A Geometric Perspective on Fusing Gaussian Distributions on Lie Groups

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 letter 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
$\mathbf {SO}(3)$
demonstrate that a novel approximation using parallel transport with curvature correction achieves similar accuracy to the state-of-the-art optimization based algorithms at a fraction of the computational cost.