Misiti M., Misiti Y., Oppenheim G., Poggi J.-M. (eds.) Wavelets and their Applications

Издательство ISTE, 2007, -352 pp.Wavelets are a recently developed signal processing tool enabling the analysis on several timescales of the local properties of complex signals that can present nonstationary zones. They lead to a huge number of applications in various fields, such as, for example, geophysics, astrophysics, telecommunications, imagery and video coding. They are the foundation for new techniques of signal analysis and synthesis and find beautiful applications to general problems such as compression and denoising.
The propagation of wavelets in the scientific community, academic as well as industrial, is surprising. First of all, it is linked to their capacity to constitute a tool adapted to a very broad spectrum of theoretical as well as practical questions. Let us try to make an analogy: the emergence of wavelets could become as important as that of Fourier analysis. A second element has to be noted: wavelets have benefited from an undoubtedly unprecedented trend in the history of applied mathematics. Indeed, very soon after the grounds of the mathematical theory had been laid in the middle of the 1980s [MEY 90], the fast algorithm and the connection with signal processing [MAL 89] appeared at the same time as Daubechies orthogonal wavelets [DAU 88]. This body of knowledge, diffused through the Internet and relayed by the dynamism of the research community enabled a fast development in numerous applied mathematics domains, but also in vast fields of application.
Thus, in less than 20 years, wavelets have essentially been imposed as a fruitful mathematical theory and a tool for signal and image processing. They now therefore form part of the curriculum of many pure and applied mathematics courses, in universities as well as in engineering schools.
By omitting the purely mathematical contributions and focusing on applications, we may identify three general problems for which wavelets have proven very powerful.
The first problem is analysis, for scrutinizing data and sounding out the local signal regularity on a fine scale. Indeed, a wavelet is a function oscillating as a wave but quickly damped. Being well localized simultaneously in time and frequency it makes it possible to define a family of analyzing functions by translation in time and dilation in scale. Wavelets constitute a mathematical zoom making it possible to simultaneously describe the properties of a signal on several timescales.
The second problem is denoising or estimation of functions. This means recovering the useful signal while we only observe a noisy version thereof. Since the denoising methods are based on representations by wavelets, they create very simple algorithms that, due to their adaptability, are often more powerful and easy to tune than the traditional methods of functional estimation. The principle consists of calculating the wavelet transform of observations, then astutely modifying the coefficients profiting from their local nature and, finally, inversing the transformation.
The third problem is compression and, in particular, compression of images where wavelets constitute a very competitive method. Due to generally very sparse representations, they make it possible to reduce the volume of information to be coded. In order to illustrate this point, we can consider two leading applications whose impact has propagated well beyond the specialists in the field. The first application relates to the storage of millions of fingerprints by the FBI and the second is linked to the new standard of image compression JPEG 2000, which is based on wavelets.
Wavelets provide particularly elegant solutions to a number of other problems. Let us quote, for example, the numerical solution of partial derivative equations or even, more to the point, the simulation of paths for fractional Brownian processes. Numerous types of software have appeared since the beginning of the 1990s and, particularly over the last few years, a complete list can be found on the website www.amara.com/current/wavesoft.html.
Our aim is to be somewhere in the space that separates the foundations and the computerized implementation. Operating with wavelets means understanding the origins of the tool and, at the same time, its application to signals. We attempt to show the link between knowing and acting by introducing a very large number of images illustrating the text (perhaps 200, often simple ones).
Who are the ideal readers? This is always difficult to say. From the most theoretical point of view, a specialist in mathematical analysis is likely not to find his due here. Other very good texts already exist. As for the applications, the programming of algorithms is not included in this book; a lot of software is available. We position ourselves somewhere between these two extremes: theoretical mathematics and applied algorithms.
This work, intended for a large audience of scientists, is directed towards learning and understanding wavelets through their applications. It can be useful by complementing the works strictly dedicated to mathematical approaches for students of engineering schools, those undertaking graduate and postgraduate research, as well as to engineers and researchers eager to have a compact yet broad view of wavelets in action.A Guided Tour.
Mathematical Framework.
From Wavelet Bases to the Fast Algorithm.
Wavelet Families.
Finding and Designing a Wavelet.
A Short 1D Illustrated Handbook.
Signal Denoising and Compression.
Image Processing with Wavelets.
An Overview of Applications.
The EZW Algorithm.