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 $({\text{RP}}^2{)}^{k-4}$, 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 ${\mathbb{R}}^3$ determine the same projective point and front–back orientation of a surface is not recorded. Oriented projective shape ($\text{OPS}$) restores this information by working on a product of $k-4$ spheres ${\text{SP}}^2$ 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 Fréchet framework for the m dimensional case from the inclusion $\text{jdir}:({\text{SP}}^m{)}^q\hookrightarrow ({\mathbb{R}}^{m+1}{)}^q,q=k-m-2$. 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 $5\mathrm{\%}$ level, confirming the concentrated data coplanarity PS result in Patrangenaru(2001)\cite{Patrangenaru2001}.