Procrustes Methods in the Statistical Analysis of Shape

SUMMARY Two geometrical figures, X and Y, in RK, each consisting of N landmark points, have the same shape if they differ by at most a rotation, a translation and isotropic scaling. This paper presents a model-based Procrustes approach to analysing sets of shapes. With few exceptions, the metric geometry of shape spaces is quite complicated. We develop a basic understanding through the familiar QR and singular value decompositions of multivariate analysis. The strategy underlying the use of Procrustes methods is to work directly with the N x K co-ordinate matrix, while allowing for an arbitrary similarity transformation at all stages of model formulation, estimation and inference. A Gaussian model for landmark data is defined for a single population and generalized to two-sample, analysis-of-variance and regression models. Maximum likelihood estimation is by least squares superimposition of the figures; we describe generalizations of Procrustes techniques to allow non-isotropic errors at and between landmarks. Inference is based on an N x K linear multivariate Procrustes statistic that, in a double-rotated co-ordinate system, is a simple but singular linear transformation of the errors at landmarks. However, the superimposition metric used for fitting, and the model metric, or covariance, used for testing, may not coincide. Estimates of means are consistent for many reasonable choices of superimposition metric. The estimates are efficient (maximum likelihood estimates) when the metrics coincide. F-ratio and Hotelling's R 2-tests for shape differences in one- and two-sample data are derived from the distribution of the Procrustes statistic. The techniques are applied to the shapes associated with hydrocephaly and nutritional differences in young rats.