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Fractional Derivatives, Splines and Tomography

M. Unser, S. Horbelt, T. Blu

Proceedings of the Tenth European Signal Processing Conference (EUSIPCO'00), Tampere, Finland, September 4-8, 2000, vol. IV, pp. 2017-2020.



We develop a spline calculus for dealing with fractional derivatives. After a brief review of fractional splines, we present the main formulas for computing the fractional derivatives of the underlying basis functions. In particular, we show that the γth fractional derivative of a B-spline of degree α (not necessarily integer) is given by the γth fractional difference of a B-spline of degree α-γ. We use these results to derive an improved version the filtered backprojection algorithm for tomographic reconstruction. The projection data is first interpolated with splines; the continuous model is then used explicitly for an exact implementation of the filtering and backprojection steps.


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