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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


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