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Student Project: Yann Barbotin
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Fast autocorrelation formula for polyharmonic B-splines.

Yann BarbotinSemester project
I&C School, EPFLJanuary 2008




Polyharmonic B-splines are a multidimensional generalization of B-splines with interesting applications such as interpolation and design of orthogonal wavelet basis [1][2].

One of the main issue is the computation of the autocorrelation sequence Aφγ necessary to build the orthogonal and interpolant version of the spline, see Figure (1) for example. Polyharmonic B-splines have infinite support in the frequency domain, and may decay slowly for γ → 1, thus direct computation of the autocorrelation formula yields an important error   O[R^(γ-1)]   considering terms at a distance less or equal to R from the origin.

building the orthogonal polyharmonic B-spline
Figure (1) - How to build an orthogonal polyharmonic B-spline (γ=3) with its autocorrelation sequence.

Some inspiration can be found in cristalography [3], namely computation of potential at the boundaries of cristal structures. Adaptation of these technics, resulted in Proposition 1.
Error after truncation is   O[e^(R^2)].

fast formula

The result is an important gain in speed and accuracy, as illustrated in Figure (2). Moreover performance of this method weakly depend of γ. A 4 pages paper is being written for publication in a signal processing publication.

performance table
Figure (2) - Comparison of time and accuracy performances between the direct formula algorithm (brute force) and the fast formula one (IG).


[1]    D. Van de Ville, T. Blu, M. Unser, "Isotropic Polyharmonic B-splines: Scaling Functions and Wavelets" , IEEE Transactions on Image Processing, 14-11, 2005.
[2]    T. Blu, M. Unser, "Fractional Splines and Wavelets" , SIAM review, 42-1, 43--67, 2000.
[6]    R. Crandall, "Fast evaluation of Epstein zeta functions" , manuscript, 1998. • 11.08.2022