Cramér-Rao Bound for Lie Group Parameter Estimation With Euclidean Observations and Unknown Covariance Matrix
Résumé
This article addresses the problem of computing a Cramér-Rao bound when the likelihood of Euclidean observations is parameterized by both unknown Lie group (LG) parameters and covariance matrix. To achieve this goal, we leverage the LG structure of the space of positive definite matrices. In this way, we can assemble a global LG parameter that lies on the product of the two groups, on which LG's intrinsic tools can be applied. From this, we derive an inequality on the intrinsic error, which can be seen as the equivalent of the Slepian-Bangs formula on LGs. Subsequently, we obtain a closedform expression of this formula for Euclidean observations. The proposed bound is computed and implemented on two realworld problems involving observations lying in R p , dependent on an unknown LG parameter and an unknown noise covariance matrix: the Wahba's estimation problem on SE(3), and the inference of the pose in SE(3) of a camera from pixel detections.
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