Book Review: An introduction to wavelet analysis
Palle E. T. Jørgensen · Bulletin of the American Mathematical Society · 2003
Michael Berry, summarizing for the Bulletin his recent AMS Gibbs Lecture [2], observes:Nowhere are the intimate connections between mathematics and physics more immediately apparent than in optics; with our own eyes, we can see through physical phenomena almost directly to the conceptual structures underlying them.Risking the wrath of philosophers, I use the term mathematical phenomena to describe these structures.Once in a while a new trend in mathematics comes along.The skeptics would call it a new fad and ask what all the fuss is about.Those who are convinced will get on the wagon and drop the infinite series they are working on.Others will be looking for the lost remainder terms.Wavelet analysis is in a sense a new trend, but it started with Alfred Haar's paper [6] almost a hundred years ago.The significance of Haar's original construction was perhaps not fully understood until much later in the mid-1980's.Some of the reasons for the wavelet craze (a favorite term of the skeptics!)have to do with the need for fast algorithms, brought about by our better understanding of connections of wavelets to signal processing, optics, data compression, turning fingerprints into digital data files, subdivision algorithms in graphics, digital cameras, high-resolution television, and the JPEG 2000 encoding of images.As a mathematical subject, the theory of wavelets draws on tools from mathematics itself, such as harmonic analysis and numerical analysis.But in addition there are exciting links to areas outside mathematics.The connections to electrical and computer engineering, and to image compression and signal processing in particular, are especially fascinating.These interconnections of research disciplines may be illustrated with the two subjects (1) wavelets and (2) subband filtering (from signal processing).While they are quite different and have distinct and independent lives, and even have different aims and different histories, they have in recent years found common ground.It is a truly amazing success story.Advances in one area have helped the other: subband filters are absolutely essential in wavelet algorithms and in numerical recipes used in subdivision schemes, for example, and especially in JPEG 2000-an important and extraordinarily successful image-compression code.JPEG uses nonlinear approximations and harmonic analysis in spaces of signals of bounded variation.Similarly, new wavelet approximation techniques have given rise to the kind of data-compression which is now used by the FBI (via a patent held by two mathematicians) in digitizing fingerprints in the U.S. It is the happy marriage of the two disciplines, signal processing and wavelets, that enriches the union of the subjects, and the applications, to an extraordinary degree.