RECURSIVE STRUCTURE OF NONCAUSAL GAUSS-MARKOV RANDOM-FIELDS

Causality is a common assumption with one-dimensional (1-D) Markov random processes. It leads to recursive descriptions and recursive filtering algorithms, such as the Kalman-Bucy filter. However, in 2-D, e.g., in physical oceanography or in image processing, noncausality is an important property, leading to Markov random field (MRF) models. To apply recursive techniques to MRF models, many authors work with subclasses that are causal or, more generally, unilateral, thus compromising an important property of the model. An alternate approach for noncausal Gauss-Markov random fields (GMRF) that enables the use of recursive procedures while retaining the noncausality of the field is developed. Recursive representations are established that are equivalent to the original field. This is achieved by first presenting a canonical representation for GMRF's that is based on the inverse of the covariance matrix, which we call the potential matrix. It is this matrix rather than the field covariance that reflects in a natural way the MRF structure. From its properties, two equivalent one-sided representations are derived, each of which is obtained as the successive iterates of a Riccati type equation. For homogeneous fields, these unilateral descriptions are symmetrized versions of each other, the study of only one Riccati equation being required. It is proven that this Riccati equation converges at a geometric rate, therefore the one-sided representations are asymptotically invariant. These unilateral representations enable us to process the fields with well-known recursive techniques such as Kalman-Bucy filters and two-point smoothers.