Thesis: Optimal Orthonormal Subband Coding and Lattice...

[Wim Sweldens (wim@bell-labs.com)]

#7 Thesis: Optimal Orthonormal Subband Coding and Lattice...

PhD Thesis : Optimal Orthonormal Subband Coding and Lattice
Quantization with Vector Dithering
Author : Ahmet Kirac
Date : July 31, 1998
Institution : California Institute of Technology
Advisor : P. P. Vaidyanathan

Web Site for Download:
http://www.systems.caltech.edu/kirac/

Abstract: In the digital era that we live in, efficient coding of
signals is an unquestionable need. This thesis is about one of the
most useful and popular technique of digital coding: subband coding.
Subband coding and its cousin wavelet-based coding are now the
preferred methods for not only speech, but also audio, image, and
video signals. Subband coding involves a linear part which is a filter
bank, and a nonlinear part which is usually a uniform scalar
quantization of each of the subbands. Subband coders are classified
according to the type of filter bank used for its transform. This
thesis is mainly about orthonormal subband coding. The ability of an
orthonormal filter bank to decompose the signal into components that
have a diverse set of signal energies is an indicator of its
efficiency for subband coding. Such a diversity in the set of the
subband energies is fully utilized by a process called bit
allocation. The traditional results on the optimality of a filter bank
for given input statistics assume that the quantizers operate at high
bit rates.

This thesis presents optimality results under more general quantizer
models without assuming high bit rates. This is accomplished by
revealing the relationship between the problems of optimal orthonormal
subband coding and principal component representation of signals. The
latter is done using what is called a principal component filter bank
(PCFB). A PCFB is one that compacts most of the energy of a signal
into smaller subsets of subbands. To date, there has not been
significant theoretical developments in the field of optimal
nonuniform subband coding, although the successful techniques of
wavelet-based coding are among the state of the art in practice. Such
techniques utilize a form of a nonuniform filter bank with a certain
structure which makes it efficient for its implementation. In this
thesis, we provide optimality results for the nonuniform orthonormal
subband coding as well. As in the uniform case, the principal
component representation of signals continues to play the key role. We
introduce nonuniform PCFB's and link them to the optimal subband
coding problem. A PCFB, in particular, contains a filter that compacts
most of the signal energy into one single channel: energy compaction
filter. The thesis goes into details of designing such filters
optimally. In particular, we propose an analytical method in the
two-channel case and a very efficient window method in the arbitrary
$M-$channel case. Multistage design of compaction filters has also
been worked out.

Finally we extend the analysis of uniform scalar quantization to
multiple dimensions. We provide an exact statistical relationship
between a lattice quantizer noise and its input vector. We then extend
the idea of dithering to the vector case. Dithering is a means of
statistically rendering the quantization noise independent of the
input. We address the optimal choice of a lattice for a given
dimension and also optimal pre- and post-filtering of a dithered
lattice quantizer.