State-space formulation of 2-D frequency transformation in Fornasini–Marchesini second model

This paper presents a state-space formulation for the two-dimensional (2-D) frequency transformation in the Fornasini–Marchesini second (FM II) model. Specifically, by introducing some new state vectors, an equivalent description of the FM II model will be first established. Then based on this new description, a state-space formulation of the 2-D frequency transformation in the FM II model is derived by revealing the substantial input and output relations among the state vectors of the prototype filter, all-pass filters and the transformed filter. The resultant formulation owns an elegant and general expression, and can be viewed as a natural extension of its counterpart in the Roesser model. Furthermore, as one of the various possible applications of the proposed formulation, a 2-D zero-phase IIR filters design procedure will be shown, by which the desired 2-D zero-phase IIR filters can be constructed in a more flexible way to avoid the kind of image distortions caused by the nonlinear phase of the used filter. Three typical image processing examples as well as the computational complexity analysis will be given to show the effectiveness and efficiency of the proposed procedure.