Implementation of 2-D digital filters by iterative methods

A two-dimensional (2-D) rational filter can be implemented by an iterative computation involving only finite-extent impulse response (FIR) filtering operations, provided a certain convergence criterion is met. In this paper, we generalize this procedure so that the convergence criterion is satisfied for any stable 2-D rational transfer function. One formulation which guarantees convergence invokes a relaxed form of the iterative computation along with prefiltering the numerator and denominator polynomials of the rational transfer function. This implementation may be applied with a frequency-varying relaxation parameter for increasing the rate of convergence. An alternative generalization uses several previously computed iterates, unlike our first modification which utilizes only the most recently computed iterate. This formulation can potentially guarantee convergence and also increase the convergence rate without the requirement of prefiltering. Another extension of the iterative computation incorporates constraints (e.g., positivity or finite extent) on the output of each iteration. Proof of convergence of such constrained iterations relies on the concept of a nonexpansive operator. In particular, the error introduced within the converging solution resulting from a finite-extent constraint is shown to satisfy a homogeneous partial difference equation. Finally, this error computation leads to an important link between our iterative implementation with constraints and an iterative solution to partial difference equations (e.g., Laplace's equation) with known boundary conditions.