NFSFT

NFSOFT - NFFT on the rotation group

These algorithms calculate the fast Fourier synthesis and its adjoint on the rotation group

for arbitrary sampling sets. They are based on the fast Fourier transform for nonequispaced nodes on the three-dimensional torus. Our algorithms evaluate the

Fourier synthesis and its adjoint, respectively, of

-bandlimited functions at

arbitrary input nodes.

This library of C functions evaluates a function $f \in \mathrm{L}^2(SO(3))$ with finite orthogonal expansion

$\displaystyle f(\alpha,\beta,\gamma)=\sum_{l=0}^{B}\sum_{m=-l}^l\sum_{n=-l}^{l}\widehat f^{m,n}_lD^{m,n}_l(\alpha,\beta,\gamma).$

in terms of Wigner-D functions $D^{m,n}_l$ on a set of arbitary nodes $\left(\alpha_d,\beta_d,\gamma_d\right)$ , $d=1,\ldots,D$ , $D \in \mathbb{N}$ , in Euler angles. Furthermore, the fast evaluation of sums

$\displaystyle \tilde{\widehat f}^{m,n}_l:= \sum_{d=1}^{D} f\left(\alpha_d,\beta_d,\gamma_d\right) \overline{ D^{m,n}_l \left(\alpha_d,\beta_d,\gamma_d\right)}$

for given function values $f\left(\alpha_d,\beta_d,\gamma_d\right) \in \mathbb{C}$ and all indices

The algorithms are implemented by Antje Vollrath in ./kernel/nfsoft. Related papers are

Gräf, M., Potts, D.
Sampling sets and quadrature formulae on the rotation group.
Numer. Funct. Anal. Optim. 30, 665 - 688, (full paper ps, pdf), 2009
Potts, D., Prestin J., Vollrath A.
A Fast algorithm for nonequispaced Fourier transforms on the rotation group.
Numer. Algorithms, 52, 355 - 384, (full paper ps, pdf), 2009
Hielscher, R., Potts, D., Prestin, J., Schaeben, H., Schmalz, M.
The Radon transform on SO(3): A Fourier slice theorem and numerical inversion.
Inverse Problems 24, 025011, (full paper ps, pdf), 2008