Bayesian Estimation for Homogeneous and Inhomogeneous Gaussian Random Fields

This paper investigates Bayesian estimation for Gaussian Markov random elds. In particular, a new class of inhomogeneous model is proposed. This inhomogeneous model uses a Markov random eld to describe spatial variation of the smoothing parameter in a second random eld which describes the spatial variation in the observed intensity image. The coupled Markov random elds will be used as prior distributions, and combined with Gaussian noise models to produce posterior distributions on which estimation will be based. All model parameters are estimated, in a fully Bayesian setting, using the Metropolis-Hastings algorithm. The models and algorithms will be illustrated using various artiicial examples. The full posterior estimation procedures using homogeneous and inhomogeneous models will be compared. For the examples considered the fully Bayesian estimation for inhomogeneous models performs very favourably when compared to methods using homogeneous models, allowing diierential smoothing and varying local textures.