EPFL | Biomedical Imaging Group

Wavelets Demystified

M. Unser

Plenary talk, Proceedings of the Summer School "New Trends and Directions in Harmonic Analysis, Approximation Theory, and Image Analysis," Inzell, Federal Republic of Germany, September 17-21, 2007, pp. 15-16.

This tutorial focuses on wavelet bases: it covers the concept of multi-resolution analysis, the construction of wavelets, filterbank algorithms, as well as an in-depth discussion of fundamental wavelet properties. The presentation is progressive—starting with the example of the Haar transform—and essentially self-contained.

We emphasize the crucial role of splines in wavelet theory, presenting a non-standard point of view that simplifies the mathematical formulation. The key point is that any wavelet (or scaling function) can be expressed as the convolution of a (fractional) B-spline and a singular distribution, and that all fundamental spline properties (reproduction of polynomials, regularity, order of approximation, etc.) are preserved through the convolution operation. A direct implication is that the wavelets have vanishing moments and that they behave like multi-scale differentiators. These latter two properties are the key for understanding why wavelets yield sparse representations of piecewise-smooth signals.

References

M. Unser, T. Blu, "Wavelet Theory Demystified," IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 470-483, February 2003.
M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 43-67, March 2000.
M. Unser, T. Blu, "Wavelet Games," Wavelet Digest, vol. 11, no. 4, April 1, 2003.

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/unser0706.html,
AUTHOR="Unser, M.",
TITLE="Wavelets Demystified",
BOOKTITLE="Summer School ``New Trends and Directions in Harmonic
	Analysis, Approximation Theory, and Image Analysis''",
YEAR="2007",
editor="",
volume="",
series="",
pages="15--16",
address="Inzell, Federal Republic of Germany",
month="September 17-21,",
organization="{M}arie {C}urie Excellence Team {MAMEBIA} funded by the
	{E}uropean {C}ommission",
publisher="",
note="Plenary talk")

© 2007 . 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 . 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.