We revisit the popular Procrustes matching procedure of landmark shape analysis and consider the situation where the landmark coordinates have a completely general covariance matrix, extending previous approaches based on factored covariance structures. Procrustes matching is used to compute the Riemannian metric in shape space and is used more widely for carrying out inference such as estimation of mean shape and covariance structure. Rather than matching using the Euclidean distance we consider a general Mahalanobis distance. This approach allows us to consider different variances at each landmark, as well as covariance structure between the landmark coordinates, and more general covariance structures. Explicit expressions are given for the optimal translation and rotation in two dimensions and numerical procedures are used for higher dimensions. Simultaneous estimation of both mean shape and covariance structure is difficult due to the inherent non-identifiability. The method requires the specification of constraints to carry out inference, and we discuss some possible practical choices. We illustrate the methodology using data from fish silhouettes and mouse vertebra images.
