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International Conference on Scale-Space Theories in Computer Vision

Scale-Space 1997: Scale-Space Theory in Computer Vision pp 101–112Cite as

Generic events for the gradient squared with application to multi-scale segmentation

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1252))

Abstract

In the Gaussian scale-space formalism, image features are often defined as loci of differential invariants. A typical behavior of these is topological stability in open intervals of the scale axis. However, it is generic that the feature topology changes at specific scales in so-called catastrophe events. In this paper, we show that the generic Gaussian scale-space catastrophe events for the gradient magnitude squared, L i L i , are the fold catastrophe and the cusp catastrophe. These results are applied to a scale-space formulation of segmentation with catchment basins/watersheds. The common problem of over-segmentation when segmenting with catchment basins of the gradient magnitude is solved by the multi-scale formulation. The necessary linking of segments across scale is based naturally on the catastrophe analysis for L i L i . Verified segmentation results on 3D medical images are presented.

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References

  1. James Damon. Local morse theory for Gaussian blurred functions. In Sporring et al. [16], chapter 12.

    Google Scholar 

  2. James N. Damon. Properties of ridges and cores for two-dimensional images, (unpublished).

    Google Scholar 

  3. Luc Florack. The Syntactical Structure of Scalar Images. PhD thesis, Universiteit Utrecht, 1993.

    Google Scholar 

  4. Robert Gilmore. Catastrophe Theory for Scientists and Engineers. Dover, 1981. ISBN 0-486-67539-4.

    Google Scholar 

  5. Lewis D. Griffin and Alan C. F. 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.

    Google Scholar 

  6. Peter Johansen. On the classification of toppoints in scale space. Mathematical Imaging and Vision, 4, 1994.

    Google Scholar 

  7. Jan J. Koenderink. The structure of images. Biological Cybernetics, 50:363–370, 1984.

    Google Scholar 

  8. Jan J. Koenderink and A.J. van Doorn. Dynamic shape. Biological Cybernetics, 53:383–396, 1986.

    Google Scholar 

  9. Lawrence M. Lifshitz and Stephen M. Pizer. A multiresolution hierarchical approach to image segmentation based on intensity extrema. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(6):529–540, June 1990.

    Google Scholar 

  10. Tony Lindeberg. Scale-space behavior of local extrema and blobs. Journal of Mathematical Imaging and Vision, 1:65–99, 1992.

    Google Scholar 

  11. F. Maes, D. Vandermeulen, P. Suetens, and G. Marchal. Computer-aided interactive object delineation using an intelligent paintbrush technique. In N. Ayache, editor, CVRMed95, pages 77–83. Springer-Verlag, 1995. Lecture Notes 905.

    Google Scholar 

  12. Laurent Najman and Michel Schmitt. Watershed of a continuous function. Signal Processing, 38(6):99–112, July 1994.

    Google Scholar 

  13. Ole Fogh Olsen. Multi-scale segmentation of grey-scale images. Technical Report 96/30, Department of Computer Science, University of Copenhagen, Nov. 1996.

    Google Scholar 

  14. T. Poston and I. N. Stewart. Taylor Expansions and Catastrophes. Pitman, 1976. ISBN 0-273-009-64-8.

    Google Scholar 

  15. J. H. Rieger. Generic evolutions of edges on families of diffused greyvalue surfaces. JMIV, 5:207–217, 1995.

    Google Scholar 

  16. Jon Sporring, Mads Nielsen, Luc Florack, and Peter Johansen, editors. Gaussian Scale-Space. Kluwer Academic Publishers, 1996.

    Google Scholar 

  17. Koen Vincken. Probabilistic Multiscale Image Segmentation by the Hyperstack. PhD thesis, Universiteit Utrecht, 1995.

    Google Scholar 

  18. Andrew P. Witkin. Scale space filtering. In Proc. of International Joint Conference on Artificial Intelligence (IJCAI), Karlsruhe, Germany, 1983.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100, Copenhagen E, Denmark

    Ole Fogh Olsen

  2. 3D-Lab, School of Dentistry, University of Copenhagen, Denmark

    Mads Nielsen

Authors
  1. Ole Fogh Olsen
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  2. Mads Nielsen
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Editor information

Bart ter Haar Romeny Luc Florack Jan Koenderink Max Viergever

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© 1997 Springer-Verlag Berlin Heidelberg

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Fogh Olsen, O., Nielsen, M. (1997). Generic events for the gradient squared with application to multi-scale segmentation. In: ter Haar Romeny, B., Florack, L., Koenderink, J., Viergever, M. (eds) Scale-Space Theory in Computer Vision. Scale-Space 1997. Lecture Notes in Computer Science, vol 1252. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63167-4_43

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  • DOI: https://doi.org/10.1007/3-540-63167-4_43

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63167-5

  • Online ISBN: 978-3-540-69196-9

  • eBook Packages: Springer Book Archive

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