On Averaging in Clifford Groups

Averaging measured data is an important issue in computer vision and robotics. Integrating the pose of an object measured with multiple cameras into a single mean pose is one such example. In many applications data does not belong to a vector space. Instead, data often belongs to a non–linear group manifold as it is the case for orientation data and the group of three–dimensional rotations SO(3). Averaging on the manifold requires the utilization of the associated Riemannian metric resulting in a rather complicated task. Therefore the Euclidean mean with best orthogonal projection is often used as approximation. In SO(3) this can be done by rotation matrices or quaternions. Clifford algebra as a generalization of quaternions allows a general treatment of such approximated averaging for all classical groups. Results for the two–dimensional Lorentz group SO(1, 2) and the related groups SL(2, IR) and SU(1, 1) are presented. The advantage of the proposed Clifford framework lies in its compactness and easiness of use.