Wavelet Introduction (5)

(5) The Great Synthesis

Jean Morlet didn’t plan to start a scientific revolution. He was merely trying to give geologists a better way to search for oil. Petroleum geologists usually locate underground oil deposits by making loud noises. Because sound waves travel through different materials at different speeds, geologists can infer what kind of material lies under the surface by sending seismic waves into the ground and measuring how quickly they rebound. If the waves propagate especially quickly through one layer, it may be a salt dome, which can trap a layer of oil underneath. Figuring out just how the geology translates into a sound wave (or vice versa) is a tricky mathematical problem, and one that engineers traditionally solve with Fourier analysis. Unfortunately, seismic signals contain lots of transients—abrupt changes in the wave as it passes from one rock layer to another. These layers, but Fourier analysis spreads that spatial information out all over the place. Morlet, an engineer for Elf-Aquitaine, developed his own way of analyzing the seismic signals to create components that were localized in space, which he called “wavelets of constant shape.” Later, they would be known as “Morlet wavelets.” Whether the components are dilated, compressed, or shifted in time, they maintain the same shape. Other families of wavelets can be built by taking a different shape, called a mother wavelet, and dilating, compressing, or shifting it in time. Researchers would find that the exact shape of the mother wavelet strongly affects the accuracy and compression properties of the approximation. Many of the differences between earlier versions of wavelets simply amounted to different choices for the mother wavelet. Morlet’s method wasn’t in the books, but it seemed to work. On his personal computer, he could separate a wave into its wavelet components, and then reassemble them into the original wave. But he wasn’t satisfied with this empirical proof, and began asking other scientists if the method was mathematically sound. Morlet found the answer he wanted from Alex Grossmann, a physicist at the Centre de Physique Théorique in Marseilles. Grossmann worked with Morlet for a year to confirm that waves could be reconstructed from their wavelet decompositions. In fact, wavelet transforms turned out to work better than Fourier transforms, because they are much less sensitive to small errors in the computation. An error or an unwise truncation of the Fourier coefficients can turn a smooth signal into a jumpy one or vice versa; wavelets avoid such disastrous consequences. Morlet and Grossmann’s paper, the first to use the word “wavelet,” was published in 1984. Yves Meyer, currently at the École Normale Supérieure de Cachan, widely acknowledged as one of the founders of wavelet theory, heard about their work in the fall of the same year. He was the first to realize the connection between Morlet’s wavelets and earlier mathematical wavelets, such as those in the work of Littlewood and Paley. (Indeed, Meyer has counted 16 separate rediscoveries of the wavelet concept before Morlet and Grossmann’s paper.) Meyer went on to discover a new kind of wavelet, with a mathematical property called orthogonality that made the wavelet transform as easy to work with and manipulate as a Fourier transform. (“Orthogonality” means that the information captured by one wavelet is completely independent of the information captured by another.) Perhaps most importantly, he became the nexus of the emerging wavelet community. In 1986, Stéphane Mallat, a former student of Meyer’s who was working on a doctorate in computer vision, linked the theory of wavelets to the existing literature on subband coding and quadrature mirror filters, which are the image processing community’s versions of wavelets. The idea of multiresolution analysis—that is, looking at signals at different scales of resolution—was already familiar to experts in image processing. Mallat, collaborating with Meyer, showed that wavelets are implicit in the process of multiresolution analysis. Thanks to Mallat’s work, wavelets became much easier. One could now do a wavelet analysis without knowing the formula for a mother wavelet. The process was reduced to simple operations of averaging groups of pixels together and taking their differences, over and over. The language of wavelets also became more comfortable to electrical engineers, who embraced familiar terms such as “filters,” “high frequencies,” and “low frequencies.” The final great salvo in the wavelet revolution was fired in 1987, when Ingrid Daubechies, while visiting the Courant Institute at New York University and later during her appointment at AT&T Bell Laboratories, discovered a whole new class of wavelets, which were not only orthogonal (like Meyer’s) but which could be implemented using simple digital filtering ideas, in fact, using short digital filters. The new wavelets were almost as simple to program and use as Haar wavelets, but they were smooth, without the jumps of Haar wavelets. Signal processors now had a dream tool: a way to break up digital data into contributions of various scales. Combining Daubechies and Mallat’s ideas, there was a simple, orthogonal transform that could be rapidly computed on modern digital computers. The Daubechies wavelets have surprising features—such as intimate connections with the theory of fractals. If their graph is viewed under magnification, characteristic jagged wiggles can be seen, no matter how strong the magnification. This exquisite complexity of detail means there is no simple formula for these wavelets. They are ungainly and asymmetric; nineteenth-century mathematicians would have recoiled from them in horror. But like the Model-T Ford, they are beautiful because they work. The Daubechies wavelets turn the theory into a practical tool that can be easily programmed and used by any scientist with a minimum of mathematical training.