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Lightweight non-uniform Fast Fourier Transform in Python

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nfft package

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The nfft package is a lightweight implementation of the non-equispaced fast Fourier transform (NFFT), implemented via numpy and scipy and released under the MIT license. For information about the NFFT algorithm, see the paper Using NFFT 3 – a software library for various nonequispaced fast Fourier transforms.

The nfft package achieves comparable performance to the C package described in that paper, without any customized compiled code. Rather, it makes use of the computational building blocks available in NumPy and SciPy. For a discussion of the algorithm and this implementation, see the Implementation Walkthrough notebook.

About

The nfft package implements one-dimensional versions of the forward and adjoint non-equispaced fast Fourier transforms;

The forward transform:

$f_j = \sum_{k=-N/2}^{N/2-1} \hat{f}_k e^{-2\pi i k x_j}$

And the adjoint transform:

$\hat{f}_k = \sum_{j=0}^{M-1} f_j e^{2\pi i k x_j}$

In both cases, the wavenumbers k are on a regular grid from -N/2 to N/2, while the data values x_j are irregularly spaced between -1/2 and 1/2.

The direct and fast version of these algorithms are implemented in the following functions:

  • nfft.ndft: direct forward non-equispaced Fourier transform
  • nfft.nfft: fast forward non-equispaced Fourier transform
  • nfft.ndft_adjoint: direct adjoint non-equispaced Fourier transform
  • nfft.nfft_adjoint: fast adjoint non-equispaced Fourier transform

Computational complexity

The direct version of each transform has a computational complexity of approximately O[NM], while the NFFT has a computational complexity of approximately O[N log(N) + M log(1/ϵ)], where ϵ is the desired precision of the result. In the current implementation, memory requirements scale as approximately O[N + M log(1/ϵ)].

Comparison to pynfft

Another option for computing the NFFT in Python is to use the pynfft package, which provides a Python wrapper to the C library referenced in the above paper. The advantage of pynfft is that, compared to nfft, it provides a more complete set of routines, including multi-dimensional NFFTs, several related extensions, and a range of computing strategies.

The disadvantage is that pynfft is GPL-licensed (and thus can't be used in much of the more permissively licensed Python scientific world), and has a much more complicated set of dependencies.

Performance-wise, nfft and pynfft are comparable, with the implementation within nfft package being up to a factor of 2 faster in most cases of interest (see Benchmarks.ipynb for some simple benchmarks).

If you're curious about the implementation and how nfft attains such performance without a custom compiled extension, see the Implementation Walkthrough notebook.

Basic Usage

import numpy as np
from nfft import nfft

# define evaluation points
x = -0.5 + np.random.rand(1000)

# define Fourier coefficients
N = 10000
k = - N // 2 + np.arange(N)
f_k = np.random.randn(N)

# non-equispaced fast Fourier transform
f = nfft(x, f_k)

For some more examples, see the notebooks in the notebooks directory.

Installation

The nfft package can be installed directly from the Python Package Index:

$ pip install nfft

Dependencies are numpy, scipy, and pytest, and the package is tested in Python versions 2.7. 3.5, and 3.6.

Testing

Unit tests can be run using pytest:

$ pytest --pyargs nfft

License

This code is released under the MIT License. For more information, see the Open Source Initiative

Support

Development of this package is supported by the UW eScience Institute, with funding from the Gordon & Betty Moore Foundation, the Alfred P. Sloan Foundation, and the Washington Research Foundation

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