Accelerating the Nonuniform Fast Fourier Transform

Abstract

The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in $O(N \log N)$ operations rather than $O(N^2)$ operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as {\em nonuniform FFTs} (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid [A. Dutt and V. Rokhlin, {\em SIAM J.\ Sci.\ Comput.}, 14 (1993), pp. 1368--1383].\\ \indent% In this paper, we observe that one of the standard interpolation or ``gridding" schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two- and three-dimensional settings, saving either $10^d N$ in storage in d dimensions or a factor of about 5--10 in CPU time (independent of dimension).

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