NEWS RELEASE

From Modems to Satellite Photos:
Mathematical Tools Help Analyze Data

For more information, please contact: Allyn Jackson, telephone 401-455-4109; fax 401-331-3842; e-mail axj@ams.org.

June 13, 1997

Providence, RI---Information---be it data coming over the modem to your computer, or a satellite photo sent to a ground-based receiver---is of no use unless you have the right tools to analyze it. Without such tools, you might have a tough time just differentiating the information you want from the random noise accompanying it. Mathematics holds the key to many such problems by providing ways of breaking complicated information into simpler pieces.

One example is the Fourier transform, a mathematical method that has been a standard tool of engineers for decades. The Fourier transform works much like the system of harmonics works for sounds: A sound can be broken up into fundamental tones which, when combined together in the proper way, will produce the original sound. Similarly, information can be broken up into component parts, represented by mathematical functions, using the Fourier transform, and the information can be recovered by recombining the functions in the proper way.

The Fourier transform is well suited to information that is repetitive, or periodic, like the strings of 0s and 1s that pulse through your modem. But it can be tremendously inefficient when confronted with irregular or transient information: The Fourier transform would not be of much use in analyzing a satellite photo of a ragged coastline.

Filling in the gap are wavelets, the recently-discovered mathematical cousin of the Fourier transform. Wavelets work in much the same way as the Fourier transform, by breaking information into fundamental components, but they differ in their ability to handle what Fourier analysis cannot: irregular, non-periodic information. Wavelets can produce highly refined analysis where it's needed---say, along the irregular edge of the coastline---but they don't waste effort on areas where there is not much going on---say, out in the flat blue of the ocean. This ability to "zoom in" and "zoom out" makes wavelets extremely efficient in a wide range of situations.

One of the most important uses of both wavelets and the Fourier transform is filtering: Once an information signal is broken into its component parts, it can be much easier to identify those parts of the signal that are relevant. For example, researchers working at potential oil recovery sites under the ocean were inundated with more data than they could handle, until they began using filtering techniques that allowed them to pluck from the torrent of incoming data the information they could use.

The Fourier transform and wavelets together provide a potent team for solving all kinds of problems in data analysis. The article "Fourier Analysis and Wavelet Analysis," to be published in the June/July 1997 issue of Notices of the American Mathematical Society, contrasts these two tools and their usefulness with different types of information.

Founded in 1888 to further mathematical research and scholarship, the 30,000-member AMS fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and everyday life.