Long correlation Gaussian random fields: parameter estimation and noise reduction

In this paper, a parametric model for Gaussian random fields (GRFs) with long-correlation feature, namely the long correlation GRF (LC-GRF), is studied. Important properties of the model are derived and used for developing new parameter estimation algorithms and for constructing an optimum noise reduction filter. In particular, a novel iterative maximum likelihood estimation (MLE) algorithm is proposed for estimating the parameters of the model from a sample image, and the expectation-maximization (EM) algorithm is proposed for estimating the signal and noise variances given a noisy image. The optimal Wiener filter is derived making use of the parametric form of the model for the noise reduction under additive white Gaussian noise (WGN). Also the theoretic performance of the filter is obtained and its behavior is analyzed in terms of the long-correlation feature of the model. The effectiveness of the presented algorithms is demonstrated through experimental results on synthetic generated GRFs. An application to the restoration of cosmic microwave background (CMB) images in the presence of additive WGN is also presented.