Testing for signals with unknown location and scale in a χ2random field, with an application to fMRI

Abstract

Siegmund and Worsley (1995) considered the problem of testing for signals 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 anN+1 dimensional ‘scale space’,Ndimensions 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 forN=2 andN=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 χ²fields. 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).