A multi-dimensional scaling approach to shape analysis

We propose an alternative to Kendall's shape space for reflection shapes of configurations in Rm with k labelled vertices, where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations. The proposed approach embeds the space of such shapes into the space P( k - 1) of ( k - 1) x ( k - 1) real symmetric positive semidefinite matrices, which is the closure of an open subset of a Euclidean space, and defines mean shape as the natural projection of Euclidean means in P( k - 1) on to the embedded copy of the shape space. This approach has strong connections with multi- dimensional scaling, and the mean shape so defined gives good approximations to other commonly used definitions of mean shape. We also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions.