Automatic Parametrisation and Block pseudo Likelihood Estimation for binary Markov random Fields

Abstract

Discrete Markov random fields play an important role in spatial statistics, and are applied in many different areas. Models which consider only pairwise interaction between sites such as the Ising model often perform well as a prior in a Bayesian setting but are generally unable to provide a realistic representation of a typical scene. Models which are defined by considering more than only two points have been shown to do well in describing many different types of textures. The specification of such models is often rather tedious, both in defining the parametric model, and in estimating the parameters. In this paper we present a procedure which in an automatic fashion defines a parametric model from a training image. On the basis of the frequencies of the different types of local configurations we define the potential function of all the different clique configurations from a relatively small number of parameters. Then we make use of a forward-backward algorithm to compute a maximum block pseudo likelihood estimator for the parametric models resulting from the automatic procedure. Then this set of methods is used to define Markov random field models from three different training images. The same algorithm which is used to calculate the block pseudo likelihood is used to implement a block Gibbs sampler. This is used to explore the properties of the models through simulation. The procedure is tested for a set of different input values. The analysis shows that the procedure is quite able to produce a reasonable presentation for one of the training images but performs insufficiently on the others. The main problem seems to be the ratio between black and white, and this seems to be a problem caused mainly by the estimator. It is therefore difficult to make a conclusion about the quality of the parametric model. We also show that by modifying the estimated potential function slightly we can get a model which is able to describe the second training image quite well.