Wavelet introduction (2)

(2) Introduction

On November 25, 1998, Walt Disney Pictures and Pixar Animation Studios released a full-length computer-animated feature film called A Bug’s Life. It was the second such collaboration for Disney and Pixar and, like its groundbreaking predecessor Toy Story three years earlier, it opened to rave reviews. A Bug’s Life, said one reviewer, “teems with beautiful visual inventions…; with intricate details that will keep adults as well as kids bug-eyed from start to finish…; and with colors teased from some new, hitherto-secret pastel spectrum…” Only the most computer graphics-savvy moviegoers would have given any thought to the mathematical modeling techniques that made it possible to develop all the characters in the animated ants’ story—not to mention their many textures, their myriad expressions, and the way they jumped, flitted, and buzzed around. As it happened, though, a particular type of modeling technique made its debut in the movie, a method of computer animation that makes use of a collection of mathematical procedures called “wavelets.” One way of thinking about wavelets is to consider how our eyes look at the world. In the real world, you can observe a forest like the one shown in the photograph on the next page from many vantage points—in effect, at different scales of resolution. From the window of a cross-country jet, for example, the forest appears to be a solid canopy of green. From the window of an automobile on the ground, the canopy resolves into individual trees, and if you get out of the car and move closer, you begin to see branches and leaves. If you then pull out a magnifying glass, you might find a drop of dew at the end of a leaf. As you zoom in at smaller and smaller scales, you can find details that you didn’t see before. Try to do that with a photograph, however, and you will be disappointed. Enlarge the photograph to get “closer” to a tree and all you’ll have is a fuzzier tree; the branch, the leaf, the drop of dew are not to be found. Although our eyes can see the forest at many scales of resolution, the camera can show only one at a time. Computers do no better than cameras; in fact, their level of resolution is inferior. On a computer screen, the photograph becomes a collection of pixels that are much less sharp than the original. Soon, however, computers everywhere will be able to do something that photographers have only been able to dream of. They will be able to display an interactive image of a forest in which the viewer can zoom in to get greater detail of the trees, branches, and perhaps even the leaves. They will be able to do this because wavelets make it possible to compress the amount of data used to store an image, allowing a more detailed image to be stored in less space. Even though as an organized research topic wavelets is less than two decades old, it arises from a constellation of related concepts developed over a period of nearly two centuries, repeatedly rediscovered by scientists who wanted to solve technical problems in their various disciplines. Signal processors were seeking a way to transmit clear messages over telephone wires. Oil prospectors wanted a better way to interpret seismic traces. Yet “wavelets” did not become a household word among scientists until the theory was liberated from the diverse applications in which it arose and was synthesized into a purely mathematical theory. This synthesis, in turn, opened scientists’ eyes to new applications. Today, for example, wavelets are not only the workhorse in computer imaging and animation; they also are used by the FBI to encode its data base of 30 million fingerprints. In the future, scientists may put wavelet analysis to work diagnosing breast cancer, looking for heart abnormalities, or predicting the weather.