Optimal Smoothing for Two Dimensional Gauss-Markov Random Fields

The smoothing problem is considered and a recursive solution is found for some particular graphical models of Gauss-Markov random fields partially observed under Gaussian correlated noise.The random fields here considered are two-dimensional (2D). With this we mean that the admissible domains which the field is defined upon, are particular kind of graphs, where the nodes are labeled using two integer indexes. Two kinds of such domains will be considered, both relevant from an application point of view: the rectangular and the spheric lattices. In a former paper it has been shown that for such field (and with a further assumption of homogeneity that we here relax) a 2D realisation can be built up. Such realisation result represents the basis for the present paper, where a 2D-recursive optimal-smoothing algorithm is derived. Various observation models are included in the setting of this paper, such as the case of observations available only on a subset of sites, as well as a variable number of process-components be measured. Even though based on the realisation result, the present paper is nevertheless self-contained.