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A Bi-Invariant Statistical Model Parametrized by Mean and Covariance on Rigid Motions
Emmanuel Chevallier1, Nicolas Guigui2
1Institut Fresnel, Aix-Marseille University, CNRS, Centrale Marseille, 13013 Marseille, France.
Abstract:
This paper aims to describe a statistical model of wrapped densities for bi-invariant statistics on the group of rigid motions of a Euclidean space. Probability distributions on the group are constructed from distributions on tangent spaces and pushed to the group by the exponential map. We provide an expression of the Jacobian determinant of the exponential map of S E ( n ) which enables the obtaining of explicit expressions of the densities on the group. Besides having explicit expressions, the strengths of this statistical model are that densities are parametrized by their moments and are easy to sample from. Unfortunately, we are not able to provide convergence rates for density estimation. We provide instead a numerical comparison between the moment-matching estimators on S E ( 2 ) and R 3 , which shows similar behaviors.
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