Finite-dimensional filters

The class of optimal nonlinear finite-dimensional recursive filters found by Benes is extended to include cases in which the drift in the state propagation equation is a general linear function plus the gradient of a scalar potential. It is shown that if the state space is one dimensional, then the deterministic systems underlying the Benes filter fall into five classes, depending on the asymptotic behaviour of the state at large times. Only two of these classes can be obtained using the Kalman filter. It is shown that an arbitrary deterministic trajectory can be approximated at small times to an accuracy of O(t(5)) by a trajectory for which the Benes filter is appropriate. The Benes construction is the starting point for the development of new finite-dimensional recursive approximations to the optimal filter. One of the new filters is applied to a simple tracking problem taken from computer vision, and its performance compared with that of the extended Kalman filter.