The detection of local shape changes via the geometry of Hotelling's T-2 fields

This paper is motivated by the problem of detecting local changes or differ ences in shape between two samples of objects via the nonlinear deformation s required to map each object to an atlas standard. Local shape changes are then detected by high Values of the random field of Hotelling's T-2 statis tics for detecting a change in mean of the Vector deformations at each poin t in the object. To control the null probability of detecting a local shape change, we use the recent result of Adler that the probability that a rand om field crosses a high threshold is very accurately approximated by the ex pected Euler characteristic (EC) of the excursion set of the random field a bove the threshold. We give an exact expression for the expected EC of a Ho telling's T-2 field, and we study the behavior of the field near local extr ema. This extends previous results for Gaussian random fields by Adler and chi(2), t and F fields by Worsley and Cao. For illustration, these results are applied to the detection of differences in brain shape between a sample of 29 males and 23 females.