# 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.
Keywords :
Document type :
Journal articles

Cited literature [48 references]

https://hal-mines-paristech.archives-ouvertes.fr/hal-00789164
Contributor : Jesus Angulo Connect in order to contact the contributor
Submitted on : Wednesday, January 21, 2015 - 4:25:37 PM
Last modification on : Thursday, September 24, 2020 - 4:38:04 PM
Long-term archiving on: : Friday, September 11, 2015 - 8:16:22 AM

### File

LpAveragingQuaternions_angulo_...
Files produced by the author(s)

### Citation

Jesus Angulo. Riemannian L p Averaging on Lie Group of Nonzero Quaternions. Advances in Applied Clifford Algebras, Springer Verlag, 2014, 24 (2), pp.355-382. ⟨10.1007/s00006-013-0432-2⟩. ⟨hal-00789164v3⟩

Record views