Image identification and estimation using the maximum entropy principle

Abstract

Image identification and estimation using a reduced update Kalman filter (RUKF) requires that a model for the generating process is available. In (Kaufman, H.,Woods, J.W., Dravida, S., Tekalp, A.M., 1983. IEEE Trans. Automat. Control AC-28 (7)), a RUKF was used for image pixel density estimation using a 2D autoregressive model (AR) and for blurred image restoration (Koch, S., Kaufman, H., Biemond, J., 1995. IEEE Trans. Image Process. 4 (4), 520–523). However, in (Kaufman et al., 1983), the AR model order and the measurement noise covariance were assumed to be known a priori. Recently, in (Kadaba, S.R., Gelfand, S.B., Kashyap, R.L., 1998. IEEE Trans. Image Process. 7 (10), 1439–1452) the authors proposed a recursive estimation algorithm for images using non-gaussian AR models. They supposed, like in (Kaufman et al., 1983), that the measurement noise covariance and the model order were a priori known. Also, the process noise density, which may be non-gaussian, is assumed to be known. In the present work, image identification and estimation using a RUKF is reconsidered. No a priori information concerning the model order, the measurement noise covariance is needed. They are determined according to the maximum entropy principle (MEP) using an exhaustive search algorithm. It is shown that the estimation error with maximum entropy corresponds to the minimum mean squared error (MSE) giving the true model order and for the true noise covariance. Experimental results on simulated and real images are given to illustrate the performance of the proposed approach.

Introduction

Digital processing of noisy images has become an interesting field of investigations. Many authors have considered the problem of image estimation and modeling. To ensure that the estimation algorithms are efficient both in computation time and storage, recursive estimators have gained much attention Kaufman et al., 1983, Koch et al., 1995, Kadaba et al., 1998, Woods et al., 1987, Wu and Kundu, 1992. The optimal estimation of images corrupted by additive noise has been considered (see for example Jain, 1989, Rajala and de Figueiredo, 1981, Kuan et al., 1985, Nahi, 1972. The first requirement is that a model is available. In (Kaufman et al., 1983), recursive identification and estimation of images has been considered. An autoregressive (AR) model was used for image modeling. Different parameter identification algorithms were compared. The identified AR model is then used in a reduced update Kalman filter (RUKF) for image estimation. However, the authors assumed that the AR model and the measurement noise covariance were a priori known. Recently, in (Kadaba et al., 1998), the authors proposed a recursive estimation algorithm inspired from the RUKF developed in (Kaufman et al., 1983) for non-gaussian AR models. They supposed that the process noise density is known as in (Kaufman et al., 1983), that the model order, the measurement noise covariance were also a priori known.

In the present work, image identification and estimation are reconsidered in the same framework as in (Kaufman et al., 1983). The only difference is that no a priori information is assumed. We consider an AR model for the given noisy image. Then the model order, the model coefficients, the measurement noise covariance are determined through an exhaustive search such that the entropy of the estimation error reaches its maximum. This state corresponds to the minimum mean squared estimation error (MSE). Experiments on simulated and real noisy images are given for illustrating the effectiveness of the proposed method. To this effect, Section 2 presents the image model. Section 3 gives a review of the RUKF used in (Kaufman et al., 1983). Section 4 briefly reviews the estimation procedures used. The maximum entropy principle (MEP) is dealt with in Section 5. Finally, Section 6 presents and discusses the experimental results.