found that with wavelets they can use magnetic resonance imaging to track the edge of the heart as it beats by sampling only a few coefficients. And wavelet compression is valuable in speeding some calculations. In the submarine detection work that Frazier and colleague Jay Epperson did for Daniel H. Wagner Associates, they were able to compress the original data by a factor of 16 with good results.
Ways to compress huge matrices (square or rectangular arrays of numbers) have been developed by Beylkin, working with Ronald Coifman and Vladimir rokhlin at Yale. The matrix is treated as a picture to be compressed; when it is translated into wavelets, "every part of the matrix that could be well represented by low-degree polynomials will have very small coefficients—it more or less disappears," Beylkin says. Normally, if a matrix has n2 entries, then almost any computation requires at least n2 calculations and sometimes as many as n3. With wavelets one can get by with n calculations—a very big difference when n is large.
Talking about numbers "more or less" disappearing, or treating very small coefficients as zero, may sound sloppy but it is "very powerful, very important"—and must be done very carefully, Grossmann says. It works only for a particular large class of matrices: "If you have no a priori knowledge about your matrix, if you just blindly use one of those things, you can expect complete catastrophe."
Just how important these techniques will prove to be is still up in the air. Daubechies predicts that "5, certainly 10 years from now you'll be able to buy software packages that use wavelets for doing big computations, in simulations, in solving partial differential equations."
Meyer is more guarded. "I'm not saying that algorithmic compression by wavelets is a dead end; on the contrary, I think it's a very important subject. But so far there is very little progress; it's just starting." Of the matrices used in turbulence, he said, only one in 10 belongs to the class for which Beylkin's algorithm works. "In fact, Rokhlin has abandoned wavelet techniques in favor of methods adapted to the particular problem; he thinks that every problem requires an ad hoc solution. If he is right, then Ingrid Daubechies is wrong, because there won't be 'prefabricated' software that can be applied to a whole range of problems, the way prefabricated doors or windows are used in housing construction."
Rokhlin works on turbulent flows in connection with aerodynamics; Marie Farge in Paris, who works in turbulence in connection with