Acknowledgments

The authors wish to express their gratitude to all the colleagues who directly or indirectly contributed to the making of the Wavelet Toolbox™ software.

Specifically

  • To Pierre-Gilles Lemarié-Rieusset (Evry) and Yves Meyer (ENS Cachan) for their help with wavelet questions

  • To Lucien Birgé (Paris 6), Pascal Massart (Paris 11), and Marc Lavielle (Paris 5) for their help with statistical questions

  • To David Donoho (Stanford) and to Anestis Antoniadis (Grenoble), who give generously so many valuable ideas

Other colleagues and friends who have helped us enormously are Patrice Abry (ENS Lyon), Samir Akkouche (Ecole Centrale de Lyon), Mark Asch (Paris 11), Patrice Assouad (Paris 11), Roger Astier (Paris 11), Jean Coursol (Paris 11), Didier Dacunha-Castelle (Paris 11), Claude Deniau (Marseille), Patrick Flandrin (Ecole Normale de Lyon), Eric Galin (Ecole Centrale de Lyon), Christine Graffigne (Paris 5), Anatoli Juditsky (Grenoble), Gérard Kerkyacharian (Paris 10), Gérard Malgouyres (Paris 11), Olivier Nowak (Ecole Centrale de Lyon), Dominique Picard (Paris 7), and Franck Tarpin-Bernard (Ecole Centrale de Lyon).

One of our first opportunities to apply the ideas of wavelets connected with signal analysis and its modeling occurred in collaboration with the team "Analysis and Forecast of the Electrical Consumption" of Electricité de France (Clamart-Paris) directed first by Jean-Pierre Desbrosses, and then by Hervé Laffaye, and which included Xavier Brossat, Yves Deville, and Marie-Madeleine Martin.

And finally, apologies to those we may have omitted.

About the Authors

Michel Misiti, Georges Oppenheim, and Jean-Michel Poggi are mathematics professors at Ecole Centrale de Lyon, University of Marne-La-Vallée and Paris 5 University. Yves Misiti is a research engineer specializing in Computer Sciences at Paris 11 University.

The authors are members of the "Laboratoire de Mathématique" at Orsay-Paris 11 University France. Their fields of interest are statistical signal processing, stochastic processes, adaptive control, and wavelets. The authors' group has published numerous theoretical papers and carried out applications in close collaboration with industrial teams. For instance:

  • Robustness of the piloting law for a civilian space launcher for which an expert system was developed

  • Forecasting of the electricity consumption by nonlinear methods

  • Forecasting of air pollution

Notes by Yves Meyer

The history of wavelets is not very old, at most 10 to 15 years. The field experienced a fast and impressive start, characterized by a close-knit international community of researchers who freely circulated scientific information and were driven by the researchers' youthful enthusiasm. Even as the commercial rewards promised to be significant, the ideas were shared, the trials were pooled together, and the successes were shared by the community.

There are lots of successes for the community to share. Why? Probably because the time is ripe. Fourier techniques were liberated by the appearance of windowed Fourier methods that operate locally on a time-frequency approach. In another direction, Burt-Adelson's pyramidal algorithms, the quadrature mirror filters, and filter banks and subband coding are available. The mathematics underlying those algorithms existed earlier, but new computing techniques enabled researchers to try out new ideas rapidly. The numerical image and signal processing areas are blooming.

The wavelets bring their own strong benefits to that environment: a local outlook, a multiscaled outlook, cooperation between scales, and a time-scale analysis. They demonstrate that sines and cosines are not the only useful functions and that other bases made of weird functions serve to look at new foreign signals, as strange as most fractals or some transient signals.

Recently, wavelets were determined to be the best way to compress a huge library of fingerprints. This is not only a milestone that highlights the practical value of wavelets, but it has also proven to be an instructive process for the researchers involved in the project. Our initial intuition generally was that the proper way to tackle this problem of interweaving lines and textures was to use wavelet packets, a flexible technique endowed with quite a subtle sharpness of analysis and a substantial compression capability. However, it was a biorthogonal wavelet that emerged victorious and at this time represents the best method in terms of cost as well as speed. Our intuitions led one way, but implementing the methods settled the issue by pointing us in the right direction.

For wavelets, the period of growth and intuition is becoming a time of consolidation and implementation. In this context, a toolbox is not only possible, but valuable. It provides a working environment that permits experimentation and enables implementation.

Since the field still grows, it has to be vast and open. The Wavelet Toolbox product addresses this need, offering an array of tools that can be organized according to several criteria:

  • Synthesis and analysis tools

  • Wavelet and wavelet packets approaches

  • Signal and image processing

  • Discrete and continuous analyses

  • Orthogonal and redundant approaches

  • Coding, de-noising and compression approaches

What can we anticipate for the future, at least in the short term? It is difficult to make an accurate forecast. Nonetheless, it is reasonable to think that the pace of development and experimentation will carry on in many different fields. Numerical analysis constantly uses new bases of functions to encode its operators or to simplify its calculations to solve partial differential equations. The analysis and synthesis of complex transient signals touches musical instruments by studying the striking up, when the bow meets the cello string. The analysis and synthesis of multifractal signals, whose regularity (or rather irregularity) varies with time, localizes information of interest at its geographic location. Compression is a booming field, and coding and de-noising are promising.

For each of these areas, the Wavelet Toolbox software provides a way to introduce, learn, and apply the methods, regardless of the user's experience. It includes a command-line mode and a graphical user interface mode, each very capable and complementing to the other. The user interfaces help the novice to get started and the expert to implement trials. The command line provides an open environment for experimentation and addition to the graphical interface.

In the journey to the heart of a signal's meaning, the toolbox gives the traveler both guidance and freedom: going from one point to the other, wandering from a tree structure to a superimposed mode, jumping from low to high scale, and skipping a breakdown point to spot a quadratic chirp. The time-scale graphs of continuous analysis are often breathtaking and more often than not enlightening as to the structure of the signal.

Here are the tools, waiting to be used.

Yves Meyer
Professor, Ecole Normale Supérieure de Cachan and Institut de France

Notes by Ingrid Daubechies

Wavelet transforms, in their different guises, have come to be accepted as a set of tools useful for various applications. Wavelet transforms are good to have at one's fingertips, along with many other mostly more traditional tools.

Wavelet Toolbox software is a great way to work with wavelets. The toolbox, together with the power of MATLAB® software, really allows one to write complex and powerful applications, in a very short amount of time. The Graphic User Interface is both user-friendly and intuitive. It provides an excellent interface to explore the various aspects and applications of wavelets; it takes away the tedium of typing and remembering the various function calls.

Ingrid C. Daubechies
Professor, Princeton University, Department of Mathematics and Program in Applied and Computational Mathematics

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