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\nor differences in shape between two samples of objects via the nonlinear\ndeformations required to map each object to an atlas standard. Local shape\nchanges are then detected by high values of the random field of\nHotelling’s $T^2$ statistics for detecting a change in mean of the\nvector deformations at each point in the object. To control the null\nprobability of detecting a local shape change, we use the recent result of\nAdler that the probability that a random field crosses a high threshold is very\naccurately approximated by the expected Euler characteristic (EC) of the\nexcursion set of the random field above the threshold. We give an exact\nexpression for the expected EC of a Hotelling’s $T^2$ field, and we\nstudy the behavior of the field near local extrema. This extends previous\nresults for Gaussian random fields by Adler and $\\chi^2$, $t$ and $F$ fields by\nWorsley and Cao. For illustration, these results are applied to the detection\nof differences in brain shape between a sample of 29 males and 23 females.

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Affiliated Institutions

• McGill University CA

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