On the Problems of Constructing Minimal Realizations for Two-Dimensional Filters

The input-output behavior of a two-dimensional linear filter is defined by a formal power series in two variables. If the power series is rational the dynamics of the filter is described by updating equations on finite dimensional local state spaces. The class of realizations considered in this paper is constituted by doubly indexed dynamical systems of reduced structure. The construction of the class of minimal realizations is based on matrix representation techniques of noncommutative power series.