State-space realization theory of two-dimensional filters

The realization problem of two-dimensional linear filters is approached from a system theoretic point of view. The input-output behavior of such a system is defined by formal power series in two variables, and a Nerode state space is constructed. This state space is, in general, infinite dimensional. If the power series is rational, the dynamics of the filter is described by updating equations on finite-dimensional local state space. The notions of local reachability and observability are defined in a natural way and an algorithm for obtaining a reachable and observable realization is given. In general, local reachability and observability do not imply the minimality of the realization.