# February Fourier Talks 2007

## Chris Brislawn

### Title:

Lifting Factorizations for Linear Phase Filter
Banks and Wavelet
Transforms

### Abstract:

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.