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

2025-11-18

AI Summary

To address the loss of surface orientation information in classical projection-based shape analysis, this work introduces the Oriented Projection Shape (OPS) space, which explicitly encodes depth direction under the pinhole camera imaging model. Building upon the extrinsic Fréchet framework, we formulate the first extrinsic total variation metric for OPS—rigorously preserving directional structure. We derive a closed-form solution for variance in the planar five-point configuration and establish a general *m*-dimensional extrinsic Fréchet analysis theory. Our method integrates spherical coordinate mean estimation, the delta method, and the generalized Slutsky theorem, coupled with leave-two-out cross-validation for coplanarity diagnosis. Evaluated on the Sope Creek rock dataset, the approach reliably detects coplanarity at the 5% significance level, demonstrating both the efficacy of orientation-aware shape analysis and its statistical robustness under high-concentration data regimes.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

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 $jdir:(SP^m)^qhookrightarrow(R^{m+1})^q,q=k-m-2$. 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}.

Problem

• Develop extrinsic total-variance framework for oriented projective shape analysis
• Address sign-blind limitation of classical projective shape methods
• Identify coplanarity in landmark configurations using oriented projective frames

Innovation

• Oriented projective shape restores surface orientation information
• Extrinsic total-variance index introduced for OPS framework
• Delta-method identifies coplanarity in oriented projective diagnostics