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Catching a Wavelet: Mathematical Tool Revolutionizes Data Analysis

September 1997

Mathematicians have found the ultimate proof of the axiom "less is more." The proof comes in the form of wavelets -- a relatively new and powerful mathematical tool that allows a researcher to quickly manipulate large sets of data, sort through them, find what is needed, and discard the rest.

Wavelets are like building blocks of different sizes that fit together because they embody compatible repeating structures. Researchers can put wavelets together like LegosTM to mimic the data and reveal inherent patterns.

Developed with research funded by NSF and others, in the last 10 years wavelet analysis has found applications in many disciplines. For geologists looking at ocean cores for possible oil exploration, the wavelet system is saving months of data analysis. Music historians use wavelets to hear cleaned-up recordings of Johannes Brahms as well as the late opera star Enrico Caruso. And wavelets have even found a home at the FBI.

In 1993, the FBI started digitizing their fingerprint database, creating 200 terabytes worth of files that, in 1994 prices, amounted to $200 million worth of disk storage. Wavelets allowed the FBI to clean up the images and compress them at a ratio of 12 to 1, saving millions of dollars in storage costs and transmission time.

While this project was startling in its day, it has already become old news in engineering circles. "Wavelet analysis went from a theory to standard fare in an engineering toolkit in less than a decade," comments Don Lewis, director of NSF's Division of Mathematical Sciences. "That's a very short period of time."

It is a process that was needed. For the better part of the 20th century, engineers have been describing data with the Fourier system, but wavelet analysis is turning out to provide more versatility.

In a recent article published by the Institute of Electrical and Electronic Engineers, David Donoho, a wavelet researcher and NSF grant recipient at Stanford University, demonstrated that describing a picture of a "sawtooth wave" would require 256 of Fourier's sine waves, but only 16 wavelets.

What's more, says Ron Coifman of Yale University, wavelets are doing things that were impossible with only Fourier analysis. During a recent presentation at NSF headquarters, Coifman used wavelet functions to describe the structures of an image. The goal was to condense an image and then reconstruct it using a small number of parameters.

For this demonstration, Coifman used a picture that has already become a classic in the short history of wavelets, a picture of a baboon--a hairy ape with humanoid features. Coifman applied wavelet and Fourier functions to the image's data and came up with two basic structures.

The Fourier method transcribed the baboon's hair, which made up the texture of the picture, while the wavelet analysis efficiently encoded the baboon's face. By taking one percent of the data from each of these two ranges (or two percent of the total data), Coifman created a reasonable imitation of the original photo.

This type of compression and transcription, as well as the work done at the FBI, is applicable to any work that has large sets of data, sets that are so big they're expensive to manage, manipulate or move.

"Transmitting radar data can be very expensive," says Coifman. vSo you clean it up and then transmit it, cutting down the transmission costs by 90 percent."

Other applications include using wavelets to analyze the effects of various drugs on the neuron system, and to clean up x-ray images.

However, says Coifman, even though wavelet analysis is changing the way engineers work, it is still only the starting point for scientists. "This data transcription mode is not science. This is just a method for organizing information in order to begin the science and describe what is going on. We must continue to develop the appropriate mathematics in order to do the science."

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