A functional realization theory for linear and non-linear recursive M-D systems

The realization problem for sampled, recursive, multidimensional filters used in image processing is solved by taking a new point-of-view. A filter now assigns a value to each path (or line). The value for a point P is then obtained by adding the values assigned to all the paths that end at P. Since the space is discretized, it is possible to code each path as a word of a free monoid. Hence the input-output behaviour of the filter is completely specified by a non-commutative formal power series. This makes it possible to develop a realization theory which exhibits similar mathematical features to that of regular (or bilinear) systems.