Biomedical Imaging GroupSTI
English only   BIG > Publications > Wavelet Families

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 All BibTeX References

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.

AUTHOR="Aldroubi, A. and Unser, M.",
TITLE="Families of Wavelet Transforms in Connection with {S}hannon's
        Sampling Theory and the {G}abor Transform",
BOOKTITLE="Wavelets: {A} Tutorial in Theory and Applications",
PUBLISHER="Academic Press",
editor="Chui, C.K.",
address="Boston MA, USA",

© 1992 Academic Press. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from Academic Press.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.