Issue No 7: November 2003
Special fund for SARS research
Q&A: The purpose of monitoring
Work on forensic DNA improves clarity of the probability factor
A closer look at meromorphic functions
Optimisation for production schedules
Reducing interference on mobile phones
Mathematical theory of fluids gives designers data on virtual models
Equation that can predict spots on a seashell
Research shows that filters for sound and images are correct
Short cut to finding best delivery route

Mathematical theory behind the quality and efficiency of digitised sound and pictures has taken a step forward with research from the City University of Hong Kong.
The scientist attributed with being the Father of the Digital Age, American Claude Shannon, wrote the blueprint for digital communications in the 1940s. But, said the CityU project’s Principal Investigator, Dr Ding-xuan Zhou, the theory could not be implemented.
Shannon’s idea was that pictures and sound could be transmitted more efficiently, or digitally, if only samples from certain points of the media were used to make up a digital signal.
Although the resulting picture or sound was different from the original, very little information would be lost.
“What scientists have been working on since then is to make the difference as small as possible,” said Dr Zhou.
With sound, for example, values can be taken at one second intervals, or at other convenient integer points and, when recomposed, it will be like the original whole signal.
“For images, values are similarly taken at integer points, or decomposed, and when recomposed they look very much like the original image.
“The benefit is that it takes less data to represent a lot of information,” said Dr Zhou. “You want to use the least amount of data as possible but you want the image or the sound to be as close to the original as possible.
“When you use this idea, of course you lose information. Even though you consider the sound or image to be identical, it isn’t.”
The main problem with Shannon’s theorem, he said, was that the time element in the signal processing was infinite. It needed to be finite, or capture values at a particular time.
Work in the 1970s used different “quadrature mirror” filters in so-called sub-band coding schemes, which divided signals into smaller bands of frequencies, therefore making them easier to process.
The quadrature mirror filters still did not produce a perfect result if the filters are required to have finite impulse responses, said Dr Zhou. Too much signal information was lost. New types of filters were devised with the latest conjugate quadrature filters eliminating any information loss.
Dr Zhou’s research has centred on the latest schemes of sub-band coding using conjugate quadrature filters and to show their foundation for wavelet analysis in applied mathematics.
“As well as knowing the scheme is perfect, we can show mathematically that it is correct.” In addition to revealing the mathematical foundation of the schemes, Dr Zhou’s work has shown how to clean up the symmetry of conjugate quadrature filters and therefore speed up their algorithm time.
Using the filters to sub-band or split a signal into smaller bands of frequencies means that the different frequency bands can be processed according to their different characteristics, he said.
The treatment of noise, an interference which appears naturally at high frequencies, is an example. If a signal is separated into sub-bands, noise in the high frequency can be eliminated without touching the remainder of the signal, he said.
With images, noise can interfere with the pixels and cause a blurring or blocking effect. Compression, he said, involves a step beyond digitisation, using part of the digitised signal. If the digitisation is bad, then the compressed signal is even worse.
“We want to keep a balance. If you have a small amount of data, then the speed of communication is fast. But if you send a large amount of data for the received message to be the same as the original, then the speed is slow. We need to achieve a balance.”
Dr Zhou has addressed conferences on the research in North America, Europe, Singapore and China.

Principal Investigator
Dr Ding-xuan Zhou :