https://hal-mines-paristech.archives-ouvertes.fr/hal-00789164v2Angulo, JesusJesusAnguloCMM - Centre de Morphologie Mathématique - Mines Paris - PSL (École nationale supérieure des mines de Paris) - PSL - Université Paris sciences et lettresRiemannian Lp Averaging on Lie Group of Nonzero QuaternionsHAL CCSD2012Lie group of zero quaternionsquaternion averagingRiemannian meanFréchet-Karcher barycenter[INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV]Angulo, Jesus2013-10-26 15:41:272022-10-18 11:02:082013-10-28 08:46:02enPreprints, Working Papers, ...https://hal-mines-paristech.archives-ouvertes.fr/hal-00789164v2application/pdf2This 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.