Efficient implementations of 2-D noncausal IIR filters

In this paper, we propose a framework for the efficient implementation of two-dimensional (2-D) noncausal infinite impulse response (IIR) filters, i.e., filter systems described implicitly by difference equations and boundary conditions. A number of common 2-D LSI filter operations, (including low-pass, high-pass, and zero-phase filters), are efficiently realized and implemented in this paper as noncausal IIR filters. The basic concepts involved in our approach include the adaptation of so-called direct methods for solving partial differential equations (PDEs), and the introduction of an approximation methodology that is particularly well suited to signal processing applications and leads to very efficient implementations. In particular, for an input and output with N×N samples, the algorithm requires only O(N²) storage and computations (yielding a per pixel computational load that is independent of image size), and has a parallel implementation (yielding a per pixel computational load that decreases with increasing image size). Also, because our approach allows for the implementation of filters with space-varying coefficients on irregularly shaped domains, it should have applications in related areas like linear estimation, geophysical signal processing, or any field requiring approximate solutions to elliptic PDEs