A Positron Named Priscilla Scientific Discovery at the Frontier (1994) / Chapter Skim
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7 The Mathematical Microscope: Waves, Wavelets, and Beyond
7 The Mathematical Microscope: Waves, Wavelets, and Beyond
Pages 196-235
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From page 196... ...
In St. Louis, Victor Wickerhauser was using the same mathematics to help the Federal Bureau of Investigation store fingerprints more economically, while at Yale University Ronald Coifman used it to coax a battered, indecipherable recording of Brahms playing the piano into yielding its secrets.
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From page 197... ...
In their initial excitement some researchers thought waveless might virtually supplant the much older and very powerful mathematical language of Fourier analysis, which you use every time you talk on the telephone or turn on a television. But now they see the two as complementary and are exploring ways to combine them or even to create more languages "beyond waveless." Different languages have different strengths and weaknesses, points out Meyer, one of the founders of the field: "French is effective for analyzing things, for precision, but bad for poetry and conveying emotion perhaps that's why the French like mathematics so much.
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From page 198... ...
His mother died when he was nine and his father the following year. Although two younger siblings were abandoned to a foundling hospital after their mother's death, Fourier continued school and in 1780 entered the Royal Military Academy of Auxerre, where at age 13 he became fascinated by mathematics and took to creeping down at night to a classroom where he studied by candlelight.
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From page 199... ...
In the seventeenth century Isaac Newton had a new insight: that forces are simpler than the motions they cause, and the way to understand the natural world is to use differential and partial differential equations to describe these forces- gravity, for example. Newton's differential equation showing how the gravitational pull between two objects is determined by their mass and the distance between them replaced countless observations, and predictive science became possible.
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From page 200... ...
. One can also treat nonperiodic functions this way, using the Fourier transform (see Box on p.
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From page 201... ...
In 1810 the French astronomer Jean-Baptiste Delambre had issued a report on mathematics expressing the fear that "the power of our methods is almost exhausted,"5 and some 30 years earlier Lagrange, comparing mathematics to a mine whose riches had all been exploited, wrote that it was "not impossible that the mathematical positions in the academies will one day become what the university chairs in Arabic are now."6 "Looking back," writes Korner In his book Fourier Analysis, we can see Fourier's memoir "as heralding the surge of new mathematical methods and results which were to mark the new century." THE EXPLANATION OF NATURAL PHENOMENA The German mathematician Carl Jacobi wrote that Fourier believed the chief goal of mathematics to be "the public good and the explanation FIGURE 7.] In 1807 Fourier showed that any function /even an irregular oneJ can be expressed as the sum of a series of sines and/or cosines The function at the top is a combination of the three functions below: sin {xJ, 0.3sin /3xJ andO.2cos {5xJ.
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From page 202... ...
For instance, the Fourier transform of a fingerprint will have large values near the "frequency" 15 ridges per centimeter. The Fourier series of a periodic function of period 2~ is given by the formula fix)
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From page 203... ...
When that function is translated into a Fourier series a remarkable thing happens: the intractable differential equation describing the evolution of the temperature decouples, becoming a series of independent differential equations, one for the coefficient of each sine or cosine making up the function. These equations tell how each Fourier coefficient varies as a function of time.
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From page 204... ...
On the "physical space" side of the Fourier transform, one can talk about an elementary particle' position; on the other side, in "Fourier space" or "momentum space," one can talk about its momentum or think of it as a wave. The modern realization that matter at very small scales behaves differently from matter on a human scale-that at small scales we cannot have precise knowledge of both sides of the transform at once, we cannot know simultaneously both the position and momentum of an elementary particle is a natural consequence of Fourier analysis.
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From page 205... ...
The basis for digital technology was given by Claude Shannon, a mathematician at Bell Laboratories whose Mathematical Theory of Communications was published in 1948; while he is not well known among the general public, he has been called a "hero to all communicators."7 Among his many contributions to information theory was the sampling theorem (discovered independently by Harry Nyquist and others)
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From page 206... ...
"But until the advent of computing machines it was a solution looking for a problem." On the other hand, the gain in speed from the FFT is greater than the gain in speed from better computers; indeed, significant gains in computer speed have come from such fast algorithms built into computer hardware. DRIVING A CAR HALF A BLOCK It could be argued that the fast Fourier transform was too successful "Because the FFT is very effective, people have used it in problems
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From page 207... ...
A Fourier transform makes it easy to see how much of each frequency a signal contains but very much harder to figure out when the various frequencies were emitted or for how long. It pretends, so to speak, that any given instant of the signal is identical to any other, even if the signal is as complex as a Bach prelude or changes as dramatically as the electrocardiogram of a fatal heart attack.
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From page 208... ...
In theory, the phases containing this time information can be calculated from the Fourier coefficients; in practice, calculating them with adequate precision is virtually impossible. In addition, the lack of time information makes Fourier transforms vulnerable to errors.
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From page 209... ...
"David Marr, who worked on artificial vision and robotics at MIT, had similar ideas. The physics community was intuitively aware of waveless dating back to a paper on renormalization by Kenneth Wilson, the Nobel prize winner, in 1971." But all these people in mathematics, physics, vision, and information processing didn't realize they were speaking the same language, partly for the simple reason that they rarely spoke to one another but also partly because the early work existed in such disparate forms.
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From page 210... ...
"lean was sent to me because I work in phase space quantum mechanics," Grossmann said. "Both in quantum mechanics and in signal processing you use the Fourier transform all the time-but then somehow you have to keep in mind what happens on both sides of the transform.
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From page 211... ...
And I don't think I could have done anything if I hadn't had a little computer and some graphics output." A MATHEMATICAL MICROSCOPE Wavelets can be seen as an extension of Fourier analysis. As with the Fourier transform, the point of waveless is not the waveless themselves; they are a means to an end.
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From page 212... ...
"With waveless, you play with the width of the wavelet in order to catch me rhythm of the signal," Meter says. "Strong correlation means that Mere is a little piece of the signal that looks like the wavelet." Constant stretches give wavelet coefficients with the value zero.
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From page 213... ...
FIGURE 7.3 In the wavelet transform a wavelet is comparer! successively to different sections of a function.
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From page 214... ...
11 ~ l ~ , Y FIGURE 7.4 Top row: In windowed Fourier analysis the size of the window is fixed and the number of oscillations varies. A small window will be "blind" to low frequencies, which are too large to fit into the window.
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From page 215... ...
Imagine covering your signal with a row of FIGURE 7.5 A small sample of waveless and scaling functions (father functionsJ. Wavelets shown in panels a and b are usecl in continuous representations; no father function is neeclecI.
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From page 216... ...
To reconstruct the signal, you add the original rough picture of the function and all the details, by multiplying each coefficient by its wavelet or father function and adding them all together, just as to reconstruct the number 78/7 you add the number 11 given by the father and all the details: 0.1, 0.04, 0.002, and so on. Of course, the number will still be in decimal form; in contrast, when you reconstruct a signal from a father function, waveless, and wavelet coefficients, you switch back to the original form of representation, out of "wavelet space." AVOIDING REDUNDANCY When Meyer took the train to Marseille to see Grossmann in 1985 the idea of multiresolution existed it originated with lean Morlet-but waveless were limited and sometimes difficult to use, compared with the
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From page 217... ...
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.
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From page 218... ...
are all perpendicular to each other. The speed of computing orthogonal wavelet coefficients is a consequence of this geometry (as is the speed of calculating Fourier coefficients; sines and cosines also form an orthogonal basis)
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From page 219... ...
while Mallat explained that the multiresolution Meyer and others were doing with waveless was the same thing that electrical engineers and people in image processing were doing under other names. "This was a completely new idea," Meyer said.
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From page 220... ...
"All those existing techniques were tricks that had been cobbled together; they had been made to work in particular cases," says Marie Farge. "Mallet helped people in the quadrature mirror filters community, for example, to realize that what they were doing was much more profound and much more general, that you had theorems, and could do a lot of sophisticated mathematics." Wavelets got a big boost because Mallat also showed how to apply to wavelet multiresolution fast algorithms that had been developed for other fields, making the calculation of wavelet coefficients fast and automatic essential if they were to become really useful.
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From page 221... ...
By the end of March 1987 I had all the results." Together, multiresolution and waveless with compact support formed the wavelet equivalent of the fast Fourier transform: again, not just doing calculations a little faster, but doing calculations that otherwise very likely wouldn't be done at all. HEISENBERG i' Multiresolution and Daubechies's waveless also made it possible to analyze the behavior of a signal in both time and frequency with unprecedented ease and accuracy, in particular, to zoom in on very brief intervals of a signal without becoming blind to the larger picture.
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From page 222... ...
One can imagine transmitting pictures electronically quickly and cheaply by sending only a coarse picture, calling up a more detailed picture only when needed. Mathematician Dennis Healy, Ir., and radiologist John Weaver of Dartmouth College are exploring the use of waveless for "adaptive" magnetic resonance imaging, in which higher resolutions would be used selectively, depending on the results already found at coarser scales.
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From page 223... ...
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 onwhether an abrupt change in the signal, an error, or noise (an unwanted signal that obscures the real message)
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From page 224... ...
"We knew that if we used waveless right, they couldn't be beaten." The method is simplicity itself: you apply the wavelet transform to your signal, throw out all coefficients below a certain size, at all frequencies or resolutions, and then reconstruct the signal. It is fast (because the wavelet transform is so fast)
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From page 225... ...
(That all orthogonal representations leave noise unchanged has been known since the 1930s.) So while noise masks the signal in "physical space," the two become disentangled in "wavelet space." In fact, Donoho said, a number of researchers at the Massachusetts Institute of Technology, Dartmouth, the University of South Carolina, and elsewhere- independently discovered that thresholding wavelet coefficients is a good way to kill noise.
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From page 226... ...
Because Fourier analysis has existed for so long, and most physicists and engineers have had years of training with Fourier transforms, interpreting Fourier coefficients is second nature to them. In addition, Meyer points out, Fourier transforms aren't just a mathematical abstraction: they have a physical meaning.
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From page 227... ...
"However, people in research groups who fine-tune the Fourier transform techniques in commercial image compressors claim they can also do something on the order of 35. So it's not really clear that we can beat the existing techniques.
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From page 228... ...
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 waveless for doing big computations, in simulations, in solving partial differential equations." Meyer is more guarded.
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From page 229... ...
We have to do a lot of experiments, get a lot of practice, develop methods, develop representations. " She uses orthogonal waveless, or related wavelet packets, for compression but continuous waveless for analysis: "I would never read the coefficients themselves in an orthogonal basis; they are too hard to read." With continuous waveless she can take a one-dimensional signal that varies in space, put it into wavelet space, and get a two-dimensional function that varies with space and scale: she can actually see, on a computer screen or a printout what is happening at different scales at .
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From page 230... ...
At the same time, there are general methods in science. So one can give a different answer depending on one's personality." FINGERPRINTS AND HUNGARIAN DANCES One contribution of waveless, Farge says, is that they have "forced people to think about what the Fourier transform is, forced them to think that when they choose a type of analysis they are in fact mixing the signal and the function used for the analysis.
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From page 231... ...
. At the other extreme it might send it to a wavelet transform (irregular signals, fractals, signals with small but important details)
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From page 232... ...
"It's as if we're trying to find the best match for each 'word' in the signal, while Best Basis is finding the best match for the whole sentence." Depending on the signal, Matching Pursuits uses one of two "dictionaries": one that contains wavelet packets and waveless, another that contains waveless and modified "windowed Fourier" waveforms. (While in standard windowed Fourier the size of the window is fixed and the number of oscillations within the window varies, in Matching Pursuits the size of the window is also allowed to vary.)
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From page 233... ...
The Fourier transform is a tool, and the wavelet transform is another tool, but very often when you have complex signals like speech, you want some kind of hybrid scheme. How can we mathematically formalize this problem?
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From page 234... ...
1993. Wavelets, Fractals, and Fourier Transforms.
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From page 235... ...
1993. Wavelets, Fractals, and Fourier Transforms.
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