Two dimensional recursive optimal smoothing of Gaussian random fields

The smoothing problem is considered for a two dimensional (2D) Gaussian Markov field defined on a finite rectangular lattice under Gaussian additive noise. The Gaussian Markov field is assumed to be generated by a (known) local correlation linking each site with the eight sites surrounding it in the lattice. 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. Even though based on the realisation result, the present paper is nevertheless self-contained.