Wavelets and applications

When we try to analyze a signal, we often establish, explicitly or implicitly, a time-frequency representation of it. The image one might have of such an operation is that of the transcription of a musical score, which indicates to the musician the notes (and therefore the frequency information) he must produce at a given moment. The Fourier transform is not the appropriate tool for this analysis, since it masks the temporal evolution of the signal. On the other hand, as we shall show, the wavelet transform and its extensions provide interesting solutions in this context.

Wavelets are the brainchild of a geophysical engineer, J. Morlet, in the 1980s. Under the impetus of scientists such as physicist A. Grossman [39] and mathematician Y. Meyer [55], wavelets have become fundamental tools of modern harmonic analysis.

From an application point of view, wavelets have had an important influence in various fields: physics, numerical analysis (for example, for solving partial differential equations), statistics, signal and image processing, computer vision...

In the context of signal processing, the link between wavelet decompositions and more traditional tools such as filter banks lends a certain legitimacy to these transformations. The filter banks under consideration act by dividing the signal spectrum logarithmically, and are therefore fairly good approximations of the way human visual or auditory perceptual systems operate. Wavelets and multi-resolution techniques have enjoyed great success in image processing for problems such as motion estimation, pattern recognition, database searching and progressive information transmission. The key property exploited in these applications is the ability to approximate images at several scales, starting from a "coarse" view and refining it in successive processing steps.

In the remainder of this article, we'll present the various forms of wavelet transformations that exist. Schematically, three types can be distinguished:

• highly redundant representations (continuous wavelet transformations) ;

• parsimonious decompositions (orthogonal or biorthogonal wavelet bases, wavelet packets, etc.);

• intermediate solutions (wavelet frames).

We'll briefly show how these concepts extend to images and multidimensional data. Finally, we will present some of our most significant applications.

It's worth noting that wavelets sometimes require advanced mathematical concepts, and we'll try to get around the thorniest points throughout our presentation, at the risk of sometimes lacking precision.