Probabilities and Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Friday, November 17, 2000 - 2:00pm - 2:30pm
Keller 3-180
Xavier Pennec (Institut National de Recherche en Informatique Automatique (INRIA))
Measurements of geometric primitives, such as rotations or rigid transformations, are often noisy and we need to use statistics either to reduce the uncertainty or to compare measurements. Unfortunately, geometric primitives often belong to manifolds and not vector spaces. We have already shown that generalizing too quickly even simple statistical notions could lead to paradoxes. Here, we develop some basic probabilistic tools to work on Riemannian manifolds: the notion of mean value, covariance matrix, normal law, Mahalanobis distance and 2 test. We also present an efficient algorithm to compute the mean value and tractable approximations of the normal and 2 laws for small variances. Finally, we present some applications in medical image analysis, mainly on the computation of the uncertainty of the registration of 3D images.