Linear Time-Frequency Analysis II: Wavelet-Type Representations
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AbstractWavelet theory was born in the mid-1980s in response to the time-frequency resolution problems of Fourier-type methods. Indeed, many non-stationary signals call for an analysis whose spectral (resp. temporal) resolution varies with the temporal (resp. spectral) localization. It is to allow this flexibility that wavelets, a new analysis concept called "multi-resolution" or "multi-scale", have been brought to light. After a brief presentation of the continuous wavelet transform, we shall focus on its discrete version, notably the Mallat algorithm, which is for the wavelet transform what the FFT is for the Fourier transform. We shall also consider the important problem of the design of wavelet generator filters (Daubechies filters, for example). Furthermore, we shall study some recent generalizations or extensions (in particular, multi-wavelets, wavelet packets and frames) that were motivated by certain limitations of wavelet theory. Finally, we shall discuss some applications that caused the present success of wavelets and, more generally, of time-scale methods (compression and denoising, aligning images, etc.). One of the aims of this chapter will thus be to demonstrate the cross-fertilization between sometimes quite theoretical approaches, where mathematics and engineering sciences are happily united. © 2008 ISTE Ltd.
All Author(s) ListBlu T., Lebrun J.
All Editor(s) Listed. by Hlawatsch, F. and Auger, F..
Detailed descriptioned. by Hlawatsch, F. and Auger, F..
Pages93 - 130
LanguagesEnglish-United Kingdom
KeywordsBi-orthogonality, Bi-orthonormality, Filter banks, Multi-resolution analysis, Multi-wavelets, Orthogonality, Orthonormality, Scale, Wavelets

Last updated on 2020-24-09 at 00:54