Valid parameter space of 2-D Gaussian Markov random fields

The valid parameter spaces of infinite- and finite-lattice (2-D noncausal) Gaussian Markov random fields (GMRFs) are investigated. For the infinite-lattice fields, the valid parameter space is shown to admit an explicit description; a procedure that yields the valid parameter space is presented. This procedure is applied to the second-order (neighborhood) 2-D GMRFs to obtain an explicit description of their valid parameter spaces. For the finite-lattice fields, it is shown that the valid parameter space does not admit a simple description; the conditions that ensure the positivity of the power spectrum are necessary, sufficient, and irreducible. The set of conditions for the infinite-lattice fields, however, serves as a good set of sufficient conditions for the finite-lattice case. The results readily extend to the class of d-D real and complex GMRFs