NonuniformFFTs.jl

Yet another package for computing multidimensional non-uniform fast Fourier transforms (NUFFTs) in Julia.

Like other existing packages, computation of NUFFTs on CPU are parallelised using threads. Transforms can also be performed on GPUs. In principle all kinds of GPU for which a KernelAbstractions.jl backend exists are supported.

Installation

NonuniformFFTs.jl can be simply installed from the Julia REPL with:

julia> ] add NonuniformFFTs

Conventions

Transform definitions

This package evaluates type-1 (non-uniform to uniform) and type-2 (uniform to non-uniform) non-uniform fast Fourier transforms (NUFFTs). These are sometimes also called the adjoint and direct NUFFTs, respectively.

In one dimension, the type-1 NUFFT computed by this package is defined as follows:

\[û(k) = ∑_{j = 1}^{M} v_j \, e^{-i k x_j} \quad \text{ for } \quad k = -\frac{N}{2}, …, \frac{N}{2} - 1\]

where the $x_j ∈ [0, 2π)$ are the non-uniform points and the $v_j$ are the input values at those points, and $k$ are the associated Fourier wavenumbers (or frequencies). Here $M$ is the number of non-uniform points, and $N$ is the number of Fourier modes that are kept (taken to be even here, but can also be odd).

Similarly, the type-2 NUFFT is defined as:

\[v_j = ∑_{k = -N/2}^{N/2 + 1} û(k) \, e^{+i k x_j}\]

for $x_j ∈ [0, 2π)$. The type-2 transform can be interpreted as an interpolation of a Fourier series onto a given location.

If the points are uniformly distributed in $[0, 2π)$, i.e. $x_j = 2π (j - 1) / M$, then these definitions exactly correspond to the forward and backward DFTs computed by FFTW.

Ordering of data in frequency space

This package follows the FFTW convention of storing frequency-space data starting from the non-negative frequencies $(k = 0, 1, …, N/2 - 1)$, followed by the negative frequencies $(k = -N/2, ..., -2, -1)$. Note that this package also allows the non-uniform data ($v_j$ values) to be purely real, in which case real-to-complex FFTs are performed and only the non-negative wavenumbers are kept (in one dimension).

One can use the fftfreq function from the AbstractFFTs package to conveniently obtain the Fourier frequencies in the right order. For real data transforms, rfftfreq should be used instead.

For complex non-uniform data, one can use fftshift and ifftshift from the same package to switch between this convention and the more "natural" convention of storing frequencies in increasing order $(k = -N/2, …, N/2 - 1)$.

Alternatively, one can pass fftshift = true to the PlanNUFFT constructor to reorder Fourier modes in increasing order of frequencies ("natural" order).

Differences with other packages

This package roughly follows the same notation and conventions of the FINUFFT library and its Julia interface, with a few differences detailed below.

Conventions used by this package

We try to preserve as much as possible the conventions used in FFTW3. In particular, this means that:

The FFT outputs are ordered starting from mode $k = 0$ to $k = N/2 - 1$ (for even $N$) and then from $-N/2$ to $-1$. Wavenumbers can be obtained in this order by calling AbstractFFTs.fftfreq(N, N). Use AbstractFFTs.fftshift to get Fourier modes in increasing order $-N/2, …, -1, 0, 1, …, N/2 - 1$. In FINUFFT, one should set modeord = 1 to get this order.
The type-1 NUFFT (non-uniform to uniform) is defined with a minus sign in the exponential. This is the same convention as the forward DFT in FFTW3. In particular, this means that performing a type-1 NUFFT on uniform points gives the same output than performing a FFT using FFTW3. In FINUFFT, this corresponds to setting iflag = -1 in type-1 transforms. Conversely, type-2 NUFFTs (uniform to non-uniform) are defined with a plus sign, equivalently to the backward DFT in FFTW3.

For compatibility with other packages such as NFFT.jl, these conventions are not applied when the AbstractNFFTs.jl interface is used. In this specific case, modes are assumed to be ordered in increasing order, and the opposite sign convention is used for Fourier transforms.

Differences with NFFT.jl

This package allows NUFFTs of purely real non-uniform data.
Different convention is used: non-uniform points are expected to be in $[0, 2π]$.

Differences with FINUFFT / cuFINUFFT / FINUFFT.jl

This package is written in "pure" Julia (besides the FFTs themselves which rely on the FFTW3 library, via their Julia interface).
This package provides a generic and efficient GPU implementation thanks to KernelAbstractions.jl meaning that many kinds of GPUs are supported, including not only Nvidia GPUs but also AMD ones and possibly more.
This package allows NUFFTs of purely real non-uniform data. Moreover, transforms can be performed on arbitrary number of dimensions.
A different smoothing kernel function is used (backwards Kaiser–Bessel kernel by default on CPUs; Kaiser–Bessel kernel on GPUs).
It is possible to use the same plan for type-1 and type-2 transforms, reducing memory requirements in cases where one wants to perform both.

Bibliography

Shih, Y.-h.; Wright, G.; Andén, J.; Blaschke, J. and Barnett, A. H. (2021), cuFINUFFT: a load-balanced GPU library for general-purpose nonuniform FFTs, arXiv:2102.08463 [cs.DC].