Group Invariant Fourier Transform Algorinthms

Abstract

Publisher Summary The design of algorithms for computing the crystallographic Fourier transform is a subject in applied group theory. This chapter presents a discussion on group invariant Fourier transform algorithms. Finite abelian groups serve as data indexing sets. A class of affine group fast Fourier transform (FFT) algorithms is introduced, which fully use data invariance with respect to subgroups of the affine group of data indexing sets. The chapter reviews all the necessary group theory. The affine group of a finite abelian group is defined. Constructs related to the action of affine subgroups on data indexing sets are introduced in the chapter. The chapter defines the Fourier transform of an abelian group and discussed its fundamental role in interchanging periodization and decimation operations (duality). The reduced transform (RT), Cooley–Tukey algorithm (CT), FFT, and Good–Thomas (GT) algorithms are presented as applications of this duality to different global decomposition strategies. Affine group FFT algorithms based on the RT algorithm are discussed, while those coming from the application of the affine group CT, FFT are introduced. The chapter describes a method of incorporating one-dimensional (1D) symmetry into FFT computations, which calls on lower order existing FFT routines using the symmetry condition. The chapter presents many examples to reflect both the theory and experience and others, over several years in writing code for the three-dimensional (3D) crystallographic FT.