A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups

Abstract

Motivated by applications in fuzzy control, robotics and vision, this paper considers the problem of computing the centre of mass (precisely, the Karcher mean) of a set of points defined on a compact Lie group, such as the special orthogonal group consisting of all orthogonal matrices with unit determinant. An iterative algorithm, whose derivation is based on the geometry of the problem, is proposed. It is proved to be globally convergent. Interestingly, the proof starts by showing the algorithm is actually a Riemannian gradient descent algorithm with fixed step size.

Optimal Data Fitting on Lie Groups: a Coset Approach

“…q −1 i · y on G/H. It is well-known that the gradient of the Karcher mean cost c( [4,7,11]. Thus the gradient of g with respect to the product metric on…”

“…The form (6) of the cost suggests the following gradient descent algorithm to minimize g as an adaption of the Karcher mean algorithm [4,7,11].…”

Distributed calibration of Camera Sensor Networks

“…, E im get m + 1. Then a more accurate estimate of the rotation can be obtained by averaging these m+1 rotations in the manifold SO(3) using the Karcher mean [23], [24].…”

Vision-based cooperative estimation via multi-agent optimization

“…Unfortunately, the algorithm is not applicable to our problem since the object's pose takes values in a Takeshi Hatanaka and Masayuki Fujita are with the Department of Mechanical and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, JAPAN, Francesco Bullo is with Department of Mechanical Engineering at the University of California at Santa Barbara non-Eucledean space. Meanwhile, [2] presents a distributed version of the computation algorithm of an average on Special Orthogonal Group called Karcher mean [12]. However, this work focuses on the averaging by assuming that the target's orientation is obtained a priori and does not mention estimation from image data.…”