Symmetric Extension Transforms
Christopher M. Brislawn · Kluwer Academic Publishers eBooks · 2006
A ubiquitous problem in subband image coding is deciding how to convolve analysis filters with finite-length input vectors. Specifically, how does one extrapolate the signal so that the convolution is well-defined when the filters “run off the ends” of the signal? The simplest idea is to extend the signal with zeros: if the analysis filters have finite impulse response (FIR) then convolving with a zero-padded, finite length vector will produce only finitely many nonzero filter outputs. Unfortunately, due to overlap at the signal boundaries, the filter output will have more nonzero values than the input signal, so even in a maximally decimated filter bank the zero-padding approach generates more transformed samples than we had in the original signal, a defect known as expansiveness. Not a good way to begin a data compression algorithm. A better idea is to take the input vector, x(n), of length and form its periodic extension, The result of applying a linear translation-invariant (LTI) filter to will also have period note that this approach is equivalent to circular convolution if the filter is FIR and its length is at most Now consider an Mchannel perfect reconstruction multirate filter bank (PR MFB) of the sort shown in Figure 1. If the filters are applied to x by periodic extension and if the decimation ratio, M, divides then the downsampled output has period This means that and therefore x, can be reconstructed perfectly from samples taken from each channel; we call such a transform a periodic extension transform (PET). Unlike the zero-padding transform, the PET is nonexpansive: it maps input samples to transform domain samples. The PET has two defects, however, that affect its use in subband coding applications. First, as with the zeropadding transform, the PET introduces an artificial discontinuity into the input