Skip to main content
SpringerLink
Log in

Scale-space behaviour of local extrema and blobs

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Elementary techniques from real analysis, and singularity theory are applied to derive analytical results for the behaviour in scale-space of critical points and related entities. The main results of the treatment comprise

  • • a description of the general nature of trajectories of critical points in scale-space

  • • estimates of the drift velocity of critical points and straight edges

  • • an analysis of the qualitative behaviour of critical points in bifurcation situations

  • • a classification of what types of blod events are possible

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Canny, “A Computational Approach to Edge Detection” IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-8, 1986, pp. 679–698.

    Google Scholar 

  2. F. Bergholm, “Edge Focusing”, IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-9, 1987, pp. 726–741.

    Google Scholar 

  3. V. Berzins, “Accuracy of Laplacian Edge Detectors”, Comput. Vis. Graph., Image Process., vol. 27, 1984, pp. 195–210.

    Google Scholar 

  4. J.J. Koenderink and A.J.van Doorn, “Dynamic Shape”, Biological Cybernetics, vol. 53, 1986, pp. 383–396.

    Google Scholar 

  5. T.P. Lindeberg, Discrete Scale-Space Theory and the Scale-Space Primal Sketch, Ph.D. dissertation 1991, TRITANA-P9108 ISSN 1101-2250, ISRN KTH/NA/P-91/8-SE, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden.

  6. T.P. Lindeberg and J.O. Eklundh, “On the Computation of a Scale-Space Primal Sketch”, J. Vis. Commun. Image Rep., vol. 2(1), 1991, pp. 55–78.

    Google Scholar 

  7. T.P. Lindeberg and J.O. Eklundh, “Scale Detection and Region Extraction from a Scale-Space Primal Sketch”, in Proceedings of the 3rd International Conference on Computer Vision, Osaka, Japan, December 4–7, 1990, pp. 416–426.

  8. T.P. Lindeberg, “Detecting Salient Blob-like Image Structures and their Scales with a Scale-Space Primal Sketch: A Method for Focus-of-Attention”, Int. J. Computer Vision, to appear.

  9. J.J. Koenderink and A.J.van Doorn, “The Structure of Images”, Biological Cybernetics, vol. 50, 1984, pp. 363–370.

    Google Scholar 

  10. J.J. Koenderink, Solid Shape, Boston: MIT Press, 1990.

    Google Scholar 

  11. J.J. Clark, “Singularity Theory and Phantom Edges in Scale-Space”, IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-10, 1988, pp. 720–727.

    Google Scholar 

  12. P. Johansen, S. Skelboe, K. Grue, and J.D. Andersen, “Representing Signals by Their Top Points in Scale-Space”, in Proceedings of the 8th International Conference on Pattern Recognition, Paris, October 27–31, 1986, pp. 215–217.

  13. A.P. Witkin, “Scale-Space Filtering”, in Proceedings of the 8th International Joint Conference on Artificial Intelligence, Karlsruhe, Federal Republic of Germany, August 8–12, 1983, pp. 1019–1022.

  14. J. Babaud, A.P. Witkin, M. Baudin, and R.O. Duda, “Uniqueness of the Gaussian Kernel for Scale-Space Filtering”, IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-8, 1986, pp. 26–33.

    Google Scholar 

  15. L.M. Lifshitz and S.M. Pizer, “A Multiresolution Hierarchical Approach to Image Segmentation Based on Intensity Extrema”, internal report, Departments of Computer Science and Radiology, University of North Carolina, Chapel Hill, NC, 1987.

  16. X. Zhuang and T.S. Huang, “Multi-Scale Edge Detection with Gaussian Filters”, in International conference on Acoustics, Speech, and Signal Processing, Tokyo, 1986, pp. 2047–2050.

  17. W. Zhang and F. Bergholm, “An Extension to Marr's Signature Based Edge Classification”, 1990, 7th Scandinavian Conference on Image Analysis, Aalborg, Denmark, August 13–16, pp. 435–443. Also available as Technical Note CVAP-TN-04, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden.

  18. T. Poston and I. Stewart, Catastrophe Theory and its Applications, London: Pitman Publishing, 1978.

    Google Scholar 

  19. C.G. Gibson, Singular Points of Smooth Mappings, Research Notes in Mathematics, vol. 25, London: Pitman Publishing, 1979.

    Google Scholar 

  20. J.W. Bruce and P.J. Giblin, Curves and Singularities, Cambridge: University Press, 1984.

    Google Scholar 

  21. V.I. Arnold, Singularity theory, Selected Papers, in London Mathematical Society Lecture Note Series, vol. 53, Cambridge: Cambridge University Press, 1981.

    Google Scholar 

  22. V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko, Singularities of Smooth Maps, volume I, Boston: Birkhäuser, 1985.

    Google Scholar 

  23. V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko, Singularities of Smooth Maps, volume II, Boston: Birkhäuser, 1988.

    Google Scholar 

  24. M. Golubitsky and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. I, Applied Mathematical Sciences, vol. 51, New York: Springer-Verlag, 1985.

    Google Scholar 

  25. Y.-C. Lu, Singularity Theory and an Introduction to Catastrophe Theory, New York: Springer-Verlag, 1976.

    Google Scholar 

  26. F. Bergholm, personal communication, 1990.

  27. T.P. Lindeberg, “Scale-Space for Discrete Signals”, IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-12, 1990, pp. 234–254.

    Google Scholar 

  28. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.

  29. D.R.J. Chillingworth, Differential Topology with a View to Applications, London: Pitman Publishing, 1976.

    Google Scholar 

  30. T.P. Lindeberg, “On the Behaviour in Scale-Space of Local Extrema and Blobs”, Proceedings of the 7th Annual Scandinavian Conference on Image Analysis, Aalborg, Denmark, August 13–16, 1991, pp. 8–17.

  31. T.P. Lindeberg, “Effective Scale: A Natural Unit for Measuring Scale-Space Lifetime”, IEEE Trans. Patt. Anal. Mach. Intell., to appear.

Download references

Author information

Authors and Affiliations

  1. Computational Vision and Active Perception Laboratory (CVAP), Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Tony Lindeberg

Authors
  1. Tony Lindeberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lindeberg, T. Scale-space behaviour of local extrema and blobs. J Math Imaging Vis 1, 65–99 (1992). https://doi.org/10.1007/BF00135225

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00135225

Key words

Search

Navigation