Estimation and Choice of Neighbors in Spatial-Interaction Models of Images

Some aspects of statistical inference for a class of spatial-interaction models for finite images are presented: primarily the simultaneous autoregressive (SAR) models and conditional Markov (CM) models. Each of these models is characterized by a set of neighbors, a set of coefficients, and a noise sequence of specified characteristics. We are concerned with two problems: the estimation of the unknown parameters in both SAR and CM models and the choice of an appropriate model from a class of such competing models. Assuming Gaussian-distributed variables, we discuss maximum likelihood (ML) estimation methods. In general, the ML scheme leads to nonlinear optimization problems. To avoid excessive computation, an iterative scheme is given for SAR models, which gives approximate ML estimates in the Gaussian case and reasonably good estimates in some non-Gaussian situations as well. Likewise, for CM models, an easily computable consistent estimate is given. The asymptotic mean-squared error (mse) of this estimate for a four-neighbor CM model is shown to be substantially less than the mse of the popular coding estimate. Asymptotically consistent decision rules are given for choosing an appropriate SAR or CM model. The usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns. Synthetic patterns obeying known SAR and CM models are generated, and the models corresponding to true and several competing neighbor sets are fitted. The estimation scheme yields estimates close to the parameters of the true models, and the decision rule for the choice of neighbors picks up the true model from the class of competing models.