Lids - P - 2294 Efficient Implementations of 2 - D Noncausal Iir Filters

In 2-D signal processing, FIR filters are the standard choice for implementing linear shift-invariant (LSI) systems, as they retain all of the advantages of their 1-D counterparts. The same is not true in general for IIR filters. In particular, with the exception of the class of causal recursively computable filters, implementation issues for 2-D IIR filters have received little attention. Moreover, the forced causality and apparent complexity in implementing even recursively computable filters has severely limited their use in practice as well. In this paper, we propose a framework for implementing 2-D LSI systems with 2-D noncausal IIR filters, i.e., filter systems described implicitly by a difference equation and boundary conditions. This framework avoids many of the drawbacks commonly associated with 2-D IIR filtering. A number of common 2-D LSI filter operations, (such as low-pass, high-pass, and fan 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 (PDE's), 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 x 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). In addition to its uses in 2-D filtering, we believe that this approach also has applications in related areas, such as geophysical signal processing and linear estimation, or any field requiring approximate solutions to elliptic PDE's.