# The Geometric Observability Index: Influence, Fisher Information, and Weak Observability in SE(3) Pose Estimation

arXiv: https://arxiv.org/abs/2602.05582 · HTML: https://mlsanity.com/paper/2602.05582/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 28 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Builds on (most cited first, 10 of 10)

- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [ORB-SLAM3: An Accurate Open-Source Library for Visual, Visual-Inertial and Multi-Map SLAM](https://mlsanity.com/paper/2007.11898/) (2020)
- [ORB-SLAM2: an Open-Source SLAM System for Monocular, Stereo and RGB-D Cameras](https://mlsanity.com/paper/1610.06475/) (2016)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [DynaSLAM: Tracking, Mapping and Inpainting in Dynamic Scenes](https://mlsanity.com/paper/1806.05620/) (2018)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [DS-SLAM: A Semantic Visual SLAM towards Dynamic Environments](https://mlsanity.com/paper/1809.08379/) (2018)
- [The invariant extended Kalman filter as a stable observer](https://mlsanity.com/paper/1410.1465/) (2014)
- [Attention and Anticipation in Fast Visual-Inertial Navigation](https://mlsanity.com/paper/1610.03344/) (2016)
- [Good Feature Matching: Towards Accurate, Robust VO/VSLAM with Low Latency](https://mlsanity.com/paper/2001.00714/) (2020)

## Abstract (arXiv)

We introduce the Geometric Observability Index (GOI), a per-feature sensitivity measure for pose estimation on SE(3): the metric norm of the pose perturbation that a single measurement induces through the (possibly rank-deficient) Gauss-Newton curvature, restricted to the observable subspace. We prove that GOI equals the norm of the M-estimator influence function, that the underlying curvature operator coincides with the Fisher information, and that its smallest observable eigenvalue governs both the worst-case amplification of a measurement's effect and a finite-sample stability radius O(sigma/sqrt(n*lambda_min)). Operationally the theory cuts both ways. GOI is the exact per-measurement attribution, predicting the true leave-one-out pose shift with log-correlation r = 1.00; yet the influence standardized by its inlier null covariance collapses exactly to the classical chi-square residual statistic. Residual gating is thus the leverage-corrected influence test -- a first-principles explanation of its robustness -- while raw-influence gating conflates a measurement's information with its harm and is predicted to over-reject high-leverage inliers in weakly observable geometry. Controlled synthetic experiments validate every quantitative claim, and studies on five TUM RGB-D sequences (four dynamic, one static control) and two KITTI odometry sequences confirm the prediction: parity of the two criteria under well-conditioned geometry, and significant degradation of raw-influence gating at cond(H) of order 10^4. All code is released for reproducibility.