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
James Damon. Local morse theory for Gaussian blurred functions. In Sporring et al. [16], chapter 12.
James N. Damon. Properties of ridges and cores for two-dimensional images, (unpublished).
Luc Florack. The Syntactical Structure of Scalar Images. PhD thesis, Universiteit Utrecht, 1993.
Robert Gilmore. Catastrophe Theory for Scientists and Engineers. Dover, 1981. ISBN 0-486-67539-4.
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.
Peter Johansen. On the classification of toppoints in scale space. Mathematical Imaging and Vision, 4, 1994.
Jan J. Koenderink. The structure of images. Biological Cybernetics, 50:363–370, 1984.
Jan J. Koenderink and A.J. van Doorn. Dynamic shape. Biological Cybernetics, 53:383–396, 1986.
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.
Tony Lindeberg. Scale-space behavior of local extrema and blobs. Journal of Mathematical Imaging and Vision, 1:65–99, 1992.
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.
Laurent Najman and Michel Schmitt. Watershed of a continuous function. Signal Processing, 38(6):99–112, July 1994.
Ole Fogh Olsen. Multi-scale segmentation of grey-scale images. Technical Report 96/30, Department of Computer Science, University of Copenhagen, Nov. 1996.
T. Poston and I. N. Stewart. Taylor Expansions and Catastrophes. Pitman, 1976. ISBN 0-273-009-64-8.
J. H. Rieger. Generic evolutions of edges on families of diffused greyvalue surfaces. JMIV, 5:207–217, 1995.
Jon Sporring, Mads Nielsen, Luc Florack, and Peter Johansen, editors. Gaussian Scale-Space. Kluwer Academic Publishers, 1996.
Koen Vincken. Probabilistic Multiscale Image Segmentation by the Hyperstack. PhD thesis, Universiteit Utrecht, 1995.
Andrew P. Witkin. Scale space filtering. In Proc. of International Joint Conference on Artificial Intelligence (IJCAI), Karlsruhe, Germany, 1983.
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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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