Procrustes Shape Analysis of Planar Point Subsets

Consider a set of points in the plane randomly perturbed about a mean configuration by Gaussian errors. In this paper a Procrustes statistic based on the shapes of subsets of the points is studied, and its approximate distribution is found for small variations. We derive various properties of the distribution including the first two moments, a central limit result and a scaled χ²–-approximation. We concentrate on the independent isotropic Gaussian error case, although the results are valid for general covariance structures. We investigate triangle subsets in detail and in particular the situation where the population mean is regular (i.e. a Delaunay triangulation of the mean of the process is comprised of equilateral triangles of the same size). We examine the variance of the statistic for differently shaped regions and provide an asymptotic result for general shaped regions. The results are applied to an investigation of regularity in human muscle fibre cross-sections.