Minimum MSE LPT Estimation with an Application to Images

Abstract

A comprehensive approach to estimation is presented that integrates Kalman estimation and minimum mean square error (MSE) linear predictive transform (LPT) signal source modeling. The approach is strictly optimum when the signal source is a linear recursive system of the Kalman type and the channel contributes additive white noise. The technique is an improvement over classical Kalman estimation for two fundamental reasons. First, it directly addresses the signal source modeling problem of Kalman estimation and, secondly, it provides an inherent transformation mechanism that allows for simplifications in the design and implementation of the estimator. These ideas are illustrated with noisy monochrome images, where it is found that simple and robust LPT estimators ( filters or smoothers) with significant signal to noise ratio (SNR) enhancement can be obtained. In particular, it is shown that when applied to images, LPT estimators are easier to design and implement, without any SNR degradation, than classical Kalman estimators by a factor that approaches four in the smoothing case. Further simplifications can be achieved with a negligible loss in SNR performance by using approximations which are suggested by inherent transformation and robustness properties of the LPT estimators. Numerous areas for further investigation are also discussed. In particular, a comprehensive approach to control is developed which integrates minimum MSE LPT signal source modeling and linear quadratic white (LQW) control, of which linear quadratic gaussian (LQG) control is a special case. The control approach is strictly optimum when the signal source is a linear recursive system of the LQW control type and the channel contributes additive white noise.