BOUNDARY CORRECTIONS FOR THE EXPECTED EULER CHARACTERISTIC OF EXCURSION SETS OF RANDOM-FIELDS, WITH AN APPLICATION TO ASTROPHYSICS

Certain images arising in astrophysics and medicine are modelled as sm ooth random fields inside a fixed region, and experimenters are intere sted in the number of 'peaks', or more generally, the topological stru cture of 'hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excur sion set is the set of points where the image exceeds a fixed threshol d, and the Euler characteristic counts the number of connected compone nts in the excursion set minus the number of 'holes'. For high thresho lds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (d ifferential topology) and IG (integral geometry) characteristics, whic h equal the Euler characteristic provided the excursion set does not t ouch the boundary of the region. Worsley (1995) finds a boundary corre ction which gives the expectation of the Euler characteristic itself i n two and three dimensions. The proof uses a representation of the Eul er characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof ta kes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the cre ation of the universe.