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CLOSURE: Fast Quantification of Pose Uncertainty Sets

Yihuai Gao, Yukai Tang, Han Qi, Heng Yang

TL;DR

This work tackles rigorous uncertainty quantification for 6D pose estimation under unknown-but-bounded noise by formulating PURSE, a pose uncertainty set in SE(3). It recasts PURSE into a minimum enclosing geodesic ball (MEGB) problem and introduces CLOSURE, a GPU-accelerated boundary-sampling algorithm that densely samples PURSE boundaries and computes an inner MEGB via miniball, with a certificate of tightness relative to state-of-the-art outer bounds. Across LM-O, 3DMatch, and LM, CLOSURE achieves tight uncertainty bounds in under 0.3 s on a single RTX 3090, outperforming slower outer-approximation methods by large factors while providing a real-time uncertainty certificate. The approach is demonstrated on both keypoint-based and direct-prediction pose estimation pipelines, and the results show substantial reductions in calibrated pose uncertainty, enabling practical real-time deployment in robotic perception systems.

Abstract

We investigate uncertainty quantification of 6D pose estimation from learned noisy measurements (e.g. keypoints and pose hypotheses). Assuming unknown-but-bounded measurement noises, a pose uncertainty set (PURSE) is a subset of SE(3) that contains all possible 6D poses compatible with the measurements. Despite being simple to formulate and its ability to embed uncertainty, the PURSE is difficult to manipulate and interpret due to the many abstract nonconvex polynomial constraints. An appealing simplification of PURSE is to find its minimum enclosing geodesic ball (MEGB), i.e., a point pose estimation with minimum worst-case error bound. We contribute (i) a geometric interpretation of the nonconvex PURSE, and (ii) a fast algorithm to inner approximate the MEGB. Particularly, we show the PURSE corresponds to the feasible set of a constrained dynamical system or the intersection of multiple geodesic balls, and this perspective allows us to design an algorithm to densely sample the boundary of the PURSE through strategic random walks. We then use the miniball algorithm to compute the MEGB of PURSE samples, leading to an inner approximation. Our algorithm is named CLOSURE (enClosing baLl frOm purSe boUndaRy samplEs) and it enables computing a certificate of approximation tightness by calculating the relative size ratio between the inner approximation and the outer approximation. Running on a single RTX 3090 GPU, CLOSURE achieves the relative ratio of 92.8% on the LM-O dataset, 91.4% on the 3DMatch dataset and 96.6% on the LM dataset with the average runtime less than 0.3 second. Obtaining comparable worst-case error bound but 398x 833x and 23.6x faster than the outer approximation GRCC, CLOSURE enables uncertainty quantification of 6D pose estimation to be implemented in real-time robot perception applications.

CLOSURE: Fast Quantification of Pose Uncertainty Sets

TL;DR

This work tackles rigorous uncertainty quantification for 6D pose estimation under unknown-but-bounded noise by formulating PURSE, a pose uncertainty set in SE(3). It recasts PURSE into a minimum enclosing geodesic ball (MEGB) problem and introduces CLOSURE, a GPU-accelerated boundary-sampling algorithm that densely samples PURSE boundaries and computes an inner MEGB via miniball, with a certificate of tightness relative to state-of-the-art outer bounds. Across LM-O, 3DMatch, and LM, CLOSURE achieves tight uncertainty bounds in under 0.3 s on a single RTX 3090, outperforming slower outer-approximation methods by large factors while providing a real-time uncertainty certificate. The approach is demonstrated on both keypoint-based and direct-prediction pose estimation pipelines, and the results show substantial reductions in calibrated pose uncertainty, enabling practical real-time deployment in robotic perception systems.

Abstract

We investigate uncertainty quantification of 6D pose estimation from learned noisy measurements (e.g. keypoints and pose hypotheses). Assuming unknown-but-bounded measurement noises, a pose uncertainty set (PURSE) is a subset of SE(3) that contains all possible 6D poses compatible with the measurements. Despite being simple to formulate and its ability to embed uncertainty, the PURSE is difficult to manipulate and interpret due to the many abstract nonconvex polynomial constraints. An appealing simplification of PURSE is to find its minimum enclosing geodesic ball (MEGB), i.e., a point pose estimation with minimum worst-case error bound. We contribute (i) a geometric interpretation of the nonconvex PURSE, and (ii) a fast algorithm to inner approximate the MEGB. Particularly, we show the PURSE corresponds to the feasible set of a constrained dynamical system or the intersection of multiple geodesic balls, and this perspective allows us to design an algorithm to densely sample the boundary of the PURSE through strategic random walks. We then use the miniball algorithm to compute the MEGB of PURSE samples, leading to an inner approximation. Our algorithm is named CLOSURE (enClosing baLl frOm purSe boUndaRy samplEs) and it enables computing a certificate of approximation tightness by calculating the relative size ratio between the inner approximation and the outer approximation. Running on a single RTX 3090 GPU, CLOSURE achieves the relative ratio of 92.8% on the LM-O dataset, 91.4% on the 3DMatch dataset and 96.6% on the LM dataset with the average runtime less than 0.3 second. Obtaining comparable worst-case error bound but 398x 833x and 23.6x faster than the outer approximation GRCC, CLOSURE enables uncertainty quantification of 6D pose estimation to be implemented in real-time robot perception applications.
Paper Structure (22 sections, 7 theorems, 68 equations, 12 figures, 5 tables, 9 algorithms)

This paper contains 22 sections, 7 theorems, 68 equations, 12 figures, 5 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Minimum Enclosing Geodesic Ball
  3. Geometric Structures
  4. The CLOSURE Algorithm
  5. Experiments
  6. Effectiveness of the Boundary Sampler
  7. Certificate of Approximation Tightness
  8. LM-O
  9. 3DMatch
  10. LM
  11. Time Decomposition
  12. Ablation: Sensitivity to Parameters
  13. How Large is the Pose Uncertainty?
  14. Conclusion
  15. Non-Gaussian Measurement Noise in Example \ref{['ex:3D3D']}
  16. ...and 7 more sections

Key Result

Proposition 4

Let $\hat{S}_{{R}} \subseteq S_{{R}}$ and $\hat{S}_{{t}} \subseteq S_{{t}}$ be nonempty subsets. Consider the optimizations and and denote their optimizers (resp. optima) to be $\hat{C}$ (resp. $\hat{D}$) and $\hat{c}$ (resp. $\hat{d}$). Then $B_{\mathrm{SO}(3)\xspace}(\hat{C},\hat{D})$ is no greater than the MEGBeq:megb-rot-ball and $B_{ { {\mathbb R}^{3} } }(\hat{c},\hat{d})$ is no greater tha

Figures (12)

  • Figure 1: Illustration of the relationship between the outer approximation (provided by the GRCC algorithm from tang23arxiv-uncertainty), inner approximation (provided by the proposed CLOSURE algorithm) and the ground truth MEGB. Blue dots are the boundary pose samples in PURSE, which are used to compute the inner approximation through the miniball algorithm gartner1999fast.
  • Figure 2: MEGB$B(C^\star,D^\star)$ of a simple rectangle $S$ and MEGB$B(\hat{C},\hat{D})$ of the subset $\hat{S} \subset S$. Note that $\hat{D} < D^\star$ but $B(\hat{C},\hat{D}) \not\subset B(C^\star,D^\star)$.
  • Figure 3: Constrained dynamical systems whose feasible sets correspond to (a) \ref{['eq:PURSE2D3D']} for Example \ref{['ex:2D3D']} and (b) \ref{['eq:PURSE3D3D']} for Example \ref{['ex:3D3D']}.
  • Figure 4: Overview of the CLOSURE algorithm in 2D.
  • Figure 5: Effectiveness of PURSE boundary sampler on (a) an example from LM-O, and (b) an example from 3DMatch. Left: rotation, Right: translation.
  • ...and 7 more figures

Theorems & Definitions (16)

  • Example 1: Keypoint-based Object Pose Estimation yang23cvpr-objectkneip14eccv-upnp
  • Example 2: Keypoint-based Point Cloud Registration yang20tro-teaserchoy2020deep
  • Example 3: Direct Pose Regression kendall15iccv-posenetxiang2017rss-posecnnwen24cvpr-foundationpose
  • Proposition 4: Inner Approximation of MEGB
  • proof
  • Proposition A5: Convexity and (Sub)gradient
  • proof
  • Lemma A6: Cosine rule in $\mathrm{SO}(3)\xspace$ hartley13ijcv-rotation
  • Definition A7: Directional derivative udriste1994convex
  • Definition A8: Subdifferential on Riemannian manifold
  • ...and 6 more