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Published online by Cambridge University Press: 04 August 2010
Abstract
A path integral representation is obtained for the optimal probability density function of the system state, conditional on the measurements. For certain non-linear systems the optimal density can be evaluated recursively using only a finite number of statistics. These systems extend the class found by Beneš, in that the drift need not be the gradient of a scalar potential.
In the one dimensional case the trajectories of the deterministic system underlying the Beneš filter fall into five classes according to their behaviour as t → ∞. It is shown that an arbitrary deterministic trajectory can be approximated at small times to an accuracy of O(t5) by a trajectory for which the Benes filter is appropriate.
Introduction
The Kalman-Bucy filter and its many variations are widely used throughout science and engineering for estimating the state of time varying systems. Applications to computer vision in particular are described in. A typical filter contains a model of the system dynamics and of the measurement process. It is given a sequence of measurements obtained over an extended time, and produces from this sequence an estimate of the probability density function for the system state, conditional on the measurements. If the filter is optimal, then the estimated density is the exact conditional density. The best known optimal filter is the Kalman-Bucy filter.
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