Abstract
In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of spatial critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe the behaviour of critical point trajectories when various different types of generic events, viz. annihilations and creations of pairs of spatial critical points, (almost) coincide. Although such events are non-generic in mathematical sense, they are not unlikely to be encountered in practice. Furthermore the behaviour leads to the observation that fine-to-coarse tracking of critical points doesn’t suffice. We apply the theory to an artificial image and a simulated MR image and show the occurrence of the described behaviour.
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
V. I. Arnold. Catastrophe Theory. Springer, Berlin, 1984.
J. Damon. Local Morse theory for solutions to the heat equation and Gaussian blurring. Journal of Differential Equations, 115(2):386–401, 1995.
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.
L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever. Cartesian differential invariants in scale-space. Journal of Mathematical Imaging and Vision, 3(4):327–348, 1993.
R. Gilmore. Catastrophe Theory for Scientists and Engineers. Dover, 1993. Originally published by John Wiley & Sons, New York, 1981.
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.
P. Johansen, M. Nielsen, and O.F. Olsen. Branch points in one-dimensional Gaussian scale space. Journal of Mathematical Imaging and Vision, 13:193–203, 2000.
J. J. Koenderink. The structure of images. Biological Cybernetics, 50:363–370, 1984.
A. Kuijper and L.M.J. Florack. Calculations on critical points under gaussian blurring. In Nielsen et al. [13], pages 318–329, 1999.
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.
L. M. Lifshitz and S. 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, 1990.
T. Lindeberg. Scale-Space Theory in Computer Vision. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, 1994.
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.
R. Thom. Structural Stability and Morphogenesis. Benjamin-Addison Wesley, 1975. translated by D. H. Fowler.
T. Wada and M. Sato. Scale-space tree and its hierarchy. In ICPR90, volume II, pages 103–108, 1990.
A. P. Witkin. Scale-space filtering. In Proceedings of the Eighth International Joint Conference on Artificial Intelligence, pages 1019–1022, 1983.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kuijper, A., Florack, L. (2002). The Relevance of Non-generic Events in Scale Space Models. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds) Computer Vision — ECCV 2002. ECCV 2002. Lecture Notes in Computer Science, vol 2350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47969-4_13
Download citation
DOI: https://doi.org/10.1007/3-540-47969-4_13
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43745-1
Online ISBN: 978-3-540-47969-7
eBook Packages: Springer Book Archive