Kevin C. Wolfe, Michael Mashner, Gregory S. Chirikjian, Bayesian Fusion on Lie Groups, JOURNAL OF ALGEBRAIC STATISTICS Vol. 2, No. 1, 2011, 75-97, DOI: 10.18409/jas.v2i1.11.
An increasing number of real-world problems involve the measurement of data, and the computation of estimates, on Lie groups. Moreover, establishing confidence in the resulting estimates is important. This paper therefore seeks to contribute to a larger theoretical framework that generalizes classical multivariate statistical analysis from Euclidean space to the setting of Lie groups. The particular focus here is on extending Bayesian fusion, based on exponential families of probability densities, from the Euclidean setting to Lie groups. The definition and properties of a new kind of Gaussian distribution for connected unimodular Lie groups are articulated, and explicit formulas and algorithms are given for finding the mean and covariance of the fusion model based on the means and covariances of the constituent probability densities. The Lie groups that find the most applications in engineering are rotation groups and groups of rigid-body motions. Orientational (rotation-group) data and associated algorithms for estimation arise in problems including satellite attitude, molecular spectroscopy, and global geological studies. In robotics and manufacturing, quantifying errors in the position and orientation of tools and parts are important for task performance and quality control. Developing a general way to handle problems on Lie groups can be applied to all of these problems. In particular, we study the issue of how to ‘fuse’ two such Gaussians and how to obtain a new Gaussian of the same form that is ‘close to’ the fused density.This is done at two levels of approximation that result from truncating the Baker-Campbell-Hausdorff formula with different numbers of terms. Algorithms are developed and numerical results are presented that are shown to generate the equivalent fused density with good accuracy