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Fractional Splines and Wavelets: From Theory to Applications

M. Unser, T. Blu

Joint IDR-IMA Workshop: Ideal Data Representation, Minneapolis MN, USA, April 9-13, 2001.



In the first part, we present the theory of fractional splines; an extension of the polynomial splines for non-integer degrees. Their basic constituents are piecewise power functions of degree α. The corresponding B-splines are obtained through a localization process similar to the classical one, replacing finite differences by fractional differences. We show that the fractional B-splines share virtually all the properties of the classical B-splines, including the two-scale relation, and can therefore be used to define new wavelet bases with a continuously varying order parameter. We discuss some of their remarkable properties; in particular, the fact that the fractional spline wavelets behave like fractional derivatives of order α + 1.

In the second part, we turn to applications. We first describe a fast implementation of the fractional wavelet transform, which is essential to make the method practical. We then present an application of fractional splines to tomographic reconstruction where we take advantage of explicit formulas for computing the fractional derivatives of splines. We also make the connection with ridgelets. Finally, we consider the use of fractional wavelets for the detection and localization of brain activation in fMRI sequences. Here, we take advantage of the continuously varying order parameter which allows us to fine-tune the localization properties of the basis functions.


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