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

We introduce the Geometric Observability Index (GOI), a per-feature sensitivity measure for pose estimation on SE(3). For a Gauss-Newton curvature matrix $H=E[J^\top WJ]$ and a Riemannian metric $G$ on the Lie algebra, the index is the $G$-norm of the influence a single measurement exerts on the estimated pose: $\mathrm{GOI}(z)=\|\mathcal{A}_{OO}^{-1}P_O\,\varphi(z)\|_G$, where $ψ(z)=J^\top Wr(z)$ is the score, $\varphi=G^{-1}ψ$ its gradient representative, $\mathcal{A}=G^{-1}H$ the curvature operator (self-adjoint in the $G$-inner product), $O=\mathrm{range}(\mathcal{A})$ the observable subspace, and $\mathcal{A}_{OO}$ its restriction. This single object (i) equals the norm of the M-estimator influence function, (ii) is governed by the Fisher information, which coincides with the curvature, (iii) exposes weak observability through the smallest eigenvalue $λ_{\min}$, which (iv) also governs finite-sample stability. Operationally the theory cuts both ways. The index is the exact per-measurement attribution: it predicts the true leave-one-out pose shift with log-correlation $r=1.00$. But we also prove that the influence standardized by its inlier null covariance collapses exactly to the classical chi-square residual statistic: residual gating is the leverage-corrected influence test, explaining its robustness from first principles, while raw-influence gating conflates a measurement's information with its harm and over-rejects high-leverage inliers in weakly observable geometry. Experiments on synthetic problems, five TUM RGB-D dynamic sequences, and two KITTI odometry sequences confirm the picture: the two criteria coincide under well-conditioned geometry, and raw-influence gating degrades significantly at $\mathrm{cond}(H)\approx 10^4$, as the leverage analysis predicts for noise-dominated weak directions. All quantitative claims are validated; code is released.

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