Abstract

Images from positron emission tomography (PET) and functional magnetic resonance imaging (fMRI) are often modelled as stationary Gaussian random fields, and a general linear model is used to test for the effect of explanatory varaibles on a set of such images (Friston {\sl et al.}, 1994; Worsley and Friston, 1995). Thompson {\sl et al.} (1996) have modelled displacements of brain surfaces as a multivariate Gaussian random field. In order to test for significant local maxima in such fields using the theory of Adler (1981) and its recent refinements (Worsley, 1994, 1995a, 1995b; Siegmund and Worsley, 1995), we need to estimate the roughness of such fields. This is defined as the variance matrix of the derivative of the random field in each dimension. Some methods have been given by Worlsey {\sl et al.} (1992) for the special case of stationary variance of the random field, and where the random field is sampled on a uniform lattice. In this note we generalise to the case of multivariate Gaussian data with unknown non-stationary variance matrix, and non-lattice sampling. This latter is particularly important for studying the displacement of brain surfaces, which will be the subject of a future publication.

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