Advertisement

Scale-Space Theories for Scalar and Vector Images

  • Luc M. J. Florack
Conference paper
Part of the Lecture Notes in Computer Science 2106 book series (LNCS, volume 2106)

Abstract

We define mutually consistent scale-space theories for scalar and vector images. Consistency pertains to the connection between the already established scalar theory and that for a suitably defined scalar field induced by the proposed vector scale-space. We show that one is compelled to reject the Gaussian scale-space paradigm in certain cases when scalar and vector fields are mutually dependent.

Subsequently we investigate the behaviour of critical points of a vector-valued scale-space image—i.e. points at which the vector field vanishes— as well as their singularities and unfoldings in linear scale-space.

Keywords

Critical Path Scalar Image Vector Image Cofactor Matrix Creation Event 
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]
    R. van den Boomgaard. The morphological equivalent of the Gauss convolution. Nieuw Archief voor Wiskunde, 10(3):219–236, November 1992.MathSciNetzbMATHGoogle Scholar
  2. [2]
    R. van den Boomgaard and A.W.M. Smeulders. The morphological structure of images, the differential equations of morphological scale-space. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(11):1101–1113, November 1994.CrossRefGoogle Scholar
  3. [3]
    J. Damon. Local Morse theory for solutions to the heat equation and Gaussian blurring. Journal of Differential Equations, 115(2):368–401, January 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    L.C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island, 1998.Google Scholar
  5. [5]
    L. Florack and A. Kuijper. The topological structure of scale-space images. Journal of Mathematical Imaging and Vision, 12(1):65–79, February 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    L.M.J. Florack. Non-linear scale-spaces isomorphic to the linear case. In B.K. Ersb∅ll and P. Johansen, editors, Proceedings of the 11th Scandinavian Conference on Image Analysis (Kangerlussuaq, Greenland, June 7-11 1999), volume 1, pages 229–234, Lyngby, Denmark, 1999.Google Scholar
  7. [7]
    L.M.J. Florack, W.J. Niessen, and M. Nielsen. The intrinsic structure of optic flow incorporating measurement duality. International Journal of Computer Vision, 27(3):263–286, May 1998.CrossRefGoogle Scholar
  8. [8]
    J.J. Koenderink. The structure of images. Biological Cybernetics, 50:363–370, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    M. Nielsen and O.F. Olsen. The structure of the optic flow field. In H. Burkhardt and B. Neumann, editors, Proceedings of the Fifth European Conference on Computer Vision (Freiburg, Germany, June 1998), volume 1407 of Lecture Notes in Computer Science, pages 281–287. Springer-Verlag, Berlin, 1998.Google Scholar
  10. [10]
    A. Simmons, S.R. Arridge, P.S. Tofts, and G.J. Barker. Application of the extremum stack to neurological MRI. IEEE Transactions on Medical Imaging, 17(3):371–382, June 1998.CrossRefGoogle Scholar
  11. [11]
    M. Spivak. Differential Geometry, volume 1. Publish or Perish, Berkeley, 1975.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Luc M. J. Florack
    • 1
  1. 1.Utrecht UniversityUtrechtThe Netherlands

Personalised recommendations