Extrinsic Total-Variance and Coplanarity via Oriented and Classical Projective Shape Analysis

Projective shape analysis provides a geometric framework for studying digital images acquired by pinhole digital cameras. In the classical projective shape (PS) method, landmark configurations are represented in (RP2)k-4, where k is the number of landmarks observed. This representation is invariant under the action of the full projective group on this space and is sign-blind, so opposite directions in R3 determine the same projective point and front--back orientation of a surface is not recorded. Oriented projective shape (OPS) restores this information by working on a product of k-4 spheres SP2 instead of projective space and restricting attention to the orientation-preserving subgroup of projective transformations. In this paper we introduce an extrinsic total-variance index for OPS, resulting in the extrinsic Fr'echet framework for the m dimensional case from the inclusion jdir:(SPm)qhookrightarrow(Rm+1)q,q=k-m-2. In the planar pentad case (m=2, q=1) the sample total extrinsic variance has a closed form in terms of the mean of a random sample of size n of oriented projective coordinates in S2. As an illustration, using an oriented projective frame, we analyze the Sope Creek stone data set, a benchmark and nearly planar example with 41 images and 5 landmarks. Using a delta-method applied to a large sample and a generalized Slutsky theorem argument, for an OPS leave-two-out diagnostic, one identifies coplanarity at the 5% level, confirming the concentrated data coplanarity PS result in Patrangenaru(2001)citePatrangenaru2001.