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 $(\RP^2)^{k-4}$, 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 $\R^{3}$ 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 $\SP^2$ 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 $\jdir:(\SP^m)^q\hookrightarrow(\R^{m+1})^q,q=k-m-2$. 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 $S^2$. 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)\cite{Patrangenaru2001}.