Particle Filtering on Riemannian Manifolds. Application to Covariance Matrices Tracking

Abstract

Recently, an extension of the particle filter has been proposed in [13]. This extension deals with tracking hidden states moving on a Riemannian manifold. In fact, in addition to the nonlinear dynamics, the system state is constrained to lie on a Riemannian manifold \({\mathcal M }\), which dimension is much lower than the whole embedding space dimension. The Riemannian manifold formulation of the state space model avoids the curse of dimensionality from which suffer most of the particle filter methods. Furthermore, this formulation is the only natural tool when the embedding Euclidean space cannot be defined (the state space is defined in an abstract geometric way) or when the constraints are not easily handled (space of positive definite matrices).