Principal component geodesics for planar shape spaces

In this paper a numerical method to compute principal component geodesics for Kendall's planar shape spaces-which are essentially complex projective spaces-is presented. Underlying is the notion of principal component analysis based on geodesics for non-Euclidean manifolds as proposed in an earlier paper by Huckemann and Ziezold [S. Huckemann, H. Ziezold, Principal component analysis for Riemannian manifolds with an application to triangular shape spaces, Adv. Appl. Prob. (SGSA) 38 (2) (2006) 299-319]. Currently, principal component analysis for shape spaces is done on the basis of a Euclidean approximation. In this paper, using well-studied datasets and numerical simulations, these approximation errors are discussed. Overall, the error distribution is rather dispersed. The numerical findings back the notion that the Euclidean approximation is good for highly concentrated data. For low concentration, however, the error can be strongly notable. This is in particular the case for a small number of landmarks. For highly concentrated data, stronger anisotropicity and a larger number of landmarks may also increase the error.