Geometric Observability Index: An Operator-Theoretic Framework for Per-Feature Sensitivity, Weak Observability, and Dynamic Effects in SE(3) Pose Estimation

We present a unified operator-theoretic framework for analyzing per-feature sensitivity in camera pose estimation on the Lie group SE(3). Classical sensitivity tools - conditioning analyses, Euclidean perturbation arguments, and Fisher information bounds - do not explain how individual image features influence the pose estimate, nor why dynamic or inconsistent observations can disproportionately distort modern SLAM and structure-from-motion systems. To address this gap, we extend influence function theory to matrix Lie groups and derive an intrinsic perturbation operator for left-trivialized M-estimators on SE(3). The resulting Geometric Observability Index (GOI) quantifies the contribution of a single measurement through the curvature operator and the Lie algebraic structure of the observable subspace. GOI admits a spectral decomposition along the principal directions of the observable curvature, revealing a direct correspondence between weak observability and amplified sensitivity. In the population regime, GOI coincides with the Fisher information geometry on SE(3), yielding a single-measurement analogue of the Cramer-Rao bound. The same spectral mechanism explains classical degeneracies such as pure rotation and vanishing parallax, as well as dynamic feature amplification along weak curvature directions. Overall, GOI provides a geometrically consistent description of measurement influence that unifies conditioning analysis, Fisher information geometry, influence function theory, and dynamic scene detectability through the spectral geometry of the curvature operator. Because these quantities arise directly within Gauss-Newton pipelines, the curvature spectrum and GOI also yield lightweight, training-free diagnostic signals for identifying dynamic features and detecting weak observability configurations without modifying existing SLAM architectures.