[Paper Review] Bayesian Pose Graph Optimization via Bingham Distributions and Tempered Geodesic MCMC

Tolga Birdal, Umut Şimşekli+2 more|arXiv (Cornell University)|May 31, 2018

Robotics and Sensor-Based Localization68 references21 citations

TL;DR

This paper proposes a novel Bayesian pose graph optimization method using Bingham distributions for rotations and Gaussian models for translations, combined with a tempered geodesic MCMC (TG-MCMC) algorithm to enable both global optimization and uncertainty quantification on the quaternion manifold. The approach achieves state-of-the-art initialization quality and provides reliable uncertainty estimates that correlate with data quality, while maintaining competitive runtime and significantly accelerating subsequent bundle adjustment.

ABSTRACT

We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm for initializing pose graph optimization problems, arising in various scenarios such as SFM (structure from motion) or SLAM (simultaneous localization and mapping). TG-MCMC is first of its kind as it unites asymptotically global non-convex optimization on the spherical manifold of quaternions with posterior sampling, in order to provide both reliable initial poses and uncertainty estimates that are informative about the quality of individual solutions. We devise rigorous theoretical convergence guarantees for our method and extensively evaluate it on synthetic and real benchmark datasets. Besides its elegance in formulation and theory, we show that our method is robust to missing data, noise and the estimated uncertainties capture intuitive properties of the data.

Motivation & Objective

• To address the lack of uncertainty quantification in pose graph optimization (PGO) initialization, especially in SLAM and SfM pipelines.
• To develop a method that provides both high-quality initial poses and informative uncertainty estimates without relying on computationally expensive bundle adjustment.
• To unify global non-convex optimization on the quaternion manifold with posterior sampling via a novel MCMC algorithm.
• To provide theoretical convergence guarantees for the proposed tempered geodesic MCMC (TG-MCMC) framework.
• To evaluate robustness to noise, missing data, and varying graph connectivity in synthetic and real-world benchmarks.

Proposed method

• Models rotations using Bingham distributions on the 3-sphere (S^3) and translations using multivariate Gaussians, enabling principled probabilistic modeling of relative pose measurements.
• Proposes a tempered geodesic MCMC (TG-MCMC) algorithm that leverages known geodesic flows on the manifold of quaternions to perform efficient sampling and optimization.
• Controls the algorithm’s behavior via a temperature parameter: at high temperatures, it performs posterior sampling for uncertainty quantification; at low temperatures, it acts as a global optimizer converging to the MAP estimate.
• Uses geodesic flows to navigate the Riemannian manifold of SE(3), avoiding issues with non-manifold projections and improving convergence stability.
• Employs a Hamiltonian Monte Carlo (HMC) framework adapted to the spherical manifold, with proper retraction and metric considerations for quaternions.
• Integrates the method into PGO initialization pipelines, enabling downstream bundle adjustment to converge faster and more reliably.

Experimental results

Research questions

• RQ1 Can a unified framework simultaneously provide high-quality PGO initialization and reliable uncertainty estimates?
• RQ2 How does the TG-MCMC algorithm perform in terms of convergence and sampling efficiency on the non-convex, Riemannian manifold of quaternions?
• RQ3 Can the uncertainty estimates produced by the method reflect intuitive data quality properties, such as distance-related triangulation error or feature matchability?
• RQ4 Does the TG-MCMC initialization improve the convergence speed and success rate of subsequent bundle adjustment compared to existing methods?
• RQ5 How robust is the method to noise, missing data, and sparse or poorly connected pose graphs?

Key findings

• The TG-MCMC algorithm achieves state-of-the-art initialization quality on both synthetic and real-world datasets, outperforming traditional methods like MinSpan and Govindu’s method in pose accuracy.
• The uncertainty estimates produced by the method correlate strongly with data quality: distant structures and hard-to-match regions (e.g., vegetation) are assigned higher uncertainty, as expected.
• On the Madrid Metropolis and South Building datasets, the uncertainty maps clearly reflect reduced triangulation accuracy with increasing distance from the camera cluster.
• The method accelerates subsequent bundle adjustment by up to 21.74× on the South Building dataset when used as initialization, demonstrating significant practical benefit.
• Runtime performance is competitive with state-of-the-art methods, and the inclusion of uncertainty estimation is a unique advantage not offered by other PGO initialization techniques.
• Visualizations show that TG-MCMC successfully evolves from random initialization to a pose graph close to ground truth, confirming robust convergence behavior.