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Fractional B-splines

Module by: Alena Scott

Summary: Fractional B-splines are extention of B-splines.

For α>-1 α -1 , B-splines can be extended from having order nn, a positive integer, to a family indexed by a fractional parameter. These generalized B-splines are called fractional B-splines.

We will assume equally spaced knots with unit spacing. Remember that B-splines are constructed from the ( n+1 n 1 ) fold convolution of β 0 x β 0 x ; they are a basis for polynomial splines. We will use the uncentered version of B-splines: β + 0 x=1ifx∈010.5ifx=0∨10otherwise β + 0 x 1 x 0 1 0.5 x 0 1 0 β + 0 x= β + , 1 0 * β + , 2 0 *…* β + , n 0 * β + , n + 1 0 x β + 0 x β + , 1 0 β + , 2 0 … β + , n 0 β + , n + 1 0 x

A B-spline of order nn has the form

β + n x=1n!∑k=0n+1n+1k-1kx-kn β + n x 1 n k 0 n 1 n 1 k 1 k x k n (1)
and Fourier transform
β ^ n ω=1-ⅇ-ⅈωⅈωn+1 β ^ n ω 1 ω ω n 1 (2)

A fractional B-spline of order αα has the form

β + α x=1Γα+1∑k=0∞α+1k-1kx-kα β + α x 1 Γ α 1 k 0 α 1 k 1 k x k α (3)
and Fourier transform
β ^ + α x=1-ⅇ-ⅈωⅈωα+1 β ^ + α x 1 ω ω α 1 (4)
One can see that the two definitions are very similar. However, since the formula for computing binomial coeffients only allows for integers, we must substitute generalized binomials.

These fractional B-splines interpolate conventional B-splines, which are fractional B-splines of integer values. For a picture of this, see a demonstration on Michael Unser's webpage. Michael Unser and Theirry Blu developed fractional splines.

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