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The crystallographic fast Fourier transform. Recursive symmetry reduction

Acta Cryst. (2007). A63, 465-480 [doi:10.1107/S0108767307047411]

[Diagonal mirror symmetry] The fractal-like asymmetric unit for diagonal mirror symmetry. Color saturation depicts depth of recursion.
We present algorithms for non-redundant crystallographic fast Fourier transform (FFT), maximally reducing the computational complexity and memory usage for all 230 symmetry groups. Previously such algorithms have been known only in several cases. The central idea of our approach is to represent a symmetric Fourier transform as a series of transforms on grids with no special points. Such transforms are reduced to P1 FFTs in one step. We provide recipes for recursive decomposition for all symmetry groups and grids.

A. Kudlicki, M. Rowicka and Z. Otwinowski