Variable Exponent Nonlocal Model with Weaker Norm in the Fidelity Term for Image Restoration

  • Fahd Karami
  • Driss Meskine
  • Khadija SadikEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10884)


In this paper, we are interested to present a variable exponent nonlocal p(x)-Laplacian model with weaker norm for image denoising. This model inherits the power of the variable exponent in reducing the execution time, besides, the benefit of using the weaker norm in the fidelity term more appropriate to represent textures and small details. At last, we present some numerical simulations and we compare the results with some existing models in the literature.


Image denoising Nonlocal p-Laplacian Variable exponent Weak norm Textured images 


  1. 1.
    Aboulaich, R., Meskine, D., Souissi, A.: New diffusion models in image processing. Comput. Math. Appl. 56(4), 874–882 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Afraites, L., Atlas, A., Karami, F., Meskine, D.: Some class of parabolic systems applied to image processing. Discrete Contin. Dyn. Syst. Ser. B 21(6), 1671–1687 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andreu, F., Mazón, J. M., Rossi, J. D., Toledo, J.: A nonlocal p-Laplacian evolution equation with non homogeneous Dirichlet boundary conditions. SIAM J. Math. Anal., 40(5):1815–1851, 2008/09MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bougleux, S., Elmoataz, A., Melkemi, M.: Discrete regularization on weighted graphs for image and mesh filtering. n 1st International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), 4485 of Lecture Notes in Computer Science:128–139, (2007)Google Scholar
  5. 5.
    Bougleux, S., Elmoataz, A., Melkemi, M.: Local and nonlocal discrete regularization on weighted graphs for image and mesh processing. International Journal of Computer Vision 84(2), 220–236 (2009)CrossRefGoogle Scholar
  6. 6.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chan, T.F., Esedolu, S.: Aspects of total variation regularized \(L^1\) function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restora- tion. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-D transform- domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17(7), 1047–1060 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Elmoataz, A., Xavier, D.: Lezoray, O: Non-local morphological pdes and p-laplacian equation on graphs with applications in image processing and machine learning. IEEE Journal of Selected Topics in Signal Processing 6, 764–779 (2012)CrossRefGoogle Scholar
  12. 12.
    Elmoataz, A., Xavier, D., Zakaria, L., Olivier, L.: Nonlocal infinity Laplacian equation on graphs with applications in image processing and machine learning. Math. Comput. Simulation 102, 153–163 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. Multiscale Model. Simul. 6(2), 595–630 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Guo, Z., Liu, Q., Sun, J., Wu, B.: Reaction-diffusion systems with \(p(x)\)-growth for image denoising. Nonlinear Anal. Real World Appl. 12(5), 2904–2918 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Guo, Z., Yin, J., Liu, Q.: On a reaction-diffusion system applied to image decomposition and restoration. Math. Comput. Modelling 53(5–6), 1336–1350 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Karami, F., Sadik, K., Ziad, L.: A variable exponent nonlocal p(x)-laplacian equation for image restoration. Computers and Mathematics with Applications 75, 534–546 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4(4), 1091–1115 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations, volume 22 of University Lecture Series. American Mathematical Society, Providence, RI, 2001. The fifteenth Dean Jacqueline B. Lewis memorial lecturesGoogle Scholar
  20. 20.
    Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM J. Numer. Anal. 40(3), 965–994 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(H^{-1}\) norm. Multiscale Model. Simul. 1(3), 349–370 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vese, L. A., Osher, S. J.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput., 19(1–3):553–572, 2003. Special issue in honor of the sixtieth birthday of Stanley OsherGoogle Scholar
  23. 23.
    Yaroslavsky, L. P.: Digital picture processing, volume 9 of Springer Series in Information Sciences. Springer-Verlag, Berlin, 1985. An introductionGoogle Scholar
  24. 24.
    Zhou, D., Schlkopf, B: Regularization on discrete spaces, in pattern recognition. Proceedings of the 27th DAGM Symposium, Berlin, Germany, pages 361–369, (2005)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.MMSC, Ecole Supérieure de Technologie d’EssaouiraCadi Ayyad UniversityEssaouiraMorocco

Personalised recommendations