# Riemannian Lp Averaging on the 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.
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https://hal-mines-paristech.archives-ouvertes.fr/hal-00789164
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Submitted on : Saturday, February 16, 2013 - 9:37:25 AM
Last modification on : Wednesday, November 17, 2021 - 12:27:12 PM
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• HAL Id : hal-00789164, version 1

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Jesus Angulo. Riemannian Lp Averaging on the Lie Group of Nonzero Quaternions. 2012. ⟨hal-00789164v1⟩

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