Local Maxima and the Expected Euler Characteristic of Excursion Sets of χ 2, F and t Fields

The maximum of a Gaussian random field was used by Worsley et al. (1992) to test for activation at an unknown point in positron emission tomography images of blood flow in the human brain. The Euler characteristic of excursion sets was used as an estimator of the number of regions of activation. The expected Euler characteristic of excursion sets of stationary Gaussian random fields has been derived by Adler and Hasofer (1976) and Adler (1981). In this paper we extend the results of Adler (1981) to χ 2 , F and t fields. The theory is applied to some three-dimensional images of cerebral blood flow from a study on pain perception.