A computable characterization of the extrinsic mean of reflection shapes and its asymptotic properties

The reflection shapes of configurations in Rm with k landmarks consist of all the geometric information that is invariant under compositions of similarity and reflection transformations. By considering the corresponding Schoenberg embedding, we embed the reflection shape space into the Euclidean space of all (k - 1) by (k - 1) real symmetric matrices. In this paper, we provide a computable formula of the extrinsic mean of the reflection shapes in arbitrary dimensions. Moreover, the asymptotic analysis of the extrinsic mean of the reflection shapes is studied. By using the differentiability of spectral operators, we obtain a central limit theorem of the sample extrinsic mean of the reflection shapes. As a direct application, the two-example hypothesis test of the reflection shapes is also derived.