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Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics

  • K. J. Worsley (a1)

Abstract

Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.

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* Postal address: Department of Mathematics and Statistics, McGill University, 805 ouest, rue Sherbrooke, Montréal, Québec, Canada H3A 2K6.

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Research supported by the Natural Sciences and Engineering Research Council of Canada, and the Fonds pour la Formation des Chercheurs et l'Aide àla Recherche de Québec.

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References

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Adler, R. J. (1977) A spectral moment estimator in two dimensions. Biometrika 64, 367373.
Adler, R. J. (1981) The Geometry of Random Fields. Wiley, New York.
Flanders, H. (1963) Differential Forms with Applications to the Physical Sciences. Academic Press, New York.
Gott, J. R., Melott, A. L. and Dickinson, M. (1986) The sponge-like topology of large scale structures in the universe. Astrophys. J. 306, 341357.
Gott, J. R., Park, C., Juskiewicz, R., Bies, W. E., Bennett, D. P., Bouchet, F. R. and Stebbins, A. (1990) Topology of microwave background fluctuations: theory. Astrophys. J. 352, 114.
Hadwiger, H. (1959) Normale Körper im euklidischen Raum und ihre topologischen und metrischen Eigenschaften. Math. Z. 71, 124140.
Kent, J. T. (1989) Continuity properties for random fields. Ann. Prob. 17, 14321440.
Knuth, D. E. (1992) Two notes on notation. Amer. Math. Monthly 99, 403422.
Morse, M. and Cairns, S. S. (1969) Critical Point Theory in Global Analysis and Differential Topology. Academic Press, New York.
Rhoads, J. E., Gott, J. R. and Postman, M. (1994) The genus curve of the Abell clusters. Astrophys. J. 421, 18.
Siegmund, D. O. and Worsley, K. J. (1995) Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist. 23, 608639.
Smoot, G. F., Bennett, C. L., Kogut, A., Wright, E. L., Aymon, J., Boggess, N. W., Cheng, E. S., De Amici, G., Gulkis, S., Hauser, M. G., Hinshaw, G., Jackson, P. D., Janssen, M., Kaita, E., Kelsall, T., Keegstra, P., Lineweaver, C., Lowenstein, K., Lubin, P., Mather, J., Meyer, S. S., Moseley, S. H., Murdock, T., Rokke, L., Silverberg, R. F., Tenorio, L., Weiss, R. and Wilkinson, D. T. (1992). Structure in the COBE differential microwave radiometer first-year maps. Astrophys. J. 396, L1L5.
Torres, S. (1994) Topological analysis of COBE-DMR cosmic microwave background maps. Astrophys. J. 423, L9L12.
Worsley, K. J. (1994) Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images. Ann. Statist. 23, 640669.
Worsley, K. J. (1995) Local maxima and the expected Euler characteristic of excursion sets of ?2, F and t fields. Adv. Appl. Prob., 26, 1342.
Worsley, K. J., Evans, A. C., Marrett, S. and Neelin, P. (1992) A three dimensional statistical analysis for CBF activation studies in human brain. J. Cerebral Blood Flow and Metabolism 12, 900918.
Worsley, K. J., Evans, A. C., Marrett, S. and Neelin, P. (1995) Detecting changes in random fields and applications to medical images. Technical Report, McGill University.

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Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics

  • K. J. Worsley (a1)

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