A Bayesian framework for noise covariance estimation using the facet model

In image processing literature, thus far, researchers have assumed the perturbation in the data to be white (or uncorrelated) having a covariance matrix σ² I, i.e., assumption of equal variance for all the data samples and that no correlation exists between the data samples. However, there have been very few attempts to estimate noise characteristics under the assumption that there is a correlation between data samples. In this work, we propose a new and a novel approach for the simultaneous Bayesian estimation of the unknown colored or correlated noise (population) covariance matrix and the hyperparameters of the covariance model using the well-known facet model. We also estimate the facet model coefficients. We use the facet model because of its simple, yet elegant, mathematical formulation. We use the generalized inverted Wishart density as the prior model for the noise covariance matrix. We place a structure on the covariance matrix using the parameters of a correlation filter. These hyperparameters are estimated by a new extension of the expectation-maximization algorithm called the generalized constrained expectation maximization algorithm that we developed.