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Article Dans Une Revue Advances in Applied Clifford Algebras Année : 2014

## Riemannian L p Averaging on Lie Group of Nonzero Quaternions

Jesus Angulo

#### Résumé

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.

#### Domaines

Informatique [cs] Traitement des images [eess.IV]
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### Dates et versions

hal-00789164 , version 1 (16-02-2013)
hal-00789164 , version 2 (26-10-2013)
hal-00789164 , version 3 (21-01-2015)

### Identifiants

• HAL Id : hal-00789164 , version 3
• DOI :

### Citer

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

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