Applied Analysis : Fourier analysis,
data compression (MPEG, JPEG), wavelets
MAT 4996/5326
January 2000
Description
Audio and video signal processing is the backbone of modern digital communication.
It all depends on fast transforms, be it the
fast Fourier, the fast cosine, or the fast wavelet transform.
The windowed Fourier transform and the windowed cosine transform localize the signal
in time and frequency.
The wavelet transform localizes the signal in time and scale. It is the
discrete versions of these transforms that are used in practice, and each
of them admits a fast implementation. No single
transform has all the good points and none of the bad points.
The "Joint Photographic Experts Group" (JPEG), the "Joint Bi-level
Imaging Group" (JBIG) and the "Moving Picture Experts Group" (MPEG)
of the International Standard Organization (ISO) have adopted
the windowed cosine transform as the basic tool to localize, quantize and compress
the signal. JBIG allows for a greater compression ratio than JPEG.
JPEG part 1-3 has a set of parameters now called JEPG utilities. The recently agreed
standard JPEG part 4, called JURA
for "JPEG Utilities Registration Authority", provides for extending the standard by
registering new utilities defined for the use of individual JPEG users. JPEG 2000 is
proposing wavelets to greatly reduce ringing artifacts present with JPEG and JBIG.
A pdf document on JPEG 2000 entitled
JPEG-2000 : Background, scope, and technical description by
Majid Rabbani, can be
downloaded
.
A widely used audio coding format known as MP3 (aka MPEG Audio Layer 3)
will be considered.
One immediately sees the impact of the decisions of the expert groups on
hardware and software design for high definition television, medical applications,
and so on. Among the many applications of wavelets stands
finger print compression by the FBI.
- The aim of the course is to present the mathematical foundation
of data compression as necessary for understanding and applying the many data
compression methods. It is not planned to go into the details
of the standards adopted by the expert groups.
- Software for signal processing exists on the web or can be bought from
software companies. To name but a few for wavelets,
The WaveLab at Stanford by David Donoho is provided over the Internet
at the FTP site
ftp://plaifair.stanford.edu/public/wavelab
and in the World-Wide-Web page
http://playfair.stanford.edu/~wavelab
It requires buying MATLAB, a product of The Mathworks company.
This company also sells the Wavelet Toobox and the Signal Processing Toolbox.
The Wavelet Explorer requires
Mathematica; these are produced and sold by Wolfram Reasearch.
- A site visit at the Communications Research Centre is being planned to meet
Canadian representatives for MPEG, JPEG and Wavelets during the course.
A few lectures will be delivered by experts in the applications in the field.
The first lecture will skim a few aspects of MPEG and JPEG,
taken from ref. no. 3 below: Digital video : an introduction to MPEG-2 .
A few lectures will be devoted to continuous and
discrete Fourier and cosine transforms.
It is expected that the course, in the main, will keep close to
the text of Strang and Nguyen.
Instructor: Rémi Vaillancourt,
Prerequisites: Third year engineering mathematics or
third year analysis for mathematics and science.
Course material:
The book, in stock at the Unversity Bookstore,
Wavelets and filter banks
of Strang and Nguyen, referenced below, will serve as the basic textbook
for the course. Theoretical and practical assignments and projects will be
part of the course.
Evaluation and grading scheme: Assignments, projects
and an open book final examination. Following engineering practice, at least
50 percents of the final grade will be for work in a controlled environment.
Classroom and schedule: Lectures will be held in KED B05 at
585 King Edward Ave. on Mondays and Wednesdays from 19:00 till 20:30.
First meeting on 5 January 2000. Rescheduling may be considered if desirable and
possible (the rule is unaninous approval of any change).
Tentative office hours: Tuesdays and Thursdays from 16:00 to 17:00 and on appointment. e-mail is the preferred medium. Telephone is also possible.
References:
- A comprehensive text for JPEG is
JPEG still image data compression standard
by William B. Pennebaker and Joan L. Mitchell, Van Nostrand Reinhold, New York,
1993.
- A text which surveys standards in chapters 7 to 10 is
Digital pictures : representation, compression, and standards
by Arun N. Netravali and Barry G. Haskell, 2nd. ed., Plenum Press, New York,
1995.
- A text which surveys MPEG-2 is
Digital video : an introduction to MPEG-2
by Barry G. Haskell, Atul Puri, and Arun N. Netravali, Chapman & Hall, London,
1997.
- A text which explains the basics behind the standards without bearing
on the details is
Image and video compression standards : algorithms and architectures
by Vasudev Bahskarna and Konstatinos Konstantinides, Kluwer Academic Publishers, Boston, 1995.
- A text for wavelets that uses both the signal processing and mathematics languages is
Wavelets and filter banks
by Gilbert Strang and Truong Nguyen, Wellesley-Cambridge Press, Box 812060,
Wellesley MA 02181, 1996.
- A reference on wavelets from an enginnering point of view with a solid
mathematical foundation is
Wavelets and subband coding by Martin Vetterli and Jelena Kovacevic,
Prentice-Hall, 1995.
- A graduate course on wavelet is
A wavelet tour of signal processing
by Stéphane Mallat, Academic Press, San Diego, 1998.
Relevent web pages:
Last modified: 24 December 1999.
