Combing a Porcupine via Stereographic Direction Difusion

  • Nir A. Sochen
  • Ron Kimmel
Conference paper
Part of the Lecture Notes in Computer Science 2106 book series (LNCS, volume 2106)


This paper addresses the problem of feature enhancement in noisy images when the feature is known to be constrained to a manifold. As an example, we study the problem of direction denoising. This problem was treated recently and several solutions were proposed. The various solutions share the same structure. They are composed of two terms: A difusion term and a projection term. Analytically, the solutions differ in the difusion part. The projection part is equivalent in all works. Yet, as it is often the case, the analytically equivalent projection terms differ from a numerical viewpoint. We present in this work a new parameterization of the problem that enables us to work always in a numerically stable way.


Noisy Image Switching Point Riemannian Structure South Hemisphere Polyakov Action 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. Chan and J. Shen, “Variational restoration of image features: Models and algorithms”, SIAM J. Appl. Math., to appear.Google Scholar
  2. 2.
    A. Cumani, “Edge detection in multi-spectral images”, CVGIP: Graphical Models and Image Processing 53 (1991) no.1 40–51.zbMATHCrossRefGoogle Scholar
  3. 3.
    R. Kimmel and N. Sochen, “Orientation Difusion or How to Comb a Porcupine?”, special issue on PDEs in Image Processing, Computer Vision and Computer Graphics, Journal of Visual Communication and Image Representation, to appear.Google Scholar
  4. 4.
    R. Kimmel and R. Malladi and N. Sochen, “Images as Embedding Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images”, International Journal of Computer Vision 39(2) (2000) 111–129.zbMATHCrossRefGoogle Scholar
  5. 5.
    E. Kreyszig, “Differential Geometry”, Dover Publications, Inc., New York, 1991.Google Scholar
  6. 6.
    A.M. Polyakov, “Quantum geometry of bosonic strings”, Physics Letters, 103B (1981) 207–210.MathSciNetGoogle Scholar
  7. 7.
    P. Perona, “Orientation Difusion” IEEE Trans. on Image Processing, 7 (1998) 457–467.CrossRefGoogle Scholar
  8. 8.
    N. Sochen and R.M. Haralick and Y.Y. Zeevi, “A Geometric functional for Derivatives Approximation” EE-Technion Report, April 1999.Google Scholar
  9. 9.
    N. Sochen and R. Kimmel and A.M. Bruckstein, “Difusions and confusions in signal and image processing”, accepted to Journal of Mathematical Imaging and Vision.Google Scholar
  10. 10.
    N. Sochen and R. Kimmel and R. Malladi, “From high energy physics to low level vision”, Report, LBNL, UC Berkeley, LBNL 39243, August, Presented in ONR workshop, UCLA, Sept. 5 1996.Google Scholar
  11. 11.
    N. Sochen and R. Kimmel and R. Malladi, “A general framework for low level vision”, IEEE Trans. on Image Processing, 7 (1998) 310–318.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    N. Sochen, “Stochastic processes in vision I: From Langevin to Beltrami”, CCIT Technion technical report No. 245Google Scholar
  13. 13.
    N. Sochen and Y.Y. Zeevi, “Representation of colored images by manifolds embedded in higher dimensional non-Euclidean space”, Proc. IEEE ICIP’98, Chicago, 1998.Google Scholar
  14. 14.
    B. Tang and G. Sapiro and V. Caselles, “Direction difusion”, International Conference on Computer Vision, 1999.Google Scholar
  15. 15.
    B. Tang and G. Sapiro and V. Caselles, “Color image enhancement via chromaticity difusion” Technical report, ECE-University of Minnesota, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Nir A. Sochen
    • 1
  • Ron Kimmel
    • 2
  1. 1.Department of Applied MathematicsUniversity of Tel AvivTel-AvivIsrael
  2. 2.Department of Computer ScienceTechnion - Israel Institute of TechnologyHaifaIsrael

Personalised recommendations