An intrinsic Bayesian bound for estimators on the Lie groups SO(3) and SE(3)

In the past decade, estimation on manifolds has been paid an increased attention for making it possible to intrinsically account for constraints on the unknown parameters. Applications are widespread from computer vision to target tracking. To evaluate the performance of the algorithms, it is of interest to derive bounds on the estimation error. This issue has been mostly investigated for parametric estimation on Riemannian manifolds. This paper addresses the problem of computing a Bayesian lower bound on the norm of the estimation error when both the unknown variables and the observations belong to specific manifolds called Lie groups (LGs). For that purpose, a metric is considered that preserves the geometric properties of the latter. We firstly derive an inequality on the correlation matrix of this intrinsic error under the mild assumption of unimodular matrix LGs. Then, we obtain a closed-form expression of the bound for the special orthogonal group SO(3) and the special Euclidean group SE(3). Finally, we detail its derivation in the case when both the prior distribution and the likelihood are concentrated Gaussian distributions. The proposed Bayesian bound is implemented and tested on two estimation problems involving both unknown parameters and observations on SE(3): the inference of the pose of a camera and that of the centroid of a cluster of space debris. •Estimation on Lie groups is being paid an increased attention in several applications for making it possible to intrinsically account for constraints on the unknown parameters.•To evaluate the performance of estimators on Lie groups, it is crucial to be able to compute lower bounds on intrinsic mean squared errors.•For that purpose, we develop a framework which is adapted to unimodular matrix Lie groups.•A tractable bound is proposed for the Lie groups SO(3) and SE(3), which are of specific interest in computer vision and tracking applications.•Its calculation is illustrated and numerically validated in the case when both the parameters of interest and the observations are distributed according to concentrated gaussian distributions.