From Box Filtering to Fast Explicit Diffusion

  • Sven Grewenig
  • Joachim Weickert
  • Andrés Bruhn
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6376)

Abstract

There are two popular ways to implement anisotropic diffusion filters with a diffusion tensor: Explicit finite difference schemes are simple but become inefficient due to severe time step size restrictions, while semi-implicit schemes are more efficient but require to solve large linear systems of equations. In our paper we present a novel class of algorithms that combine the advantages of both worlds: They are based on simple explicit schemes, while being more efficient than semi-implicit approaches. These so-called fast explicit diffusion (FED) schemes perform cycles of explicit schemes with varying time step sizes that may violate the stability restriction in up to 50 percent of all cases. FED schemes can be motivated from a decomposition of box filters in terms of explicit schemes for linear diffusion problems. Experiments demonstrate the advantages of the FED approach for time-dependent (parabolic) image enhancement problems as well as for steady state (elliptic) image compression tasks. In the latter case FED schemes are speeded up substantially by embedding them in a cascadic coarse-to-fine approach.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)MATHGoogle Scholar
  2. 2.
    Höcker, C., Fehmers, G.: Fast structural interpretation with structure-oriented filtering. Geophysics 68(4), 1286–1293 (2003)CrossRefGoogle Scholar
  3. 3.
    Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.P.: Image compression with anisotropic diffusion. Journal of Mathematical Imaging and Vision 31(2-3), 255–269 (2008)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 629–639 (1990)CrossRefGoogle Scholar
  5. 5.
    Lu, T., Neittaanmäki, P., Tai, X.C.: A parallel splitting up method and its application to Navier-Stokes equations. Applied Mathematics Letters 4(2), 25–29 (1991)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Weickert, J., ter Haar Romeny, B.M., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing 7(3), 398–410 (1998)CrossRefGoogle Scholar
  7. 7.
    Drblíková, O., Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing. SIAM Journal on Numerical Analysis 46(1), 37–60 (2007)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Weickert, J., Scharr, H.: A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. Journal of Visual Communication and Image Representation 13(1/2), 103–118 (2002)CrossRefGoogle Scholar
  9. 9.
    Welk, M., Steidl, G., Weickert, J.: Locally analytic schemes: A link between diffusion filtering and wavelet shrinkage. Applied and Computational Harmonic Analysis 24, 195–224 (2008)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gentzsch, W., Schlüter, A.: Über ein Einschrittverfahren mit zyklischer Schrittweitenänderung zur Lösung parabolischer Differentialgleichungen. ZAMM, Zeitschrift für Angewandte Mathematik und Mechanik 58, T415–T416 (1978)Google Scholar
  11. 11.
    Gentzsch, W.: Numerical solution of linear and non-linear parabolic differential equations by a time discretisation of third order accuracy. In: Hirschel, E.H. (ed.) Proceedings of the Third GAMM-Conference on Numerical Methods in Fluid Mechanics, pp. 109–117. Friedr. Vieweg & Sohn (1979)Google Scholar
  12. 12.
    Alexiades, V., Amiez, G., Gremaud, P.A.: Super-time-stepping acceleration of explicit schemes for parabolic problems. Communications in Numerical Methods in Engineering 12, 31–42 (1996)MATHCrossRefGoogle Scholar
  13. 13.
    Bornemann, F., Deuflhard, P.: The cascadic multigrid method for elliptic problems. Numerische Mathematik 75, 135–152 (1996)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Weickert, J.: Nonlinear diffusion filtering. In: Jähne, B., Haußecker, H., Geißler, P. (eds.) Handbook on Computer Vision and Applications. Signal Processing and Pattern Recognition, vol. 2, pp. 423–450. Academic Press, San Diego (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Sven Grewenig
    • 1
  • Joachim Weickert
    • 1
  • Andrés Bruhn
    • 1
  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

Personalised recommendations