Haecceity: An Ontological Essay

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1. Introduction

Sign In. Access provided by: anon Sign Out. An introduction to wavelets Abstract: Wavelets were developed independently by mathematicians, quantum physicists, electrical engineers and geologists, but collaborations among these fields during the last decade have led to new and varied applications. What are wavelets, and why might they be useful to you? The fundamental idea behind wavelets is to analyze according to scale.

Indeed, some researchers feel that using wavelets means adopting a whole new mind-set or perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. Most of the basic wavelet theory has now been done.

The mathematics have been worked out in excruciating detail, and wavelet theory is now in the refinement stage. This involves generalizing and extending wavelets, such as in extending wavelet packet techniques. By definition, a wavelet has an integral of zero—half the area enclosed by the curve of the wavelet is above zero and half is below. Multiplying a wavelet by a constant changes both the positive and the negative components equally, so the integral remains zero. Wavelets can also be made that give coefficients of zero when they meet linear and quadratic stretches and even higher polynomials.

The more zero coefficients, the greater the compression of the signal, which makes it cheaper to store or transmit, and can simplify calculations. A typical signal may have about , values, but only 10, wavelets are needed to express it; parts of the signal that give coefficients of zero are automatically disregarded. Using wavelets that are "blind" to linear and quadratic stretches, and higher polynomials, also makes it easier to detect very irregular changes in a signal. Such wavelets react violently to irregular changes, giving big coefficients that stand out against the background of very small coefficients and zero coefficients indicating regular changes.

As a result, wavelets adapt automatically to the different components of a signal, using a big "window" to look at long-lived components of low frequency and progressively smaller windows to look at short-lived components of high frequency. The procedure is called multiresolution; the signal is studied at a coarse resolution to get an overall picture and at higher and higher resolutions to see increasingly fine details. Wavelets have in fact been called a "mathematical microscope";. The product of the section and the wavelet is itself a curve; the area under that curve is the wavelet coefficient.

Sections of the function that look like the wavelet give big coefficients, as seen in panels c and d. Two negative functions multiplied together give a positive function. Slowly changing sections produce small coefficients, as seen in panels e and f. Courtesy of John Hubbard, Cornell University.

And unlike a Fourier transform, which treats all parts of a signal equally, wavelets only encode changes in a function. That is, unchanging stretches of a signal give coefficients with the value zero, which can be ignored. This makes them good for "seeing" changes—peaks in a signal, for example, or edges in a picture—and also means that they can be effective for compressing information. In wavelet analysis a function is represented by a family of little waves: what the French call a "mother" wavelet, a "father," and a great many babies of various sizes.

But the French terminology "shows a scandalous misunderstanding of human reproduction," objects Cornell mathematician Robert Strichartz. But the father function plays an important role. If you were looking at changes in temperature, you might be interested in broad changes over millions of years the ice ages, for example , fluctuations over the past hundred years, or changes between day and night.

If your real interest were the effect of climate on wheat production in the nineteenth century, you might look at temperatures on the scale of a year, a decade, or possibly a century; you wouldn't bother with changes on the scale of a. A small window will be "blind" to low frequencies, which are too large to fit into the window. If a large window is used, information about a brief change will be submerged in the information about the entire section of the signal corresponding to the window. Bottom row: In wavelet analysis, a "mother wavelet" left is stretched or compressed to change the window size, making it possible to analyze a signal at different scales.

The wavelet transform has been called a "mathematical microscope": wide wavelets give an overall picture,while smaller and smaller wavelets are used to "zoom in" on small details. Wavelets shown in panels a and b are used in continuous representations; no father function is needed. The Morlet wavelet in panel a is complex valued; the dotted line gives the imaginary part.

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The "Mexican hat" shown in panel b is the second derivative of a Gaussian function bell curve. The "order" of a wavelet is a measure of its regularity or smoothness; the Daubechies wavelet in panel h is more regular than the one in panel f. Courtesy of Marie Farge and Eric Goirand. The father function now more often referred to as the "scaling function" gives the starting point see Figure 7. To do this, you construct a very rough approximation of your signal, using only father functions.

Imagine covering your signal with a row of. You then multiply each one by the corresponding section of the signal and integrate measure the space under the resulting curve.

Mathematical Theory of Subdivision: Finite Element and Wavelet Methods

The resulting numbers, or coefficients, give the information you would need to reconstitute a very rough picture of your function. Wavelets are then used to express the details that you would have to add to that first rough picture in order to reconstitute the original signal. At the coarsest resolution, perhaps a hundred fat wavelets are lined up next to each other on top of the signal. The wavelet transform see Box on p. At the next resolution the next generation of wavelets—twice as many, half as wide, and with twice the frequency—is put on top of the signal, and the process is repeated.

Each step gives more details; at each step the frequency doubles and the wavelets are half as wide. Typically, up to five different resolutions are used. The farther you go in layers of detail the more accurate the approximation will be, and each time you will be using a tool of the appropriate size. At the end, your signal has been neatly divided into different-frequency components—but unlike Fourier analysis, which gives you global amounts of different frequencies for the whole signal, wavelets give you a different frequency breakdown for different times on your signal.

Of course, the number will still be in decimal form; in contrast, when you reconstruct a signal from a father function, wavelets, and wavelet coefficients, you switch back to the original form of representation, out of "wavelet space. When Meyer took the train to Marseille to see Grossmann in the idea of multiresolution existed—it originated with Jean Morlet—but wavelets were limited and sometimes difficult to use, compared with the.

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,, 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. 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.

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 , by constructing precisely the kind of wavelet he had thought didn't exist. The practical consequences of orthogonality for wavelets are both good and bad.

Orthogonal wavelets encode information efficiently, and their coefficients can be calculated very rapidly. But the coefficients can be hard to interpret and the representation of the signal is not "shift invariant"—where you start encoding a signal makes a difference. Mathematically, orthogonality refers to a geometrical relationship.

Over the past years or so mathematicians have come to think of functions geometrically, as single points in an infinite dimensional space. You can define a point on a line with one number, a point in a place with two numbers, and a point in three-dimensional space with three numbers, but to define an entire function you need to know all its values: you need infinitely many numbers.

A function—a wavelet, for example—becomes a single point in an infinite dimensional space. Mathematicians can't really picture infinite dimensional space, but they learn to use their intuition about ordinary three-dimensional space to think about it. To get an idea of what orthogonality means geometrically, think of ordinary space.

Any point in space can be defined by a vector. Vectors are orthogonal if they form right angles with each other. In three-dimensional space, no more than three vectors can be mutually perpendicular, and in the plane only two.

Wavelet Theory and Its Applications

But in an infinite-dimensional space, you can choose infinitely many vectors, each of which is perpendicular to all the others. Families of orthogonal wavelets form such systems: the vectors formed by one mother wavelet and all its "babies" its "translates" and "dilates" 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.

That smooth orthogonal wavelets exist is not obvious; Yves Meyer, who found that they did, originally set out to prove the opposite. The Haar function , together with its translates and dilates, forms an orthogonal system, but some wavelet researchers do not consider it a true wavelet because it is so jerky:. An intermediary construction between the Haar function and Meyer's wavelets was made by J.

The following year, in the fall of , while Meyer was giving a course on wavelets at the University of Illinois at Urbana, he received several telephone calls from a persistent year-old graduate student in computer vision at the University of Pennsylvania in Philadelphia. Mallat had found this system absurd and had decided that he would do what he wanted.

Mallat had heard about Meyer's work on wavelets from a friend in the summer of while he was vacationing in St. Tropez; to him it sounded suspiciously familiar. So on returning to the United States, he called Meyer, who agreed to meet him at the University of Chicago. The two spent 3 days holed up in a borrowed office "I kept telling Mallat that he absolutely had to go to the Art Institute in Chicago, but we never had time," Meyer says while Mallat explained that the multiresolution Meyer and others were doing with wavelets was the same thing that electrical engineers and people in image processing were doing under other names.

In 3 days the two worked out the mathematical details; since Meyer was already a full professor, at his insistence the resulting paper,. The paper made it clear that work that existed in many different guises and under many different names—the pyramid algorithms used in image processing, the quadrature mirror filters of digital speech processing, zero-crossings, wavelets—were at heart all the same. For using wavelets to look at a signal at different resolutions can be seen as applying a succession of filters: first filtering out everything but low frequencies, then filtering out everything but frequencies twice as high, and so on.

And, in accordance with Shannon's sampling theorem, wavelets automatically "sample" high frequencies more often than low frequencies, since as the frequency doubles the number of wavelets doubles. The realization benefited everyone. A whole mathematical literature on wavelets existed by , some of it developed before the word wavelet was even coined; this mathematics could now be applied to other fields.

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. And he paved the way for the development by Daubechies of a new kind of regular orthogonal wavelet that was easier and faster to use: wavelets with "compact support. Daubechies, who is Belgian, was trained as a mathematical physicist; she had worked with Grossmann in France on her Ph.

She is the recipient of a 5-year MacArthur fellowship. She is able to speak to engineers, to mathematicians; she is trained as a physicist and one sees her training in quantum mechanics. Daubechies had heard about Meyer and Mallat's multiresolution work very early on. I had been thinking about some of these issues, and I got very interested," she said. The orthogonal wavelets Meyer had constructed trial off at the sides, never actually ending; this meant that calculating a single wavelet coefficient was a lot of work.

I didn't know Yves Meyer so very well at the time. When I had the first construction, he had gotten very excited, and somebody told me he had given a seminar on it. I knew he was a very strong mathematician and I thought, oh my God, he's probably figuring out things much faster than I can. By the end of March I had all the results.

Together, multiresolution and wavelets 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. Multiresolution and Daubechies's wavelets 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.

But although one mathematician hearing Daubechies lecture objected that she seemed "to be beating the uncertainty principle," the Heisenberg uncertainty principle still holds. You cannot have perfect knowledge of both time and frequency. Just as you cannot know simultaneously both the position and momentum of an elementary particle if you could hold an electron still long enough to figure out where it was, you would no longer know how fast it would have been going if you hadn't stopped it.

The product of the two uncertainties or spreads of possible values is always at least a certain minimum number. One must always make a compromise; knowledge gained about time is paid for in frequency, and vice versa. At very high frequency I have very narrow wavelets, and I localize very well in time but not so well in frequency," Daubechies said. This imprecision about frequency results from the increasing range of frequencies at high frequencies: as we have seen, frequency doubles each time one goes up an octave.

This widening spread of frequencies can be seen as a barrier to precision, but it's also an opportunity that engineers have learned to exploit. It is the reason why the telephone company shifts voices up into higher frequencies, not down to lower ones, when it wants to fit a lot of voices on one line: there's a lot more room up there. It also explains the advantage of fiber optics, which carry high-frequency light signals, over conventional telephone wires. Wavelets appear unlikely to have the revolutionary impact on pure mathematics that Fourier analysis has had.

British scientists are using wavelets to study ocean currents around the Antarctic, and researchers and mechanics are exploring their use in detecting faults in gears by analyzing vibrations. Multiresolution lends itself to a variety of applications in image processing. 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, Jr. Multiresolution is also useful in studying the large-scale distribution of matter in the universe, which for years was thought to be random but which is now seen to have a complicated structure, including "voids" and "bubbles. 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. 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. 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.

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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.

Among all edges of panel a, a computer program has selected the ''important" ones. Panel b displays these edges at a fine scale. Panel c is reconstructed from edges selected at different scales, with an algorithm developed by Mallat and Zhifeng Zhang. The edge selection removes the noise and small irregular structures.

The skin is now smoother. A simple way to avoid this blind slaughter has been found by a group of statisticians. David Donoho of Stanford University and the University of California at Berkeley and his colleague Iain Johnstone of Stanford had proved mathematically that if a certain kind of orthogonal basis existed, it would do the best possible job of extracting a signal from white noise.

A basis is something with which you can represent any possible function in a given space; each mother wavelet provides a different basis, for example, since any function can be represented by it and its translates and dilates. This result was interesting but academic since Donoho and Johnstone did not know whether such a basis existed. But in the summer of , when Donoho was in St. Flour, in France's Massif Central, to teach a course in probability, he heard Dominique Picard of the University of Paris-Jussieu give a talk on the possibility of using wavelets in statistics.

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 , and it works for a variety of kinds of signals. The astonishing thing is that it requires no assumptions about the signal. The traditional view is that one has to know, or assume, something about the signal one wants to extract from noise—that, as Grossmann put it, "if there is absolutely no a priori assumption you can make about your signal, you may as well go to sleep.

On the other hand, you don't want to put your wishes into your algorithm and then be surprised that your wishes come out. The wavelet method stands this traditional wisdom on its head. Making no assumptions about the signal, Donoho says, "you do as well as someone who makes correct assumptions, and much better than someone who makes wrong assumptions. The trick is that an orthogonal wavelet transform makes a signal look very different while leaving noise alone. That all orthogonal representations leave noise unchanged has been known since the s.

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.

Among those was Mallat, who uses a somewhat different approach. Donoho's method works for a whole range of functions but isn't necessarily optimal for each. When it is applied to blurred images, for example, it damages some of the edges; the elimination of small coefficients creates ripples that can be annoying. Mallat and graduate student Wen Liang Hwang developed a way to avoid this by computing the wavelet transform of the signal and selecting the points where the correlation between the curve and the wavelet is greatest, compared to nearby points.

These maximum values, or wavelet maxima , are kept if the points are thought to belong to a real edge and discarded if they are thought to correspond to noise. That decision is made automatically, but it requires more calculations than Donoho's method; it is based on the existence and size of maxima at different resolutions. Although Donoho and Johnstone's technique is simple and automatic, wavelets aren't always foolproof. With orthogonal wavelets it can matter where one starts encoding the signal: shifting over a little can change the coefficients completely, making pattern analysis hazardous.

This danger does not exist with continuous wavelets, but they have their own pitfalls; what looks like a correlation of coefficients different coefficients "seeing" the same part on the signal may sometimes be an. Generally, using wavelets takes practice. Farge has had similar experiences, but Meyer thinks the problem is real.

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. But wavelets don't exist in nature; that's why it is harder to interpret wavelet coefficients," he said.

Curiously, though, both our ears and our eyes appear to use wavelet techniques in the first stages of processing information. The work on "wavelets" and hearing goes back to the s, Daubechies said. You might miss important things, but you would miss things that our ear would miss too. One way to cope with an ever-increasing volume of signals is to widen the electronic highways—for example, by moving to higher frequencies. Another, which also reduces storage and computational costs, is to compress the signal temporarily, restoring it to its original form when needed.

In fact, only a small number of all possible "signals" are capable of being compressed, as the Russian mathematician Andrei Kolmogorov pointed out in the s. A compressible signal can by definition be expressed. It is easy to see that, using any given language such as the computer language Pascal , the number of short sequences is much smaller than the number of long sequences: most long sequences cannot be encoded by anything shorter than themselves.

Even a highly efficient encoding scheme like a library card catalog cannot cope with an infinite number of books; eventually, the only way to distinguish one book from another would be to print the entire book in the card catalog. Like Heisenberg with his uncertainty principle, Kolmogorov has set an absolute limit that mathematicians and scientists cannot overcome, however clever they are.

Accepting some loss of information makes more compression possible, but a limit still remains. In practice, however, many signals people want to compress have a structure that lends itself to compression; they are not random. For instance, any point in such a signal might be likely to be similar to points near it. In a picture of a white house with a blue door, a blue point is likely to be surrounded by other blue points, a white point by other white points. Wavelets lend themselves to such compression; because wavelet coefficients indicate only changes, areas with no change or very small change are automatically ignored, reducing the number of figures that have to be kept to encode the information.

So far, Daubechies says, image compression factors of about 35 or 40 have been achieved with wavelets with little loss. But wavelets alone cannot achieve those compression factors; an even more important role is played by clever quantization methods, mathematical ways of giving more weight in the encoding to information that is important for human perception edges, for example than information that is less so. So it's not really clear that we can beat the existing techniques. I do not think that image compression—for instance, television image compression—is really the place where wavelets will have the greatest impact.

But the fact that wavelets concentrate the information of a signal in relatively few coefficients makes them good at detecting edges in images, which may result in improved medical tests. Healy and Weaver have. And wavelet compression is valuable in speeding some calculations. 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 n 2 entries, then almost any computation requires at least n 2 calculations and sometimes as many as n 3.

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. But so far there is very little progress; it's just starting. 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. She was working on her doctoral thesis when she heard about wavelets from Alex Grossmann in Much later she learned that some turbulence researchers in the former Soviet Union, in Perm, had been working with similar techniques completely independently since So when Alex showed me that wavelets were objects that allowed one to unfold the representation both in physical space and in scale, I said to myself, this is it, now we're going to get somewhere.

I was shocked by their reaction to his talk. Now some of the people who were the most skeptical are completely infatuated and insist that everyone should use wavelets. It's just as ridiculous. It's a new tool, and one cannot force it into problems as shapeless as turbulence if it isn't calibrated first on academic signals that we know very well. We have to do a lot of experiments, get a lot of practice, develop methods, develop representations.

She uses orthogonal wavelets, or related wavelet packets, for compression but continuous wavelets for analysis: "I would never read the coefficients themselves in an orthogonal basis; they are too hard to read. Orthogonal wavelets also give time-scale information, of course although in a rougher form, since one doubles the scale each time, ignoring intermediate scales. The difference is largely one of legibility. Once, Meyer says, Jean Jacques Rousseau invented a musical notation based on numbers rather than notes on a staff, only to be told that it would never catch on, that musicians wanted to see the shape and movement of music on the page, to see the patterns formed by notes.

The coefficients of orthogonal wavelets correspond to Rousseau's music by numbers; continuous wavelets to the musical notation we know. Farge compares the current state of turbulence research to "prescientific zoology. Possible candidates are ill-defined creatures called "coherent structures" a tornado, for example, or the vortex that forms when you drain the bath. She uses wavelets to isolate them and to see how many exist at different scales or whether a single structure exists at a whole range of scales. Identifying the dynamically important structures would tell researchers "where we should invest lots of calculations and where we can skimp," Farge said.

For studying turbulence requires calculations that defy the most powerful computers. The Reynolds number for interactions of the atmosphere—a measure of its turbulence—ranges from 10 9 to 10 12 ; direct computer simulations of turbulence can now handle Reynolds numbers on the order of 10 2 or 10 3. But so far the results have been disappointing, Meyer says: "There should be something between turbulence and wavelets, everyone thinks so, but so far no one has a real scientific fact to offer. What can one hope for from methods that don't take the particular problem into account? At the same time, there are general methods in science.

So one can give a different answer depending on one's personality. One contribution of wavelets, 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. Often when people use the same technique for several scientific generations, they become blind to it. As work with wavelets progressed, it became clear that if Fourier analysis had limitations, wavelet analysis did also. As David Marr wrote in Vision , "Any particular representation makes certain information explicit at the expense of information that is pushed into the background and may be quite hard to recover.

So Coifman, Meyer, and Wickerhauser developed an information-compression scheme to take advantage of the strengths of both Fourier and wavelet methods: the "Best Basis" algorithm. In Best Basis a signal enters a computer like a train entering a switchyard in a train station. The computer analyzes the signal and decides what basis could encode it most efficiently, with the smallest possible amount of information. At one extreme it might send the signal to Fourier analysis for signals that resemble music, with repeating patterns.

At the other extreme it might send it to a wavelet transform irregular signals, fractals, signals with small but important details. Signals that don't fall clearly into either group are represented by "wavelet packets" that combine features of both Fourier analysis and wavelets. Loosely speaking, a wavelet packet is the product of a wavelet by a wiggle, an oscillating function. The wavelet itself can then react to abrupt changes, while the wiggle inside can react to regular oscillations.

Since the choice of wiggles is infinite, "it gives a family that is very rich. Because Best Basis was being patented, Wickerhauser said, the FBI did not adopt it but instead custom-made a similar technique. So far, the wavelet technique is intended only to compress fingerprints for storage or transmission, reconstructing them before identification by people or machines. But the FBI plans to hold a competition for automatic identification systems.

In studies with military helicopters, Best Basis has been used to simplify the calculations needed to decide, from radar signals, whether a possible target is a tank or perhaps just a boulder. In trials the Best Basis algorithm could compress the 64 original numbers produced by the radar system to 16 and still give "identical or better results than the original 64, especially in the presence of noise," Wickerhauser said. But probably the most unusual use of Best Basis has been in removing noise from a battered recording of Brahms playing his own work, recorded in on Thomas Edison's original phonograph machine, which used tinfoil and wax cylinders to record sound.

The Yale School of Music had entrusted it to Coifman after all else had failed. Then it was converted to a 78 record. That was the condition in which Yale had it—beaten to death. Coifman's approach was to say that noise can be defined as everything that is not well structured and that "well structured" means easily expressed, with very few terms, with something like the Best Basis algorithm. So the idea is to use Best Basis to decompose the signal and to remove anything that is left over.

Wavelet Methods In Mathematical Analysis And Engineering - Google книги

The result was not musical—no one hoped for that, from a recording that contained perhaps 30 times as much noise as signal—but they were able to identify the music as variations on Hungarian dances. For some purposes, however, Best Basis is not ideal. Because it treats the signal as a whole, it has trouble dealing with highly nonstationary signals—signals that change unpredictably. Depending on the signal, Matching Pursuits uses one of two "dictionaries": one that contains wavelet packets and wavelets, another that contains wavelets 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.

First, the appropriate dictionary is scanned and the "word" chosen that best matches the signal; then that "word'' is subtracted out, and the best match is chosen for the remaining part of the signal, and so on. To some. Because the waveforms used are not orthogonal to each other, the system is slower than Best Basis: n 2 calculations compared to n log n for Best Basis. On the other hand, the lack of orthogonality means that it doesn't matter where on the signal you start encoding.

This makes it better suited for pattern recognition, for encoding contours and textures, for example. But the quest for new ways to encode information is far from over. If your dictionary is too small, you'll need a lot of words to express one idea," Mallat said. 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? In some cases the task goes beyond mathematics; the ultimate judge of effective compression of a picture, or of speech, is the human eye or ear, and developing the right mathematical representation is often intimately linked to human perception.

Information is not all equal. Even a young child can draw a recognizable outline of a cat, for example, while the very notion of a drawing without edges is perplexing, like the Cheshire cat who vanished, leaving only his smile. Other differences are less well understood. People have no trouble differentiating textures, for example, while "after 20 years of research on texture, we still don't really know what it is mathematically," Mallat said. Wavelets may help with this, especially since some wavelet-like techniques are used in human vision and hearing, but any illusions researchers may have had that wavelets will solve all problems unsuited to Fourier analysis have long since vanished; the field has become wide open.

It may not be the least of the contributions made by wavelets that they have inspired both a closer and a broader look at mathematical languages for expressing information: a more judicious look at Fourier analysis, which was often used reflexively "The first thing any engineer. It opens up your eyes to a much broader universe.

Herivel , p. Cousin, V.

Fourier Biographical notes to follow praise of Mr. Delambre, Jean-Baptiste. Encyclopedia Britannica, 11 th ed. Harmonic analysis. Farge, M. Hunt and J. Vassilicos eds. Wavelets, Fractals, and Fourier Transforms. Clarendon Press, Oxford. Fourier, J. Healy, D.

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Herivel, J. Joseph Fourier — the Man and the Physicist. Fourier Analysis. Cambridge University Press, Cambridge.

Easy Introduction to Wavelets

Meyer, Y.