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

Abstract

This paper is motivated by the problem of detecting local changes or differences in shape between two samples of objects via the nonlinear deformations 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 $T^2$ statistics for detecting a change in mean of the vector deformations at each point 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 random field crosses a high threshold is very accurately approximated by the expected Euler characteristic (EC) of the excursion set of the random field above the threshold. We give an exact expression for the expected EC of a Hotelling’s $T^2$ field, and we study the behavior of the field near local extrema. 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.