Modeling 2-D AR Processes With Various Regions of Support

We show that there exists a causal 2-D linear process in the nonsymmetric half-plane having the same autocorrelations as a noncausal 2-D linear process in the whole-plane; this property is called the autocorrelation equivalence relation, and can be used for practical fitting and modeling of 2-D processes. Some causal 2-D autoregressive (AR) models with various regions of support are considered such as half-cross, half-diamond, quarter-plane square, half-square, half-hexagon, half-octagon, and half-circle. Considerations of parsimony in 2-D model fitting are then focused not only on the number of parameters in our model, but also most importantly on the optimal shape of the region of support. Their 2-D Yule-Walker equations are derived, and a computationally efficient order-recursive algorithm is proposed to solve them. The autocorrelation equivalence relation and the order-recursive algorithm are utilized to specify a noncausal 2-D AR process as well as its spectrum from a given realization of a random field