More Accurate Digital Tunes, Images May Result From New Mathematical Theory

These three scenes have something in common. The people they portray are all dealing with digital representations of one kind or another. Such representations are finding increasing use in modern society because they are so easy to store, manipulate, display and transmit.

In the foreseeable future, digital music recordings may be audibly clearer, digital pictures may be much sharper and MRI scans more precise due to a new mathematical theory developed by mathematics professors Akram Aldroubi from Vanderbilt University and Karlheinz Gröchenig from the University of Connecticut. Writing in the Dec. 5 issue of the Society of Industrial and Applied Mathematics Review, Aldroubi and Gröchenig describe a new mathematical theory for producing digital representations of complex signals that overcomes a number of the limitations of current methods.

Not only does the theory have application in areas such music, photography and medical imaging, but it also promises improvements in areas as disparate as astronomy, geophysics and communications.

Most real-world signals, such as voices, are "analog" in nature. That is, they vary continuously over time or space. As a result, they contain a tremendous amount of information. Before these signals can be saved and manipulated in computers, however, they must be converted to digital form. This is done by a process called sampling. The strength of the analog signal, say the sound of a clarinet, is measured, or sampled, at regular intervals.

As a result, digital depictions are not perfect. But the shorter the sampling interval and the more accurate the measurements, the more realistic the resulting digital representation tends to be. Still, the string of ones and zeros burned onto the surface of a music CD does not completely capture the sound of a live band. Nor can a digital picture reproduce all the features of a landscape that the human eye perceives. Similarly, an MRI image of the brain provides only an approximate representation of the contours of gray matter hidden beneath the skull.

"Our theory - which is based on a lot of beautiful new mathematics - can produce more accurate digital representations of all kinds of samples, including those that classical methods handle poorly or cannot handle at all," says Aldroubi. "It generates algorithms [sets of mathematical procedures] that are fast, efficient, stable and robust."

Traditional sampling procedures date back half a century to the work of Claude Shannon, the Bell Labs mathematician who laid the foundation of modern information theory. These methods generally require that the sampling be done at regular intervals. Classical techniques also require that the original signal be "band limited" - a technical term meaning that the signal must stay within certain, defined limits. Take the case of music. Because human hearing does not extend above 20,000 hertz (cycles per second), extremely accurate digital representations can be made with digital representations that totally ignore sounds above this frequency.

The new theory, however, handles situations where the sampling is non-uniform and the signal is not band-limited.

A number of important applications stand to benefit. Oil and gas exploration, for example, makes heavy use of geophysical data collected at locations that frequently deviate from a regular grid pattern due to factors like variations in surface terrain. This is analogous to the intermittent nature of astronomical observations when light from a given star is blocked by clouds passing overhead.

Much of the impetus for developing the new sampling theory comes from the need for improving the quality of medical imaging. Aldroubi worked on sampling problems at the National Institutes of Health before coming to Vanderbilt. One of the figures in the paper shows three MRI images. The first is the original image. The second is the same image peppered with black splotches where 50 percent of the data has been removed. The third image was produced by one of the new reconstruction algorithms, which accurately restored all the major features in the original image.

In an attempt to make the new theory apply to the widest range of possible applications, Aldroubi and Gröchenig also designed it to take into account two other important factors that are not covered by the classical theory. It deals explicitly with the real-world problem of noise. Similarly, it takes the actual characteristics of the sampling device into account, unlike current methods that assume the measurements are perfect.

The new approach can produce some startling effects. In one exercise, for example, Aldroubi takes a large MRI image, shrinks it down to a small size and then expands it back to the same size as the original. The original and reconstructed images look almost identical, and a pixel-by-pixel comparison shows that only a minute amount of information has been lost.