A sparse data fast Fourier transform (SDFFT) - Algorithm and implementation

Abstract

An algorithm that efficiently Fourier transforms sparse spatial data to sparse spectral data with controllable error is presented. Unlike the ordinary nonuniform last Fourier transform (NUFFT), which becomes O(N 2) for sparse k-space and sparse k-space data, the sparse data fast Fourier transform (SDFFT) presented herein decreases the cost to O(N log N) while preserving the O(N log N) memory complexity. The algorithm can be readily employed in general signal processing applications where only part of the k-space is to be computed - regardless of whether it is a regular region like an angular section of the Ewald's sphere or it consists completely of arbitrary points. Among its applications in electromagnetics are back-projection tomography, diffraction tomography, synthetic aperture radar imaging, and the computation of far field patterns due to general aperture antennas and antenna arrays.