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

arXiv:2602.05582v2 Announce Type: replace Abstract: 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^op 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: GOI(z)=|A_{OO}^{-1}P_O,arphi(z)|G, where psi(z)=J^op Wr(z) is the score, arphi=G^{-1}psi its gradient representative, A=G^{-1}H the curvature operator (self-adjoint in the G-inner product), O=range(A) the observable subspace, and 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 lambda_{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 cond(H)approx 10^4, as the leverage analysis predicts for noise-dominated weak directions. All quantitative claims are validated; code is released.

Source: arXiv cs.CV | 2026-07-07