Bicycle chain shape models

In this paper we introduce landmark-based pre-shapes which allow mixing of anatomical landmarks and pseudo-landmarks, constraining consecutive pseudo-landmarks to satisfy planar equidistance relations. This defines naturally a structure of Riemannian manifold on these preshapes, with a natural action of the group of planar rotations. Orbits define the shapes. We develop a geodesic generalized procrustes analysis procedure for a sample set on such a preshape spaces and use it to compute principal geodesic analysis. We demonstrate it on an elementary synthetic example as well on a dataset of manually annotated vertebra shapes from x-ray. We re-landmark them consistently and show that PGA captures the variability of the dataset better than its linear counterpart, PCA.