Abstract: This 2-hour 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 multiscale differentiators. These latter two properties are the key for understanding why wavelets yield sparse representations of piecewise-smooth signals.

• Slides of the presentation (PDF 7.8Mb)

Bibliography

• M. Unser, M. Unser, "Wavelet Games," Wavelet Digest, vol. 11, no. 4, April 1, 2003.
• S.G. Mallat, "A theory of multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-11, no. 7, pp. 674-693, 1989.
• M. Unser, T. Blu, "Wavelets Theory Demystified," IEEE Trans. Signal Processing, vol. 51, no. 2, pp. 470-483, 2003.
• M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 43-67, 2000.
• M. Unser, A. Aldroubi, "A Review of Wavelets in Biomedical Applications," Proc. IEEE, vol. 84, no. 4, pp. 626-638, 1996.

• Abstract
• Slides of the presentation Part I (PDF 5Mb)
• Slides of the presentation Part II (PDF 5 Mb)

• To provide a tutorial on splines that is geared to a signal processing audience.
• To gather all their important properties, and to provide an overview of the mathematical and computational tools available; i.e., a road map for the practitioner with references to the appropriate literature.
• To review the primary applications of splines in signal and image processing.