Efficient Calculations of 3-D FFTs on Spiral Contours

This paper proposes a fast algorithm, called the SpiralFFT, that computes samples of the 3-D discrete Fourier transform of an object of interest along spiral contours in frequency space. This type of sampling geometry is prevalent in 3-D magnetic resonance imaging, as spiral sampling patterns allow for rapid, uninterrupted scanning over a large range of frequencies. We show that parameterizing the spiral contours in a certain way allows us to decompose the computation into a series of 1-D transforms, meaning that the 3-D transform is effectively separable, while still yielding spiral sampling patterns that are geometrically faithful and provide dense coverage of 3-D frequency space. We present a number of simulations which demonstrate that the SpiralFFT compares favorably to a state-of-the-art algorithm for computing general non-uniform discrete Fourier transforms.