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A Geometric Perspective on Fusing Gaussian Distributions on Lie Groups

Yixiao Ge, Pieter van Goor, Robert Mahony

TL;DR

This work tackles the problem of fusing independent concentrated Gaussians defined on Lie groups by mapping distributions to a common exponential-coordinate frame and applying Gaussian fusion. It introduces an extended concentrated Gaussian model that decouples the reference point from the distribution mean, and develops a three-step fusion algorithm that uses either the exact Jacobian, Taylor approximations, or curvature-based parallel transport to align covariances. A curvature-corrected parallel transport approach yields accuracy comparable to state-of-the-art BCH-based optimizers on $SO(3)$ but with significantly lower computational cost, highlighting its practical viability for real-time manifold-valued state estimation. The results establish a general, geometry-aware framework for fusion on Lie groups that can leverage standard matrix-exponential computations in common libraries.

Abstract

Stochastic inference on Lie groups plays a key role in state estimation problems such as; inertial navigation, visual inertial odometry, pose estimation in virtual reality, etc. A key problem is fusing independent concentrated Gaussian distributions defined at different reference points on the group. In this paper we approximate distributions at different points in the group in a single set of exponential coordinates and then use classical Gaussian fusion to obtain the fused posteriori in those coordinates. We consider several approximations including the exact Jacobian of the change of coordinate map, first and second order Taylor's expansions of the Jacobian, and parallel transport with and without curvature correction associated with the underlying geometry of the Lie group. Preliminary results on SO(3) demonstrate that a novel approximation using parallel transport with curvature correction achieves similar accuracy to the state-of-the-art optimisation based algorithms at a fraction of the computational cost.

A Geometric Perspective on Fusing Gaussian Distributions on Lie Groups

TL;DR

This work tackles the problem of fusing independent concentrated Gaussians defined on Lie groups by mapping distributions to a common exponential-coordinate frame and applying Gaussian fusion. It introduces an extended concentrated Gaussian model that decouples the reference point from the distribution mean, and develops a three-step fusion algorithm that uses either the exact Jacobian, Taylor approximations, or curvature-based parallel transport to align covariances. A curvature-corrected parallel transport approach yields accuracy comparable to state-of-the-art BCH-based optimizers on but with significantly lower computational cost, highlighting its practical viability for real-time manifold-valued state estimation. The results establish a general, geometry-aware framework for fusion on Lie groups that can leverage standard matrix-exponential computations in common libraries.

Abstract

Stochastic inference on Lie groups plays a key role in state estimation problems such as; inertial navigation, visual inertial odometry, pose estimation in virtual reality, etc. A key problem is fusing independent concentrated Gaussian distributions defined at different reference points on the group. In this paper we approximate distributions at different points in the group in a single set of exponential coordinates and then use classical Gaussian fusion to obtain the fused posteriori in those coordinates. We consider several approximations including the exact Jacobian of the change of coordinate map, first and second order Taylor's expansions of the Jacobian, and parallel transport with and without curvature correction associated with the underlying geometry of the Lie group. Preliminary results on SO(3) demonstrate that a novel approximation using parallel transport with curvature correction achieves similar accuracy to the state-of-the-art optimisation based algorithms at a fraction of the computational cost.
Paper Structure (13 sections, 2 theorems, 44 equations, 3 figures)

This paper contains 13 sections, 2 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Lie groups
  4. Connection, curvature and parallel transport
  5. Problem Formulation
  6. Extended Concentrated Gaussian
  7. Fusion problem
  8. Coordinate Representation of Concentrated Gaussian Distribution
  9. Changing reference
  10. Approximation with Curvature
  11. Fusion on Lie groups
  12. Simulation
  13. Conclusion

Key Result

Lemma 4.1

Given an extended concentrated Gaussian distribution $p(g) = \mathbf{N}_{x_1}(\mu_1,\Sigma_1)$ on $\mathbf{G}$ and a point $x_2 \in \mathbf{G}$ then the concentrated Gaussian $q(g) = \mathbf{N}_{x_2}(\mu_2,\Sigma_2)$ with parameters minimises the Kullback-Leibler divergence of $q(g)$ with respect to $p(g)$ up to second-order linearisation error.

Figures (3)

  • Figure 1: Comparative study of the average error \ref{['eq:err_metric']} and the relative processing time of each fusion method. Relative processing time is measured as a ratio of algorithm run-time with respect to run-time for the naive fusion ($\diamond$) algorithm. "Jac Full($\triangle$)" refers to using the analytic form of the Jacobian of the transition functions. "Jac 1st($\triangle$)" and "Jac 2nd($\triangle$)" refer to first and second order Taylor approximations of the Jacobian. "PT($\star$)" refers to parallel transport and "PTC($\star$)" to parallel transport with curvature correction. "BCH 1st($\square$)" and "BCH 2nd($\square$)" refer to the Baker-Campbell-Hausdorff approximation methods proposed in wolfeBayesianFusionLie2011.
  • Figure 2: Estimation error using different approximation methods with fixed $\gamma = 1.0$ or $\xi=1.0$.
  • Figure 3: Heatmaps showing the estimation error with different approximation methods. The full Jacobian and PTC perform identically, while second-order BCH method has slightly better performance for small $\gamma$.

Theorems & Definitions (5)

  • Lemma 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • proof