Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-19T10:11:57.976Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

K. J. Worsley
K. J. Worsley*
Affiliation:
McGill University
*
* Postal address: Department of Mathematics and Statistics, McGill University, 805 ouest, rue Sherbrooke, Montréal, Québec, Canada H3A 2K6.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

DIFFERENTIAL TOPOLOGYINTEGRAL GEOMETRYIMAGE ANALYSISMORSE THEORY

MSC classification

Primary: 60G15: Gaussian processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 4 , December 1995 , pp. 943 - 959
Copyright
Copyright © Applied Probability Trust 1995 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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.

References

Adler, R. J. (1977) A spectral moment estimator in two dimensions. Biometrika 64, 367–373.Google Scholar
Adler, R. J. (1981) The Geometry of Random Fields. Wiley, New York.Google Scholar
Flanders, H. (1963) Differential Forms with Applications to the Physical Sciences. Academic Press, New York.Google Scholar
Gott, J. R., Melott, A. L. and Dickinson, M. (1986) The sponge-like topology of large scale structures in the universe. Astrophys. J. 306, 341–357.CrossRefGoogle Scholar
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, 1–14.Google Scholar
Hadwiger, H. (1959) Normale Körper im euklidischen Raum und ihre topologischen und metrischen Eigenschaften. Math. Z. 71, 124–140.Google Scholar
Kent, J. T. (1989) Continuity properties for random fields. Ann. Prob. 17, 1432–1440.Google Scholar
Knuth, D. E. (1992) Two notes on notation. Amer. Math. Monthly 99, 403–422.Google Scholar
Morse, M. and Cairns, S. S. (1969) Critical Point Theory in Global Analysis and Differential Topology. Academic Press, New York.Google Scholar
Rhoads, J. E., Gott, J. R. and Postman, M. (1994) The genus curve of the Abell clusters. Astrophys. J. 421, 1–8.Google Scholar
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, 608–639.Google Scholar
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, L1–L5.CrossRefGoogle Scholar
Torres, S. (1994) Topological analysis of COBE-DMR cosmic microwave background maps. Astrophys. J. 423, L9–L12.CrossRefGoogle Scholar
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, 640–669.Google Scholar
Worsley, K. J. (1995) Local maxima and the expected Euler characteristic of excursion sets of ?2, F and t fields. Adv. Appl. Prob., 26, 13–42.CrossRefGoogle Scholar
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, 900–918.Google Scholar
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.Google Scholar