Probabilistic image processing by extended Gauss-Markov random fields

We propose an extension of the Gauss-Markov random field (GMRF) models by introducing next-nearest neighbour interactions. The values of the next-nearest neighbour interactions are set to positive real numbers with the expectation that this will lead to some noise reduction while preserving the edges. Values for the hyperparameters in the proposed model are determined by using the EM algorithm in order to maximize the marginal likelihood. In addition, a measure of mean squared error, which quantifies the statistical performance of our proposed model, is derived analytically as an exact explicit expression by means of the multi-dimensional Gaussian integral formulas.