Neural filtering of colored noise based on Kalman filter structure

In this paper, adaptive filtering approaches of colored noise based on the Kalman filter structure using neural networks are proposed, which need not extend the dimensions of the filter. The colored measurement noise is first modeled from a Gaussian white noise through a shaping filter. The Kalman filtering model of colored noise is then built by adopting an equivalent observation equation, which can avoid the dimension extension and complicated computations. An observation correlation-based algorithm is suggested to estimate the variance of the measurement noise by use of a single layer neural network. The Kalman gain can be obtained when a perfect knowledge of the plant model and noise variances is given. However, in some cases, the difficulties of the correlative method and the Kalman filter equations are the amount of computations and memory requirements. A neural estimator based on the Kalman filter structure is also analyzed as an alternative in this paper. The Kalman gain is replaced by a feedforward neural network whose weight adjustment permits minimization of the estimation error. The estimator has the capability of estimating the states of the plant in a stochastic environment without knowledge of noise statistics. If the noise of the plant is white and Gaussian and its statistics are well known, the neural estimator and the Kalman filter produce equally good results. The neural filtering approaches of colored noise based on the Kalman filter structure are applied to restore the cephalometric images of stomatology. Several experimental results demonstrate the feasibility and good performances of the approaches.