February Fourier Talks 2007

Lifting Factorizations for Linear Phase Filter Banks and Wavelet Transforms

Linear phase FIR filter banks, or discrete wavelet transforms, form an integral part of the ISO/IEC JPEG 2000 image coding standard, where they provide spatial decorrelating transforms that enable a variety of progressive transmission capabilities. Filter banks are specified in the JPEG 2000 standard in terms of lifting factorizations, a cascade-form decomposition of the filter bank into a ladder-like structure. Mathematically, lifting factorizations correspond to factorizations of the polyphase transfer matrix for the filter bank into alternating lower and upper triangular lifting matrices. It was shown by Daubechies and Sweldens that any FIR perfect reconstruction filter bank (i.e., any 2x2 invertible matrix polynomial) has such a decomposition (many of them, in fact), obtained via a straightforward application of the Euclidean algorithm. The talk will discuss recent research by the author, inspired by his work on the JPEG 2000 standard, into the structure of lifting factorizations for linear phase filter banks. It will be shown that any whole-sample symmetric (FIR type 1 linear phase) filter bank can always be factored completely into a cascade involving half-sample symmetric (FIR type 2 linear phase) lifting filters. The structure theory for half-sample symmetric filter banks is more complicated, involving whole-sample antisymmetric (FIR type 3 linear phase) lifting filters and an equal-length base filter bank. Very recent results based on a new theory of "group lifting structures" show that these linear phase filter bank factorizations are also unique, a rather surprising result given the general nonuniqueness of lifting factorizations. All of these results make use of the group-theoretic structure of lifting factorizations and linear phase filter banks.