A parallel non-uniform fast Fourier transform library based on an "exponential of semicircle" kernel

TL;DR

This paper introduces FINUFFT, a parallel library for nonuniform Fourier transforms using a novel 'exponential of semicircle' kernel, offering high efficiency, minimal memory use, and no precomputation, suitable for various applications.

Contribution

The paper presents a new kernel-based approach for NUFFT, with an efficient parallel implementation that improves speed and memory efficiency over existing libraries.

Findings

Favorable speed compared to existing libraries

Low memory footprint, especially in 3D

Achieves exponential error bounds similar to Kaiser--Bessel kernel

Abstract

The nonuniform fast Fourier transform (NUFFT) generalizes the FFT to off-grid data. Its many applications include image reconstruction, data analysis, and the numerical solution of differential equations. We present FINUFFT, an efficient parallel library for type 1 (nonuiform to uniform), type 2 (uniform to nonuniform), or type 3 (nonuniform to nonuniform) transforms, in dimensions 1, 2, or 3. It uses minimal RAM, requires no precomputation or plan steps, and has a simple interface to several languages. We perform the expensive spreading/interpolation between nonuniform points and the fine grid via a simple new kernel---the `exponential of semicircle' $e^{\beta \sqrt{1-x^2}}$ in $x\in[-1,1]$---in a cache-aware load-balanced multithreaded implementation. The deconvolution step requires the Fourier transform of the kernel, for which we propose efficient numerical quadrature. For types 1 and 2, rigorous error bounds asymptotic in the kernel width approach the fastest known exponential rate, namely that of the Kaiser--Bessel kernel. We benchmark against several popular CPU-based libraries, showing favorable speed and memory footprint, especially in three dimensions when high accuracy and/or clustered point distributions are desired.