In this paper the Cooley-Tukey fast Fourier transform (FFT) algorithm is generalized to the multidimensional case in a natural way which allows for the evaluation of discrete Fourier transforms of rectangularly or hexagonally sampled signals or of signals which are sampled on an arbitrary periodic grid in either the spatial or Fourier domain. This general algorithm incorporates both the traditional rectangular row-column and vector-radix algorithms as special cases. This FFT algorithm is shown to result from the factorization of an integer matrix; for each factorization of that matrix, a different algorithm can be developed. This paper presents the general algorithm, discusses its computational efficiency, and relates it to existing multi-dimensional FFT algorithms.

A unified treatment of Cooley-Tukey algorithms for the evaluation of the multidimensional DFT

In this paper, the prime factor algorithm for the evaluation of a one-dimensional discrete Fourier transform is generalized to the evaluation of multidimensional discrete Fourier transforms defined on arbitrary periodic sampling lattices. It is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete Fourier transform. As a sidelight to the derivation of the algorithm, a Chinese remainder theorem is derived for integer lattices.

Fast algorithms for the multidimensional discrete Fourier transform

The authors present transposition algorithms for matrices that do not fit in main memory. Transposition is interpreted as a permutation of the vector obtained by mapping a matrix to linear memory. Algorithms are derived from factorizations of this permutation, using a class of permutations related to the tensor product. Using this formulation of transposition, the authors first obtain several known algorithms and then they derive a new algorithm which reduces the number of disk accesses required. The new algorithm was compared to existing algorithms using an implementation on the Intel iPSC/860. This comparison shows the benefits of the new algorithm.

Efficient transposition algorithms for large matrices

http://xrm.phys.northwestern.edu/research/pdf_papers/1991/vanheel_ultramic_1991.pdf

A fast algorithm for transposing large multidimensional image data sets

This paper addresses the problem of parallel transposition of large out-of-core arrays. Although algorithms for out-of-core matrix transposition have been widely studied, previously proposed algorithms have sought to minimise the number of I/O operations and the in-memory permutation time. We propose an algorithm that directly targets the improvement of overall transposition time. The I/O characteristics of the system are used to determine the read, write and communication block sizes such that the total execution time is minimised. We also provide a solution to the array redistribution problem for arrays on disk. The solutions to the sequential transposition problem and the parallel array redistribution problem are then combined to obtain an algorithm for the parallel out-of-core transposition problem.

Efficient parallel out-of-core matrix transposition

We consider the problem of performing a two-dimensional fast Fourier transform (FFT) on a very large matrix in limited core memory. We propose a decomposition of the Cooley-Tukey algorithm to allow efficient utilization of core memory and mass storage. The number of input/output operations is greatly reduced, with no increase in the computational burden. The method is suitable for nonsquare matrices and arrays of three or more dimensions.

G. Anderson

Papers

A stepwise approach to computing the multidimensional fast Fourier transform of large arrays