`Wavelets' Are Causing Ripples Everywhere

As head radar maven for Martin Marietta Corp.'s missiles division, Charles Stirman wants to transform today's "smart" weapons into budding Einsteins that can discern in an instant the difference between, say, an enemy troop carrier and an ambulance. The heart of his task is to find faster and better ways of interpreting the jumble of data "seen" by the radar eyes of missiles and other weapons. That's why he was intrigued in 1990 by a scientific article about a new mathematical theory, dubbed wavelets, that could be used to analyze data. "As soon as I saw it," recalls Stirman, "I said, `This is what we've been looking for."'

After months of testing, wavelets have proved to be even more powerful than Stirman imagined. Wavelet-based analyses of radar signals require only one-tenth the computing time of traditional techniques. And tests show that the new method can sharpen radar's ability to find camouflaged tanks. Now, Martin Marietta is starting a project to slip wavelet processors into actual missiles--and Stirman thinks the company has a jump on its competition. "This technology is so new," he says, "not many people have heard of it."

They will. Wavelets are making a splash among the mathematical cognoscenti. The technique could help detect stealthy submarines, provide cruise missiles or autos with detailed maps, and enable phone lines to carry movies and other video images more efficiently. Researchers also are enlisting wavelets to probe the human body and solve complicated equations describing such phenomena as turbulent air and fluid flows--a key to predicting the weather or designing more streamlined jets. Says University of Colorado mathematician Gregory Beylkin: "If it works, the potential payoff is tremendous."

The actual mathematics involves forbidding equations, new mathematical theorems, and elaborate computations. But the basic idea is simple. In the computer age, the trick is to transform everything from radar signals to photographs into equations that make it far easier to analyze or manipulate the data. Engineers can use equations, for example, to identify the most important features of a picture, such as the edges between light and dark, then compress the data representing the image into a fraction of its original size. The goal: to transmit this compressed representation, then reconstruct the image so that it looks identical to the original. The technique is key to a variety of emerging technologies. For instance, "without compression, there's no hope of transmitting a picture over a picture-phone," notes Howard L. Resnikoff, president of Aware Inc., a Cambridge (Mass.) company that makes wavelet-based computer chips for industrial use.

BETTER WAY. Turning information into equations isn't a new trick. For decades, scientists performed these transformations using a method invented by French mathematician Joseph Fourier in 1822. So-called Fourier analysis represents everything from the sound of a violin to the jumble of a radar echo as a collection of identical repeating waves--in mathematical parlance, sine and cosine curves.

But Fourier analysis has a hard time representing signals with erratic patterns, such as speech or seismic tests. That's why French geophysicist Jean Morlet began searching in the 1980s for a better way to analyze the sound echoes used in oil prospecting. Tapping into esoteric mathematical theories, he experimented with equations that substituted individual waves--or wavelets--for the endless series of sine and cosine curves.

The idea was promising, and by 1987 AT&T Bell Labs mathematician Ingrid Daubechies and others had worked out many of the kinks. So, when City University of New York mathematician Louis Auslander began a stint at the Defense Advanced Research Projects Agency in 1990, he says, he "was willing to bet DARPA dollars that this technology would fly."

Propelled by $4 million in DARPA funds since 1990, by companies such as Aware, and by growing interest among mathematicians, wavelets have soared. Though complex, the wavelet equations nevertheless lend themselves to rapid computer calculations. "That's why this got off the ground so fast," explains Yale University mathematician Ronald Coifman. "Otherwise, it would have just been a mathematician's toy."

`JIGSAW PUZZLE.' Coifman and his Yale colleagues pushed the idea further with a concept called wavelet packets. Where the early work relied on a single wavelet shape to analyze a set of data, this scheme picks from a library of them. "It's like pieces of a jigsaw puzzle," he says. "We try to select the minimal collection of shapes that will add together to match the image." Coifman has co-founded Fast Mathematical Algorithms & Hardware Corp. in Hamden, Conn., to produce wavelet-packet software for a variety of uses. Switching from traditional Fourier analysis to Coifman's packets opens up "thousands and thousands of possibilities," says Martin Marietta's Stirman.

Although engineers are just beginning to explore commercial applications, wavelets are often proving superior to existing methods. French researchers have been able to distinguish the clicking noises of shrimp from nearly identical sounds of lobsters, a key test for submarine-detection gear. At Dartmouth College, mathematician Dennis M. Healy Jr. and radiologist John B. Weaver produce magnetic-resonance images in one-third the usual time and with fewer false images caused by pulsating blood vessels. Colorado's Beylkin is developing algorithms--numerical schemes--that could help solve many scientific problems faster. And Aware can compress a digital movie by a factor of 150, then generate an image that is nearly indistinguishable from the original. "A lot of work in academia doesn't get transferred to industry," says Yale mechanical engineer Katepalli R. Sreenivasan. "This will almost certainly get there."

The ultimate impact of wavelets still is unclear. Many leading researchers worry that the theory is becoming a bit too fashionable. "We can't expect miracles," warns AT&T's Daubechies. "Wavelets are just a tool." But if they fulfill even some of the high hopes mathematicians and scientists hold for them, they could help industry sail into the next phase of the Information Age.

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