Revisiting the Absolutely Minimal Realization for Two-dimensional Digital Filters

The open problem of absolutely minimal state-space realization for two-dimensional (2D) digital filters is revisited. A thorough investigation is carried out for the class of first order 2D digital filters. It is found that the canonical form proposed by Kung et al. three decades ago is incomplete, probably due to an incorrect assumption, which is also pointed out for the first time in this paper. Three new canonical forms for absolutely minimal state-space realization are proposed which can realize any 2D causal digital filters of first order. Several examples are illustrated