Rotation and scale space random fields and the Gaussian kinematic formula

TL;DR

This paper introduces a new method using the Gaussian kinematic formula to analyze rotation and scale space random fields, extending previous results to higher dimensions and non-Gaussian cases in fMRI data analysis.

Contribution

It provides a novel approach that generalizes existing results to broader classes of random fields and higher dimensions, simplifying calculations and enabling future research.

Findings

Extended results to non-Gaussian random fields

Achieved higher-dimensional analysis without computer algebra

Provided a clearer solution framework for related problems

Abstract

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.