Riemannian L p Averaging on Lie Group of Nonzero Quaternions

Abstract

This paper discusses quaternion $L^p$ geometric weighting averaging working on the multiplicative Lie group of nonzero quaternions $\mathbb{H}^{*}$, endowed with its natural bi-invariant Riemannian metric. Algorithms for computing the Riemannian $L^p$ center of mass of a set of points, with $1 \leq p \leq \infty$ (i.e., median, mean, $L^p$ barycenter and minimax center), are particularized to the case of $\mathbb{H}^{*}$.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 $\mathbf{1}$, associated to the Lie algebra of $\mathbb{H}^{*}$. In the tangent space, Euclidean algorithms for $L^p$ 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 Fr\'{e}chet-Karcher barycenter ($p=2$), the quaternion Fermat-Weber point ($p=1$) and the quaternion Riemannian 1-center ($p=+\infty$). Besides giving explicit forms of these algorithms, their application for quaternion image processing is shown by introducing the notion of quaternion bilateral filtering

Extrinsic calibration for motion estimation using unit quaternions and particle filtering

“…Gradients were formulated in terms of the Riemannian metric on the SO(3) manifold, and the Karcher mean was obtained with a Riemannian gradient descent method, which was shown to converge to the Karcher mean under reasonable conditions. Similar results were presented in (Angulo, 2014) for nonzero quaternions. In this work, the nonzero quaternions were treated as a Lie group with a Riemannian metric, and a range of optimization problems was discussed, including the Karcher mean for nonzero quaternions.…”

“…The set of unit quaternions H u is a Lie group where the group action is the quaternion product (Angulo, 2014). The corresponding Lie algebra g is the set of vectors u ∈ R 3 , while the elements of the tangent plane T q H u at q ∈ H u are given by q ⊗ u.…”

“…In this section the corresponding solution for quaternions will be presented. This was also treated in (Angulo, 2014) for nonzero quaternions, which are quaternions that are not necessarily unit quaternions. The problem is formulated as follows.…”

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