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Testing for signals with unknown location and scale in a χ2 random field, with an application to fMRI

Part of: General convexity Inference from stochastic processes Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Keith J. Worsley
Keith J. Worsley*
Affiliation:
McGill University
*
∗ Postal address: Department of Mathematics and Statistics, McGill University, 805 ouest, rue Sherbrooke, Montréal, Québec, Canada H3A 2K6. Email address: worsley@math.mcgill.ca
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Abstract

Siegmund and Worsley (1995) considered the problem of testing for signals with unknown location and scale in a Gaussian random field defined on ℝN. The test statistic was the maximum of a Gaussian random field in an N+1 dimensional ‘scale space’, N dimensions for location and 1 dimension for the scale of a smoothing filter. Scale space is identical to a continuous wavelet transform with a kernel smoother as the wavelet, though the emphasis here is on signal detection rather than image compression or enhancement. Two methods were used to derive an approximate null distribution for N=2 and N=3: one based on the method of volumes of tubes, the other based on the expected Euler characteristic of the excursion set. The purpose of this paper is two-fold: to show how the latter method can be extended to higher dimensions, and to apply this more general result to χ2 fields. The result of Siegmund and Worsley (1995) then follows as a special case. In this paper the results are applied to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

Keywords

Euler characteristicdifferential topologyintegral geometryimage analysiswavelet thresholdingadaptive filteringmultiscalemultiresolution

MSC classification

Primary: 60G60: Random fields 62M09: Non-Markovian processes
Secondary: 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 773 - 793
Copyright
Copyright © Applied Probability Trust 2001 

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Footnotes

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

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