Wavelets Demystified
M. Unser
Plenary talk, Summer School "New Trends and Directions in Harmonic Analysis, Approximation Theory, and Image Analysis," September 17-21, 2007, Inzell, Germany.
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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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M. Unser, T. Blu, "Wavelet Theory Demystified," IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 470-483, February 2003.
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M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 43-67, March 2000.
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M. Unser, T. Blu, "Wavelet Games," Wavelet Digest, vol. 11, no. 4, April 1, 2003.