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

@article{Barnett2019APN, title={A parallel non-uniform fast Fourier transform library based on an "exponential of semicircle" kernel}, author={Alex H. Barnett and Jeremy F. Magland and Ludvig af Klinteberg}, journal={SIAM J. Sci. Comput.}, year={2019}, volume={41}, pages={C479-C504} }

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…

## Figures, Tables, and Topics from this paper

## 49 Citations

Aliasing error of the exp(β√(1-z2)) kernel in the nonuniform fast Fourier transform

- Computer Science, MathematicsArXiv
- 2020

An aliasing error estimate is proved which bounds the error of the one-dimensional NUFFT of types 1 and 2 in exact arithmetic and new connections are drawn between the above kernel, Kaiser–Bessel, and prolate spheroidal wavefunctions of order zero, which all appear to share an optimal exponential convergence rate.

A fast Petrov-Galerkin spectral method for the multi-dimensional Boltzmann equation using mapped Chebyshev functions

- Computer Science, MathematicsArXiv
- 2021

A Petrov-Galerkin spectral method for the Boltzmann equation in the unbounded domain is introduced and is able to construct a fast algorithm with the help of the non-uniform fast Fourier transform (NUFFT).

Fast Ewald summation for electrostatic potentials with arbitrary periodicity.

- Medicine, MathematicsThe Journal of chemical physics
- 2021

It is shown that removing periodic boundary conditions from one or two directions out of three will only moderately increase the total runtime, and in the free-space case, the runtime is around four times that of the triply periodic case.

A high-order integral equation-based solver for the time-dependent Schrodinger equation

- Computer Science, MathematicsArXiv
- 2020

A numerical method is introduced for the solution of the time-dependent Schrodinger equation with a smooth potential, based on its reformulation as a Volterra integral equation, which avoids the need for artificial boundary conditions, admits simple, inexpensive high-order implicit time marching schemes, and naturally includes time- dependent potentials.

Periodic Fast Multipole Method

- Computer Science, MathematicsArXiv
- 2021

The approach extends to the oscillatory equations of mathematical physics, including the Helmholtz and Maxwell equations, but will address these in a companion paper, since the nature of the problem is somewhat different and includes the consideration of quasiperiodic boundary conditions and resonances.

Factorization of the translation kernel for fast rigid image alignment

- Mathematics, Computer Science
- 2019

TheFTK, an optimal interpolation method which represents images in a Fourier--Bessel basis and uses a rank-$H$ approximation of the translation kernel via an operator singular value decomposition (SVD) is proposed, which proves that H = \mathcal{O}(Hn(n + N)$ per image pair.

Numerical reparametrization of periodic planar curves via curvature interpolation

- Mathematics, Computer ScienceArXiv
- 2021

A novel static algorithm is proposed for numerical reparametrization of periodic planar curves. The method identifies a monitor function of the arclength variable with the true curvature of an open…

Optical homogeneity measurement of parallel plates using nonuniform fast Fourier transform based on low-rank approximation and Taylor expansion

- Engineering
- 2021

Abstract. When the optical homogeneity of parallel plates is measured using wavelength phase shifting interferometers, the nonlinearity of phase shifts caused by nonlinear wavelength tuning results…

Efficient Long-Range Convolutions for Point Clouds

- Computer Science, MathematicsArXiv
- 2020

A novel neural network layer is presented that directly incorporates long-range information for a point cloud and leverages the convolutional theorem coupled with the non-uniform Fourier transform, and can be performed in nearly-linear time asymptotically with respect to the number of input points.

Efficient Fourier representations of families of Gaussian processes

- Computer Science, MathematicsArXiv
- 2021

A class of algorithms for constructing Fourier representations of Gaussian processes in 1 dimension that are valid over ranges of hyperparameter values and generalize mathematically to higher dimensions, though they suffer from the standard curse of dimensionality.

## References

SHOWING 1-10 OF 90 REFERENCES

Fast Fourier Transforms for Nonequispaced Data, II

- Mathematics
- 1995

Abstract A group of algorithms generalizing the fast Fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval [-π, π] is presented. The schemes of this paper…

Accelerating the Nonuniform Fast Fourier Transform

- Mathematics, Computer ScienceSIAM Rev.
- 2004

This paper observes 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, of particular value in two- and three- dimensional settings.

gNUFFTW : Auto-Tuning for High-Performance GPU-Accelerated Non-Uniform Fast Fourier Transforms by Teresa

- 2017

Non-uniform sampling of the Fourier transform appears in many important applications such as magnetic resonance imaging (MRI), optics, tomography and radio interferometry. Computing the inverse often…

Optimized Least-Square Nonuniform Fast Fourier Transform

- Mathematics, Computer ScienceIEEE Transactions on Signal Processing
- 2009

A memory efficient approximation to the nonuniform Fourier transform of a support limited sequence is derived based on the theory of shift-invariant representations and an exact expression for the worst-case mean square approximation error is derived.

Nonuniform fast Fourier transforms using min-max interpolation

- Mathematics, Computer ScienceIEEE Trans. Signal Process.
- 2003

This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm and indicates that the proposed method easily generalizes to multidimensional signals.

Short Note: The type 3 nonuniform FFT and its applications

- Mathematics
- 2005

The nonequispaced or nonuniform fast Fourier transform (NUFFT) arises in a variety of application areas, including imaging processing and the numerical solution of partial differential equations. In…

Fast Fourier Transforms for Nonequispaced Data

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 1993

A group of algorithms is presented generalizing the fast Fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval $[ - \pi ,\pi ]$. The schemes of this paper…

Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT

- Computer Science, MedicineOptics express
- 2010

GPU-NUFFT provides an accurate approximation to GPU-NUDFT in terms of image quality, but offers >10 times higher processing speed and improved sensitivity roll-off, higher local signal-to-noise ratio and immunity to side-lobe artifacts caused by the interpolation error.

A Nonuniform Fast Fourier Transform Based on Low Rank Approximation

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2018

A fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT) is proposed, which is essentially the FFT, and is competitive with state-of-the-art algorithms.

Direct inversion of the nonequispaced fast Fourier transform

- MathematicsLinear Algebra and its Applications
- 2019

Various applications such as MRI, solution of PDEs, etc. need to perform an inverse nonequispaced fast Fourier transform (NFFT), i. e., compute $M$ Fourier coefficients from given $N$ nonequispaced…