Reproducing Polynomials with B-Splines
A B-Spline of order is known to be able to reproduce any polynomial up to order :
In words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order . This is needed for different applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework. In this case any kernel reproducing polynomials (that is, satisfying the Strang-Fix conditions) can be used. However, among all possible kernels, the B-Splines have the smallest possible support.
An important question is how to obtain the coefficients for the reproduction-formula. In this small article, I describe one way.
the coefficients can be obtained using the dual of ,  (I set for consistency with my notes):
However, even if the dual would be known, solving the infinite integral is only feasible when the dual has finite support. This is the case with the B-Spline itself but not with its dual!
A closer look at the formula tells that this is nothing more than a convolution (under the assumption that is symmetric which is the case):
Now, this can be transformed to fourier domain:
Writing the inverse of this expression yields:
It is known that:
Now the whole procedure has been reduced to calculating the derivative of and set the result to zero.
An open question is how to obtain the dual of . As the reproduction formula spans a vector space, must be at least bi-orthogonal to . This translates in fourier domain to:
The fourier transform of a B-Spline of order is (e.g. ):
The following derivation of the sum is borrowed from . For this derivation to work, I set temporarily:
and because is always even:
Because of the periodicity it is known that
And finally the following relation is used:
in order to finally obtain:
and with :
Therefore, together with this yields:
and finally substituting for :
As this function is not well defined it is better to use the limit:
Examples for a cubic spline
For a cubic spline (N=3) the coefficients are:
- ↑ P.L. Dragotti, M. Vetterli, T.Blu: "Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang-Fix", IEEE Transactions on Signal Processing, vol. 55, No. 5, May 2007
- ↑ http://en.wikipedia.org/wiki/Dirac_delta_function
- ↑ S. Mallat: "A Wavelet Tour of Signal Processing", Academic Press 1999
- ↑ M.Unser: "Splines - A Perfect Fit for Signal and Imaging Processing", IEEE Signal Processing Magazine Nov. 1999
- ↑ M.J.C.S. Reis, P.J.S.G. Ferreira, S.F.S.P. Soares: "Linear combinations of B-splines as generating functions for signal approximation", Elsevier Digital Signal Processing 15, 2005
- ↑ L.V. Ahlfors: "Complex Analysis", McGraw-Hill, 1979
- ↑ I.J. Schoenberg: "Cardinal interpolation and spline functions", J. Approx. Theory volume 2, pp. 167-206, 1969