A recursive filter for linear systems on Riemannian manifolds

We present an online, recursive filtering technique to model linear dynamical systems that operate on the state space of symmetric positive definite matrices (tensors) that lie on a Riemannian manifold. The proposed approach describes a predict-and-update computational paradigm, similar to a vector Kalman filter, to estimate the optimal tensor state. We adapt the original Kalman filtering algorithm to appropriately propagate the state over time and assimilate observations, while conforming to the geometry of the manifold. We validate our algorithm with synthetic data experiments and demonstrate its application to visual object tracking using covariance features.