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.

翻译：本文提出了一种统一的算子理论框架，用于分析李群 SE(3) 上相机位姿估计的逐特征敏感性。经典敏感性分析工具——条件数分析、欧几里得扰动论证以及费舍尔信息界——既无法解释单个图像特征如何影响位姿估计，也无法说明为何动态或不一致的观测会不成比例地扭曲现代 SLAM 与运动恢复结构系统。为弥补这一不足，我们将影响函数理论推广至矩阵李群，并推导出 SE(3) 上左平凡化 M 估计量的本征扰动算子。由此得到的几何可观测性指数通过曲率算子与可观测子空间的李代数结构，量化了单个测量值的贡献。GOI 允许沿可观测曲率主方向进行谱分解，揭示了弱可观测性与放大敏感性之间的直接对应关系。在总体统计意义上，GOI 与 SE(3) 上的费舍尔信息几何相吻合，从而导出了克拉美-罗界的单测量类比形式。同一谱机制解释了经典退化情形（如纯旋转与消失视差）以及沿弱曲率方向的动态特征放大现象。总体而言，GOI 通过曲率算子的谱几何，为测量影响提供了几何一致的描述，统一了条件数分析、费舍尔信息几何、影响函数理论以及动态场景可检测性。由于这些量可直接在高斯-牛顿流程中计算，曲率谱与 GOI 还能为识别动态特征和检测弱可观测配置提供轻量级、无需训练的诊断信号，且无需修改现有 SLAM 架构。