Recursive Estimation of Gauss-Markov Random Fields Indexed over 1-D Space

This paper presents optimal recursive estimators for vector valued Gauss-Markov random \emph{fields} taking values in $\R^M$ and indexed by (intervals of) $\R$ or $\Z$. These 1-D fields are often called reciprocal processes. They correspond to two point boundary value fields, i.e., they have boundary conditions given at the end points of the indexing interval. To obtain the recursive estimators, we derive two classes of recursive representations for reciprocal processes. The first class transforms the Gaussian reciprocal process into a Gauss-Markov \emph{process}, from which we derive forward and backwards recursive representations. The second representation folds the index set and transforms the original \emph{field} taking values in $\R^M$ into a Markov \emph{process} taking values in $\R^{2M}$. The folding corresponds to recursing the reciprocal process from the boundary points and telescoping inwards. From these recursive representations, Kalman filters and recursive smoothers are promptly derived.