Wavelet Introduction (7)

Wavelets in the Future

With the foundations of wavelet theory securely in place, the field has grown rapidly over the last decade. A distribution list on wavelets that began with 40 names in 1990 is now an online newsletter with more than 17,000 subscribers. Moreover, it has continued to evolve through a healthy mix of theory and practice. Engineers are always trying new applications, and for mathematicians, there are still important theoretical questions to be answered. Although wavelets are best known for image compression, many researchers are interested in using wavelets for pattern recognition. In weather forecasting, for example, they might slim down the data-bloated computer models that are now in use. Traditionally, such models sample the barometric pressure (for instance) at an enormous number of grid points and use this information to predict how the data will evolve. However, this approach uses up a lot of computer memory. A model of the atmosphere that uses a 1000-by-1000-by-1000 grid requires a billion data points—and it’s still a fairly crude model. However, most of the data in the grid are redundant. The barometric pressure in your town is probably about the same as the barometric pressure a mile down the road. If the weather models used wavelets, they could view the data the same way weather forecasters do, concentrating on the places where abrupt changes occur—warm fronts, cold fronts and the like. Other problems in fluid dynamics have been tackled the same way. At Los Alamos National Laboratory, for example, wavelets are used to study the shock waves produced by a bomb explosion. As demonstrated by the recent spate of full-length computer-animated films, wavelets also have a promising future in the movies. Because the wavelet transform is a reversible process, it is just as easy to synthesize an image (build it up out of wavelets) as it is to analyze it (break it down into wavelet components). This idea is related to a new computer animation method called subdivision surfaces, basically a multiresolution analysis run in reverse. To draw a cartoon character, the animator only has to specify where a few key points go, creating a low-resolution version of the character. The computer can then do a reverse multiresolution analysis, making the character look like a real person and not a stick figure. Subdivision surfaces debuted in the 1998 movie A Bug’s Life, replacing a more clumsy method called NURBs (for non-uniform rational B splines) that had been used in the first Toy Story movie in 1995. Interestingly, NURBs and subdivision methods coexisted in 1999’s Toy Story 2, where the characters that appeared in the first Toy Story remained NURBs, but where the new characters were based on the subdivision method. The next frontier for subdivision surfaces may be the video game industry, where they could eliminate the blocky look of today’s graphics. Meanwhile, on the theoretical side, mathematicians are still looking for better kinds of wavelets for two- and three-dimensional images. Although the standard wavelet methods are good at picking up edges, they do it one pixel at a time—an inefficient way of representing something that may be a very simple curve or line. David Donoho and Emmanuel Candès of Stanford University have proposed a new class of wavelets called “ridgelets,” which are specifically designed to detect discontinuities along a line. Other researchers are studying “multiwavelets,” which can be used to encode multiple signals traveling through the same line, such as color images in which three color values (red, green, and blue) have to be transmitted at once. When asked to justify the value of mathematics, mathematicians often point out that ideas developed to solve a pure mathematical problem can lead to unexpected applications years later. But the story of wavelets paints a more complicated and somewhat more interesting picture. In this case, specific applied research led to a new theoretical synthesis, which in turn opened scientists’ eyes to new applications. Perhaps the broader lesson of wavelets is that we should not view basic and applied sciences as separate endeavors: Good science requires us to see both the theoretical forest and the practical trees.