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Families of Wavelet Transforms in Connection with Shannon's Sampling Theory and the Gabor Transform

A. Aldroubi, M. Unser

Wavelets: A Tutorial in Theory and Applications, C. K. Chui, Ed., Academic Press, Boston MA, USA, vol. 2, chap. VI, pp. 509-528, 1992.



In this chapter, we look at the algebraic structure of nonorthogonal scaling functions, the multiresolutions they generate, and the wavelets associated with them. By taking advantage of this algebraic structure, it is possible to create families of multiresolution representations and wavelet transforms with increasing regularity that satisfy some desired properties. In particular, we concentrate on two important aspects. First, we show how to generate sequences of scaling functions that tend to the ideal lowpass filter and for which the corresponding wavelets converge to the ideal bandpass filter. We give the conditions under which this convergence occurs and provide the link between Mallat's theory of multiresolution approximations and the classical Shannon Sampling Theory. This offers a framework for generating generalized sampling theories. Second, we construct families of nonorthogonal wavelets that converge to Gabor functions (modulated Gaussians). These latter functions are optimally concentrated in both time and frequency and are therefore of great interest for signal and image processing. We obtain the conditions under which this convergence occurs; thus allowing us to create whole classes of wavelets with asymptotically optimal time-frequency localization. We illustrate the theory using polynomial splines.


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