Journal of Mathematical Imaging and Vision

, Volume 33, Issue 1, pp 25–37 | Cite as

Two New Nonlinear Nonlocal Diffusions for Noise Reduction



Two new nonlocal nonlinear diffusion models for noise reduction are proposed, analyzed and implemented. They are both a close relative of the celebrated Perona-Malik equation. In a way, they can be viewed as a new regularization paradigm for Perona-Malik. They do preserve and enhance the most cherished features of Perona-Malik while delivering well-posed equations which admit a stable natural discretization. Unlike other regularizations, however, certain piecewise smooth functions are (meta)stable equilibria and, as a consequence, their dynamical behavior and that of their discrete implementations can be fully understood and do not lead to any “paradox”. The presence of nontrivial equilibria also explains why blurring is kept in check. One of the models has been proved to be well-posed. Numerical experiments are presented that illustrate the main features of the new models and that provide insight into their interesting dynamical behavior as well as demonstrate their effectiveness as a denoising tool.


Nonlinear diffusion Nonlocal diffusion De-noising 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge-detection by non-linear diffusion. II. SIAM J. Numer. Anal. 29, 845–866 (1992) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Amann, H.: Time-delayed Perona-Malik problems. Acta Math. Univ. Comen. 76, 15–38 (2007) MATHMathSciNetGoogle Scholar
  3. 3.
    Arnoldi, W.E.: The principle of minimized iteration in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9, 17–25 (1951) MATHMathSciNetGoogle Scholar
  4. 4.
    Belahmidi, A.: Equations aux dérivées partielles appliquées à la restoration et à l’agrandissement des images. Ph.D. Thesis, Université Paris-Dauphine, Paris (2003) Google Scholar
  5. 5.
    Belahmidi, A., Chambolle, A.: Time-delay regularization of anisotropic diffusion and image processing. M2AN Math. Model. Numer. Anal. 39, 231–251 (2005) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bellettini, G., Novaga, M., Paolini, M.: Convergence for long-times of a semidiscrete Perona-Malik equation in one dimension. Preprint (2008) Google Scholar
  7. 7.
    Bellettini, G., Novaga, M., Paolini, M., Tornese, C.: Classification of the equilibria for the semi-discrete Perona-Malik equation. Preprint (2008) Google Scholar
  8. 8.
    Buades, A., Coll, B., Morel, J.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4, 490–530 (2005) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Catté, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge-detection by non-linear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen, Y., Bose, P.: On the incorporation of time-delay regularization into curvature-based diffusion. J. Math. Imaging Vis. 14, 149–164 (2001) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Cottet, G., Ayyadi, M.E.: A Volterra type model for image processing. IEEE Trans. Image Process. 7, 292–303 (1998) CrossRefGoogle Scholar
  12. 12.
    Didas, S., Weickert, J., Burgeth, B.: Stability and local feature enhancement of higher order nonlinear diffusion filtering. Pattern Recogn. 3663, 451–458 (2005) CrossRefGoogle Scholar
  13. 13.
    Esedoglu, S.: An analysis of the Perona-Malik scheme. Commun. Pure Appl. Math. 54, 1442–1487 (2001) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Geman, S., McClure, D., Geman, D.: A nonlinear filter for film restoration and other problems in image processing. Graph. Models Image Process. 54, 281–289 (1992) CrossRefGoogle Scholar
  15. 15.
    Ghisi, M., Gobbino, M.: Gradient estimates for the Perona-Malik equation. Math. Ann. 3, 557–590 (2007) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Guidotti, P.: A new well-posed nonlinear nonlocal diffusion. Submitted Google Scholar
  17. 17.
    Kichenassamy, S.: The Perona-Malik paradox. SIAM J. Appl. Math. 57, 1328–1342 (1997) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lysaker, M., Lundervold, A., Tai, X.: Noise removal using fourth order differential equations with applications to medical magnetic resonance images in space-time. IEEE Trans. Image Process. 12, 1579–1590 (2003) CrossRefGoogle Scholar
  19. 19.
    Matthieu, B., Melchior, P., Outstaloup, A., Ceyral, C.: Fractional differentiation for edge detection. Signal Process. 83, 2421–2432 (2003) CrossRefGoogle Scholar
  20. 20.
    Nitzberg, M., Shiota, T.: Nonlinear image smoothing with edge and corner enhancement. Tech. Report 90-2, Harvard University, Cambridge, MA (1990) Google Scholar
  21. 21.
    Nitzberg, M., Shiota, T.: Nonlinear image filtering with edge and corner enhancement. IEEE Trans. Pattern Anal. Mach. Intell. 14, 826–833 (1992) CrossRefGoogle Scholar
  22. 22.
    Perona, P., Malik, J.: Scale-space and edge detection using anistotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 161–192 (1990) CrossRefGoogle Scholar
  23. 23.
    Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P.: Scale-space and edge detection using anistotropic diffusion. IEEE Trans. Image Process. 12, 1338–1351 (2003) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856–869 (1986) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Tan, S., Jiao, L.: Multishrinkage: analytical form for a Bayesian wavelet estimator based on the multivariate Laplacian model. Opt. Lett. 32, 2583–2585 (2007) CrossRefGoogle Scholar
  26. 26.
    Weickert, J.: Anisotropic diffusion in image processing. Ph.D. Thesis, Universität Kaiserslautern, Kaiserslautern (1996) Google Scholar
  27. 27.
    Weickert, J.: Anisotropic Diffusion in Image Processing. ECMI Series. Teubner, Stuttgart (1998) Google Scholar
  28. 28.
    You, Y., Kaveh, M.: Fourth order partial differential equations for noise removal. IEEE Trans. Image Process. 9, 1723–1730 (2000) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

Personalised recommendations