A New Extension of Kalman Filter to Non-Gaussian Priors

In the Kalman filter, the state dynamics is specified by the state equation while the measurement equation characterizes the likelihood. In this paper, we propose a generalized methodology of specifying state dynamics using the conditional density of the states given its neighbors without explicitly defining the state equation. In other words, the typically strict linear constraint on the state dynamics imposed by the state equation is relaxed by specifying the conditional density function and using it as the prior in predicting the state. Based on the above idea, we propose a sampling-based Kalman Filter (KF) for the image estimation problem. The novelty in our approach lies in the fact that we compute the mean and covariance of the prior (possibly non-Gaussian) by importance sampling. These apriori mean and covariance are fed to the update equations of the KF to estimate the aposteriori estimates of the state. We show that the estimates obtained by the proposed strategy are superior to those obtained by the traditional Kalman filter that uses the auto-regressive state model.