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Complex B-Splines and Wavelets

B. Forster, T. Blu, M. Unser

Second International Conference on Computational Harmonic Analysis, Nineteenth Annual Shanks Lecture (CHA'04), Nashville TN, USA, May 24-30, 2004.

B-spline multiresolution analyses have proven to be an adequate tool for signal analysis. But for some applications, e.g. in speech processing and digital holography, complex-valued scaling functions and wavelets are more favourable than real ones, since they allow to deduce the crucial phase information.

In this talk, we extend the classical resp. fractional B-spline approach to complex B-splines. We perform this by choosing a complex exponent, i.e., a complex order z of the B-spline, and show that this does not influence the basic properties such as smothness and decay, recurrence relations and others. Moreover, the resulting complex B-splines satisfy a two-scale relation and generate a multiresolution analysis of L2(R). We show that the complex B-splines as well as the corresponding wavelets converge to Gabor functions as ℜ(z) increases and ℑ(z) is fixed. Thus they are approximately optimally time-frequency localized.

AUTHOR="Forster, B. and Blu, T. and Unser, M.",
TITLE="Complex \mbox{{B}-Splines} and Wavelets",
BOOKTITLE="Second International Conference on Computational Harmonic
        Analysis, Nineteenth Annual Shanks Lecture ({CHA'04})",
address="Nashville TN, USA",
month="May 24-30,",

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