Keskeiset käsitteet
Fusing Gaussian distributions on Lie groups using geometric methods achieves accuracy at a lower computational cost.
Tiivistelmä
This preprint discusses the fusion of concentrated Gaussian distributions on Lie groups from a geometric perspective. The paper introduces approximations for fusing independent Gaussians defined at different reference points on the group. It explores various methods, including Jacobian approximations, parallel transport with curvature correction, and optimization algorithms. Results show that parallel transport with curvature correction achieves similar accuracy to optimization-based algorithms but at a fraction of the computational cost. The methodology involves transforming distributions into a unified set of coordinates, applying classical Gaussian fusion, and resetting the fused distribution around a new mean.
Abstract:
Stochastic inference on Lie groups is crucial for state estimation problems.
Approximating distributions in exponential coordinates simplifies fusion.
Various approximation methods are explored for accurate fusion at low computational cost.
Introduction:
Bayes theorem applied to parametric distributions like Gaussians.
Rise in interest in manifolds and Lie groups for robotics and avionics systems.
Extended Kalman filter methods conduct fusion in tangent spaces using local coordinates.
Preliminaries:
Definitions related to Lie groups, translations, adjoint maps, exponential mapping.
Jacobi field application for computing Jacobian approximations.
Concentrated Gaussian Distribution:
Construction of concentrated Gaussian distribution on Lie group.
Extension to allow offset mean in the Lie algebra for modeling purposes.
Changing Reference:
Introduction of extended concentrated Gaussian distribution around non-coincident means.
Formulation provided through lemma and proof for minimizing Kullback-Leibler divergence.
Approximation with Curvature:
Proposal to approximate Jacobian using geometric structure of the Lie group.
Theorem application linking Jacobian approximation with Jacobi field.
Fusion on Lie Groups:
Methodology proposed for fusing multiple concentrated Gaussians on Lie groups.
Steps involving choosing reference point, applying approximation methods, and resetting fused estimate.
Simulation:
Evaluation of proposed methods through simulation using SO(3) as the Lie group of interest.
Comparison of different approximation methods and their performance metrics.
Tilastot
"Relative Processing Time 10−1"
"Average Error Naive"
"BCH 1st"
"BCH 2nd"
"Jac 1st"
"Jac 2nd"
"Jac Full"
"PT"
"PTC"
Lainaukset
"The closer ˆx is to the correct group-mean, the less approximation error will be incurred before the full fusion process is undertaken."
"The parallel transport method can also be thought of as a first order method."
"The second order BCH method achieves the lowest average error in our simulations."