Gradient Schemes for Image Processing

  • Robert EymardEmail author
  • Angela Handlovičová
  • Raphaèle Herbin
  • Karol Mikula
  • Olga Stašová
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 4)


We present a gradient scheme (which happens to be similar to the MPFA finite volume O-scheme) for the approximation to the solution of the Perona-Malik model regularized by a time delay and to the solution of the nonlinear tensor anisotropic diffusion equation. Numerical examples showing properties of the method and applications in image filtering are discussed.


advection equation semi-implicit scheme finite volume method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. Aavatsmark, T. Barkve, O. Boe, and T. Mannseth. Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys., 127(1):2–14, 1996.zbMATHCrossRefGoogle Scholar
  2. 2.
    F. Catté, P.L. Lions, J.M. Morel and T. Coll. Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29:182–193,1992.Google Scholar
  3. 3.
    A. Handlovičová and Z. Krivá. Error estimates for finite volume scheme for Perona - Malik equation. Acta Math. Univ. Comenianae,74,(1):79–94, 2005.Google Scholar
  4. 4.
    R. Eymard, A. Handlovičová, R. Herbin, K. Mikula and O. Stašová. Applications of approximate gradient schemes for nonlinear parabolic equations. in preparation, 2011.Google Scholar
  5. 5.
    R. Eymard, R. Herbin. Gradient Scheme Approximations for Diffusion Problems. these proceedings, 2011.Google Scholar
  6. 6.
    O. Drblíková and K. Mikula. Convergence Analysis of Finite Volume Scheme for Nonlinear Tensor Anisotropic Diffusion in Image Processing. SIAM Journal on Numerical Analysis, 46 (1): 37–60,2007.MathSciNetCrossRefGoogle Scholar
  7. 7.
    O. Drblíková A. Handlovičová and K. Mikula. Error estimates of the Finite Volume Scheme for the Nonlinear Tensor -Driven Anisotropic Diffusion. Applied Numerical Mathemtaics, 59: 2548–2570,2009.zbMATHCrossRefGoogle Scholar
  8. 8.
    K. Mikula and N. Ramarosy. Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing. Numerische Mathematik, 89, (3):561–590,2001.Google Scholar
  9. 9.
    P. Perona, J. Malik. Scale space and edge detection using anisotropic diffusion. In: Proc. IEEE Computer Society Workshop on Computer Vision (1987).Google Scholar
  10. 10.
    N. J. Walkington. Algorithms for computing motion by mean curvature. SIAM J. Numer. Anal., 33(6):2215–2238, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    J. Weickert. Coherence-enhancing diffusion filtering. Int. J. Comput. Vision, 31: 111–127, 1999.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Robert Eymard
    • 1
    Email author
  • Angela Handlovičová
    • 3
  • Raphaèle Herbin
    • 2
  • Karol Mikula
    • 3
  • Olga Stašová
    • 3
  1. 1.Université Paris-EstMarne la ValléeFrance
  2. 2.Centre de Mathmatiques et InformatiqueUniversité de ProvenceMarseille 13France
  3. 3.Department of MathematicsSlovak University of TechnologyBratislavaSlovakia

Personalised recommendations