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 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).

Variability of fMRI‐response patterns at different spatial observation scales

“…First, in scale search for brain responses using multi-filter analysis, multiple statistical tests are performed for different spatial scales, these multiple tests need to be taken into account to control the false-positive rate. To this aim, scale-space searches have been integrated with Gaussian random field theory [Shafie et al, 2003;Siegmund and Worsley, 1995;Worsley, 2001;Worsley et al, 1996Worsley et al, , 1997. In contrast, in the present study, we performed correction for multiple tests only for the spatial but not for the scale domain because we aimed not to actually perform a scale search, but rather investigate the differences between typical single-scale fMRI analyses when performed at different spatial scales, i.e., using different filter widths.…”

“…A unified Pvalue for local maxima in 4D scale-space searches (three spatial dimensions and one extra dimension for scale) was presented [Worsley et al, 1996] and later extended to higher dimensions and to v 2 random fields [Worsley, 2001]. Scale-space searches were further generalized to rotating filters with a better detection power for ellipsoidally shaped brain responses [Shafie et al, 2003] and to cortical surface-constrained analysis of fMRI data [Andrade et al, 2001]. When examining each scale separately, multiple clustered peaks may be blurred together and detected as a single wide peak at a larger scale [Worsley et al, 1996].…”

A Global Bayes Factor for Observations on an Infinite-Dimensional Hilbert Space, Applied to Signal Detection in fMRI

“…This data set has been analyzed by several researchers using the χ 2 scale space method (Worsley 2001), rotation space random field method (Shafie, Sigal, Siegmund, and Worsley 2003), and Bayesian method (Rohani et al 2006). To apply the random field methodology, these researchers fitted a sine wave at each fixed pixel and extracted two images.…”

“…The Matched Filter Theorem justifies smoothing the image before any analysis with a filter of the form of signal. The smoothing assumption is essential for methods proposed by Worsley (2001) and Shafie et al (2003). In order to get a smooth random field, the image is smoothed before analyzing data with a Gaussian filter.…”

Adaptive spatial smoothing of fMRI images

“…While the use of a fixed Gaussian kernel is by far the most common approach towards smoothing fMRI data, a number of other studies have suggested alternative approaches. For example, Gaussians of varying width (Poline and Mazoyer, 1994;Worsley et al, 1996) and rotations (Shafie et al, 2003) have been proposed, as well as both wavelets (Van De Ville, Blu, and Unser, 2006) and prolate spheroidal wave functions (Lindquist and Wager, 2008;Lindquist et al, 2006). A common theme in all these methods is that the amount of smoothing is chosen a priori and independently of the data.…”