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The Spline Foundation of Wavelet Theory

M. Unser, T. Blu

Plenary talk, proceedings of the International Conference on Wavelets and Splines (EIMI-WS'03), Saint Petersburg, Russia, July 3-8, 2003, pp. 98-99.



Recently, we came up with two interesting generalizations of polynomial splines by extending the degree of the generating functions to both real and complex exponents. While these may qualify as exotic constructions at first sight, we show here that both types of splines (fractional and complex) play a truly fundamental role in wavelet theory and that they lead to a better understanding of what wavelets really are.

To this end, we first revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to re-derive the main results of the classical theory—including some new extensions for fractional orders—in a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multi-scale differentiation, and smoothness (regularity) of the basis functions.

Second, we show that any scaling function can be expanded as a sum of harmonic splines (a particular subset of the splines with complex exponents); these play essentially the same role here as the Fourier exponentials do for periodic signals. This harmonic expansion provides an explicit time-domain representation of scaling functions and wavelets; it also explains their fractal nature. Remarkably, truncating the expansion preserves the essential multiresolution property (two-scale relation). Keeping the first term alone yields a fractional-spline approximation that captures most of the important wavelet features; e.g., its general shape and smoothness.

References


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