences