Open Access
December 2012 Rotation and scale space random fields and the Gaussian kinematic formula
Robert J. Adler, Eliran Subag, Jonathan E. Taylor
Ann. Statist. 40(6): 2910-2942 (December 2012). DOI: 10.1214/12-AOS1055


We provide a new approach, along with extensions, to results in two important papers of Worsley, Siegmund and coworkers closely tied to the statistical analysis of fMRI (functional magnetic resonance imaging) brain data. These papers studied approximations for the exceedence probabilities of scale and rotation space random fields, the latter playing an important role in the statistical analysis of fMRI data. The techniques used there came either from the Euler characteristic heuristic or via tube formulae, and to a large extent were carefully attuned to the specific examples of the paper.

This paper treats the same problem, but via calculations based on the so-called Gaussian kinematic formula. This allows for extensions of the Worsley–Siegmund results to a wide class of non-Gaussian cases. In addition, it allows one to obtain results for rotation space random fields in any dimension via reasonably straightforward Riemannian geometric calculations. Previously only the two-dimensional case could be covered, and then only via computer algebra.

By adopting this more structured approach to this particular problem, a solution path for other, related problems becomes clearer.


Download Citation

Robert J. Adler. Eliran Subag. Jonathan E. Taylor. "Rotation and scale space random fields and the Gaussian kinematic formula." Ann. Statist. 40 (6) 2910 - 2942, December 2012.


Published: December 2012
First available in Project Euclid: 8 February 2013

zbMATH: 1296.60132
MathSciNet: MR3097964
Digital Object Identifier: 10.1214/12-AOS1055

Primary: 60G15 , 60G60
Secondary: 52A22 , 60D05 , 60G70 , 62M30

Keywords: Euler characteristic , fMRI , Gaussian kinematic formula , Lipschitz–Killing curvatures , Random fields , Rotation space , scale space , thresholding

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 6 • December 2012
Back to Top