choices available today (which are still changing rapidly). For one thing computing wavelet coefficients was rather slow. For another the wavelet transforms that existed then were all continuous. Imagine a wavelet slowly gliding along the signal, new wavelet coefficients being computed as it moves. (The process is repeated at all possible frequencies or scales; instead of brutally changing the size of the wavelets by a factor of 2, you stretch or compress it gently to get all the intermediate frequencies.)

In such a continuous representation, there is a lot of repetition, or redundancy, in the way information is encoded in the coefficients. (The number of coefficients is in fact infinite, but in practice "infinite may mean 10,000, which is not so bad," Grossmann says.) This can make it easier to analyze data, or recognize patterns. A continuous representation is shift invariant: exactly where on the signal one starts the encoding doesn't matter; shifting over a little doesn't change the coefficients. Nor is it necessary to know the coefficients with precision.

"It's like drawing a map," says Ingrid Daubechies, professor of mathematics at Princeton and a member of the technical staff at AT&T Bell Laboratories, who has worked with wavelets since 1985. "Many men draw these little lines and if you miss one detail you can't find your way. Most women tend to put lots of detail—a gas station here, a grocery store there, lots and lots of redundancy. Suppose you took a bad photocopy of that map, if you had all that redundancy you still could use it. You might not be able to read the brand of gasoline on the third corner but it would still have enough information. In that sense you can exploit redundancy: with less precision on everything you know, you still have exact, precise reconstruction."

But if the goal is to compress information in order to store or transmit it more cheaply, redundancy can be a problem. For those purposes it is better to have a different kind of wavelet, in an orthogonal transform, in which each coefficient encodes only the information in its own particular part of the signal; no information is shared among coefficients (see Box on p. 218).

At the time, though, Meyer wasn't thinking in terms of compressing information; he was immersed in the mathematics of wavelets. A few years before it had been proved that it is impossible to have an orthogonal representation with standard windowed Fourier analysis; Meyer was convinced that orthogonal wavelets did not exist either (more precisely, infinitely differentiable orthogonal wavelets that soon get close to zero on either side). He set out to prove it—and failed, in the summer of 1985, by constructing precisely the kind of wavelet he had thought didn't exist.



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