Advertisement

On Manifolds in Gaussian Scale Space

  • Arjan Kuijper
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2695)

Abstract

In an ordinary 2D image the critical points and the isophotes through the saddle points provide sufficient information for classifying the image into distinct regions belonging to the extrema (i.e. a collection of bright and dark blobs), together with their nesting due to the saddle isophotes. For scale space images, obtained by convolution of the image with a Gaussian filter at a continuous range of widths for the Gaussian, things are more complicated. Here only scale space saddle points occur. They are related to spatial saddle points and spatial extrema and can thus provide a scale space based segmentation and hierarchy. However, a spatial extremum can be related to multiple scale space saddles. The key to solve this ambiguity is the investigation of both the scale space saddles and the iso-intensity manifolds (the extension of isophotes in scale space) through them. I will describe the different situations that one can encounter in this investigation, which scale space saddles are relevant, give examples and show the difference between selecting the relevant and the non-relevant (“void”) scale space saddles.

Keywords

Saddle Point Scale Space Initial Image Critical Curve Hierarchy Tree 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Damon. Local Morse theory for solutions to the heat equation and Gaussian blurring. Journal of Differential Equations, 115(2):386–401, 1995.CrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Damon. Local Morse theory for Gaussian blurred functions. In M. Nielsen, L. M. J. Florack, and P. Johansen, editors. Gaussian Scale-Space Theory, volume 8 of Computational Imaging and Vision Series. Kluwer Academic Publishers, Dordrecht, second edition Sporring et al. [27], pages 147–162, 1997.Google Scholar
  3. 3.
    R. Duits, L. M. J. Florack, J. de Graaf, and B. M. ter Haar Romeny. On the axioms of scale space theory, 2001. submitted.Google Scholar
  4. 4.
    D. Eberly. A differential geometric approach to anisotropic diffusion. In ter Haar Romeny, [12], pages 371–392, 1994.Google Scholar
  5. 5.
    M. Fidrich. Iso-surface extraction in 4-D with applications related to scale space. In Proceedings of DGCI’96, 6th International Conference on Discrete Geometry for Computer Imagery (Lyon, France, November 1996), pages 257–268, 1996. Lecture Notes in Computer Science 1176.Google Scholar
  6. 6.
    M. Fidrich. Following feature lines across scale. In L. M. J. Florack, J. J. Koenderink, and M. A. Viergever, editors. Scale-Space Theory in Computer Vision: Proceedings of the First International Conference, Scale-Space’97, Utrecht, The Netherlands, volume 1252 of Lecture Notes in Computer Science. Springer-Verlag, Berlin ter Haar Romeny et al. [13], pages 140–151, 1997.Google Scholar
  7. 7.
    M. Fidrich. Iso-surface extraction in n-D applied to tracking feature curves across scale. Image and Vision Computing, 16(8):545–556, 1998.CrossRefGoogle Scholar
  8. 8.
    L. M. J. Florack. Image Structure, volume 10 of Computational Imaging and Vision Series. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.Google Scholar
  9. 9.
    L. M. J. Florack and A. Kuijper. The topological structure of scale-space images. Journal of Mathematical Imaging and Vision, 12(1):65–80, February 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever. Linear scale-space. Journal of Mathematical Imaging and Vision, 4(4):325–351, 1994.CrossRefMathSciNetGoogle Scholar
  11. 11.
    L. D. Griffin and A. Colchester. Superficial and deep structure in linear diffusion scale space: Isophotes, critical points and separatrices. Image and Vision Computing, 13(7):543–557, September 1995.CrossRefGoogle Scholar
  12. 12.
    B. M. ter Haar Romeny, editor. Geometry-Driven Diffusion in Computer Vision, volume 1 of Computational Imaging and Vision Series. Kluwer Academic Publishers, Dordrecht, 1994.zbMATHGoogle Scholar
  13. 13.
    B. M. ter Haar Romeny, L. M. J. Florack, J. J. Koenderink, and M. A. Viergever, editors. Scale-Space Theory in Computer Vision: Proceedings of the First International Conference, Scale-Space’97, Utrecht, The Netherlands, volume 1252 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, July 1997.Google Scholar
  14. 14.
    T. Iijima. Basic theory of pattern normalization (for the case of a typical one dimensional pattern). Bulletin of the Electrotechnical Laboratory, 26:368–388, 1962. (in Japanese).Google Scholar
  15. 15.
    M. Kerckhove, editor. Scale-Space and Morphology in Computer Vision, volume 2106 of Lecture Notes in Computer Science. Springer-Verlag, Berlin Heidelberg, 2001.zbMATHGoogle Scholar
  16. 16.
    J. J. Koenderink. The structure of images. Biological Cybernetics, 50:363–370, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Kuijper. The Deep Structure of Gaussian Scale Space Images. PhD thesis, Utrecht University, 2002.Google Scholar
  18. 18.
    A. Kuijper and L. M. J. Florack. Hierarchical pre-segmentation without prior knowledge. In Proceedings of the 8th International Conference on Computer Vision (Vancouver, Canada, July 9–12, 2001), pages 487–493, 2001.Google Scholar
  19. 19.
    A. Kuijper and L. M. J. Florack. Logical filtering in scale space. Technical Report UU-CS-2002-018, Department of Computer Science, Utrecht University, 2002. Accepted for publication as “The hierarchical structure of images” in IEEE Transactions on Image Processing.Google Scholar
  20. 20.
    A. Kuijper and L. M. J. Florack. The relevance of non-generic events in scale space. In Proceedings of the 7th European Conference on Computer Vision (Copenhagen, Denmark, May 28–31, 2002), pages 190–204, 2002.Google Scholar
  21. 21.
    A. Kuijper and L. M. J. Florack. Understanding and modeling the evolution of critical points under Gaussian blurring. In Proceedings of the 7th European Conference on Computer Vision (Copenhagen, Denmark, May 28–31, 2002), pages 143–157, 2002.Google Scholar
  22. 22.
    A. Kuijper, L. M. J. Florack, and M. A. Viergever. Scale space hierarchy. Journal of Mathematical Imaging and Vision, 18(2):169–189, April 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    T. Lindeberg. Scale-Space Theory in Computer Vision. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, 1994.Google Scholar
  24. 24.
    M. Loog, J. J. Duistermaat, and L. M. J. Florack. On the behavior of spatial critical points under Gaussian blurring, a folklore theorem and scale-space constraints. In Kerckhove [15], pages 183–192, 2001.Google Scholar
  25. 25.
    M. Nielsen, P. Johansen, O. Fogh Olsen, and J. Weickert, editors. Scale-Space Theories in Computer Vision, volume 1682 of Lecture Notes in Computer Science. Springer-Verlag, Berlin Heidelberg, 1999.Google Scholar
  26. 26.
    T. Poston and I. N. Stewart. Catastrophe Theory and its Applications. Pitman, London, 1978.zbMATHGoogle Scholar
  27. 27.
    J. Sporring, M. Nielsen, L. M. J. Florack, and P. Johansen, editors. Gaussian Scale-Space Theory, volume 8 of Computational Imaging and Vision Series. Kluwer Academic Publishers, Dordrecht, second edition, 1997.zbMATHGoogle Scholar
  28. 28.
    J. A. Weickert. Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.zbMATHGoogle Scholar
  29. 29.
    A. P. Witkin. Scale-space filtering. In Proceedings of the Eighth International Joint Conference on Artificial Intelligence, pages 1019–1022, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Arjan Kuijper
    • 1
  1. 1.Image groupIT-University of CopenhagenCopenhagenDenmark

Personalised recommendations