Fractional Derivatives, Splines and Tomography

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 $γ ^{ th }$ fractional derivative of a B-spline of degree α (not necessarily integer) is given by the $γ ^{ th }$ 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.