International Journal of Analysis and Applications ISSN 2291-8639 (Online)

Research Article

Image Restoration by a Fractional Reaction-Diffusion Process

  • Hana Matallah1*
  • Messouad Maouni
  • Hakim Lakhal
  1. Laboratory of Applied Mathematics and History and Didactics of Mathematics "LAMAHIS" \\Department of Mathematics, University 20 august 1955 Skikda

* Corresponding author

Abstract

We propose new approaches to the investigation of a reaction-diffusion model of fractional order in which we apply the fractional derivative in the sense of the Caputo by contribution to time on the model proposed by Nourddine Alaa in 2014, this study is based on the restoration of digital image such that a digital result is given on a noisy image in which this model is found to be effective in eliminating noise.

How to Cite

H. Matallah, M. Maouni, H. Lakhal, Image Restoration by a Fractional Reaction-Diffusion Process, Int. J. Anal. Appl. 19 (2021), 709–724. https://doi.org/10.28924/2291-8639-19-2021-709

BibTeX
@article{Matallah2021,
  author  = {Matallah, Hana and Maouni, Messouad and Lakhal, Hakim},
  title   = {Image Restoration by a Fractional Reaction-Diffusion Process},
  journal = {Int. J. Anal. Appl.},
  volume  = {19},
  year    = {2021},
  pages   = {709--724},
  doi     = {10.28924/2291-8639-19-2021-709}
}

References

  • [1] A. Aarab, N. Alaa, H. Khalfi. Generic reaction-diffusion model with application to image restoration and enhancement, Electron. J. Differ. Equ. 2018 (2018), 125.
  • [2] N. Alaa, M. Aitoussous, W. Bouarifi, D. Bensikaddour, Image restoration using a reaction-diffussion process, Electron. J. Differ. Equ. 2014 (2014), 197.
  • [3] N.E. Alaa, M. Zirhem, Bio-inspired reaction diffusion system applied to image restoration, Int. J. Bio-Inspired Comput. 12 (2018), 128-137.
  • [4] B. Al-Hamzah, N. Yebri. Global existence in reaction diffusion nonlinear parabolic partial differential equation in image procrssing, Glob. J. Adv. Eng. Technol. Sci. 3 (2016), 5.
  • [5] F. Catt ´e, P.L. Lions, J-M. Morel, T. Coll. Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29 (1992), 182-193.
  • [6] J. Chen. An implicit approximation for the Caputo fractional reaction-dispersion equation. J. Xiamen Univ. (Nat. Sci.) 46 (2007), 616-619. (in Chinese).
  • [7] C. Gong, W. Bao, G. Tang, Y. Jiang, J. Liu, A domain decomposition method for time fractional reaction-diffusion equation, Sci. World J. 2014 (2014), 681707.
  • [8] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65-70.
  • [9] A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. volume 204 of North-Holland Mathematics Studies. Elsevier, Amesterdam, 2006.
  • [10] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65-70.
  • [11] S. Lecheheb, M. Maouni, H. Lakhal, Existence of the solution of a quasilinear equation and its application to image denoising, Int. J. Comput. Sci. Commun. Inform. Technol. 7 (2019), 1-6.
  • [12] S. Lecheheb, M. Maouni, H. Lakhal, Image restoration using nonlinear elliptic equation, Int. J. Comput. Sci. Commun. Inform. Technol. 6 (2019), 32-37.
  • [13] S. Lecheheb, M. Maouni, H. Lakhal, Image restoration using a novel model combining the Perona-Malik equation and the heat equation, Int. J. Anal. Appl. 19 (2021), 228-238.
  • [14] M. Maouni, F.Z. Nouri, Image restoration based on p-gradient model, Int. J. Appl. Math. Stat. 41 (2013), 48-57.
  • [15] S. Morfu. On some applications of diffusion processes for image processing. Phys. Lett. A, 373 (2009), 24-44.
  • [16] P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990), 629-639.
  • [17] B. Tellab, R ´esolution des ´equations diff ´erentielles fractionnaires. Universit ´e des Fr`eres Mentouri Constantine-1, 2018.
  • [18] Feng Xing. Methode de decomposition de domaines pour l'equation de Schrodinger. Analyse numerique [math.NA]. Universite Lille 1 - Sciences et Technologies, 2014.
  • [19] J. Simon, Compact sets in the spaceL p (O,T; B), Ann. Mat. Pura Appl. 146 (1986), 65-96.
  • [20] J. Weickert. Anisotropic Diffusion in Image Processing. PhD Thesis, Kaiserslautern University, Kaiserslautern, Germany, 1996.
  • [21] F.Z. Zeghbib, M. Maouni, F.Z. Nouri, Overlapping and nonoverlapping domain decomposition methods for image restoration, Int. J. Appl. Math. Stat. 40 (2013), 123-128.