Bartusiak, Marcia F., Burke, Barbara, Chaikin, Andrew, Greenwood, Addison, Heppenheimer, T.A., Hoffman, Michelle, Holzman, David, Maggio, Elizabeth J., Moffat, Anne Simon. "7 The Mathematical Microscope: Waves, Wavelets, and Beyond." A Positron Named Priscilla: Scientific Discovery at the Frontier. Washington, DC: The National Academies Press, 1994.
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A Positron Named Priscilla: Scientific Discovery at the Frontier
that subcluster was identified as an x-ray source. "Wavelets were like a telescope pointing to the right place," Meyer said. And at the Centre de Recherche Paul Pascal in Pessac (near Bordeaux), Alain Arnéodo and colleagues have exploited "the fascinating ability of the wavelet transform to reveal the construction rule of fractals"11.
In addition, it can be instructive to compare wavelet coefficients at different resolutions. Zero coefficients, which indicate no change, can be ignored, but nonzero coefficients indicate that something is going on—whether an abrupt change in the signal, an error, or noise (an unwanted signal that obscures the real message). If coefficients appear only at fine scales, they generally indicate the slight but rapid variations characteristic of noise. "The very fine scale wavelets will try to follow the noise," Daubechies explains, while wavelets at coarser resolutions are too approximate to pick up such slight variations.
But coefficients that appear at the same part of the signal at all scales indicate something real. If the coefficients at different scales are the same size, it indicates a jump in the signal; if they decrease, it indicates a singularity—an abrupt, fleeting change. It is even possible to use scaling to sharpen a blurred signal. If the coefficients at coarse and medium scales suggest there is a singularity, but at high frequencies noise overwhelms the signal, one can project the singularity into high frequencies by restoring the missing coefficients—and end up with something better than the original.
CUT THE WEEDS AND SPARE THE DAISIES
Wavelets also made possible a revolutionary method for extricating signals from pervasive white noise ("all-color," or all-frequency, noise), a method that Meyer calls a "spectacular application" with great potential in many fields, including medical scanning and molecular spectroscopy.
An obvious problem in separating noise from a signal is knowing which is which. If you know that a signal is smooth—changing slowly—and that the noise is fluctuating rapidly, you can filter out noise by averaging adjacent data to kill fluctuations while preserving the trend. Noise can also be reduced by filtering out high frequencies. For smooth signals, which change relatively slowly and therefore are mostly lower frequency, this will not blur the signal too much.
But many interesting signals (the results of medical tests, for example) are not smooth; they contain high-frequency peaks. Killing all high frequencies mutilates the message—"cutting the daisies along with the weeds," in the words of Victor Wickerhauser of Washington University in St. Louis.