Riemannian L Averaging on the Lie Group of Nonzero Quaternions

This paper discusses quaternion L geometric weighting averaging working on the multiplicative Lie group of nonzero quaternionsH∗, endowed with its natural bi-invariant Riemannian metric. Algorithms for computing the Riemannian L center of mass of a set of points, with 1 ≤ p ≤ ∞ (i.e., median, mean, L barycenter and minimax center), are particularized to the case of H∗. Two different approaches are considered. The first formulation is based on computing the logarithm of quaternions which maps them to the Euclidean tangent space at the identity 1, associated to the Lie algebra of H∗. In the tangent space, Euclidean algorithms for L center of mass can be naturally applied. The second formulation is a family of methods based on gradient descent algorithms aiming at minimizing the sum of quaternion geodesic distances raised to power p. These algorithms converges to the quaternion Fréchet-Karcher barycenter (p = 2), the quaternion Fermat-Weber point (p = 1) and the quaternion Riemannian 1-center (p = +∞). Besides giving explicit forms of these algorithms, their application for quaternion image processing is shown by introducing the notion of quaternion bilateral filtering.