Computing the Mean of Geometric Features Application to the Mean Rotation
Résumé
The question we investigate in this article is: what is the mean value of a set of geometric features and how can we compute it? We use as a guiding example one of the most studied type of features in computer vision and robotics: 3D rotations. The usual techniques on points consist of minimizing the least-square criterion, which gives the barycenter, the weighted least-squares or the sum of (squared) Mahalanobis distances. Unfortunately, these techniques rely on the vector space structure of points and generalizing them directly to other types of features could lead to paradoxes \cite{Pennec:JMIV:97}. For instance, computing the barycenter of rotations using rotation matrices, unit quaternions or rotation vectors gives three different results. We present in this article a thorough generalization of the three above criterions to homogeneous Riemannian manifolds that rely only on intrinsic characteristics of the manifold. The necessary condition for the mean rotation, independently derived in \cite{denney96}, is obtained here as a particular case of a general formula. We also propose an intrinsic gradient descent algorithm to obtain the minimum of the criterions and show how to estimate the uncertainty of the resulting estimation. These algorithms prove to be not only accurate but also efficient: computations are only 3 to 4 times longer for rotations than for points. The accuracy prediction of the results is within 1%, which is quite remarkable. The striking similarity of the algorithms' behavior for general features and for points stresses the validity of our approach using Riemannian geometry and lets us anticipate that other statistical results and algorithms could be generalized to manifolds in this framework.
Domaines
Autre [cs.OH]
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