Rotation space random fields with an application to fMRI data

Abstract

Siegmund and Worsley considered the problem of testing for a signal with unknown location and scale in a Gaussian random field defined on~$\mathbb{R}^N$. The test statistic was the maximum of a Gaussian random field in an $(N+1)$-dimensional "scale space," N dimensions for location and one dimension for the scale of a smoothing kernel. Siegmund and Worsley used two methods, one involving the expected Euler characteristic of the excursion set and the other involving the volume of tubes, to derive an approximate null distribution. The purpose of this paper is to extend the scale space result to the rotation space random field when N=2, where the maximum is taken over all rotations of the filter as well as scales. We apply this result to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).