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.

Basic usage

Type-1 (or adjoint) NUFFT in one dimension

using NonuniformFFTs

N = 256   # number of Fourier modes
Np = 100  # number of non-uniform points

# Generate some non-uniform random data
T = Float64                # non-uniform data is real (can also be complex)
xp = rand(T, Np) .* T(2π)  # non-uniform points in [0, 2π]
vp = randn(T, Np)          # random values at points

# Create plan for data of type T
plan_nufft = PlanNUFFT(T, N; m = HalfSupport(4))  # larger support increases accuracy

# Set non-uniform points
set_points!(plan_nufft, xp)

# Perform type-1 NUFFT on preallocated output
ûs = Array{Complex{T}}(undef, size(plan_nufft))
exec_type1!(ûs, plan_nufft, vp)

Type-2 (or direct) NUFFT in one dimension

using NonuniformFFTs

N = 256   # number of Fourier modes
Np = 100  # number of non-uniform points

# Generate some uniform random data
T = Float64                        # non-uniform data is real (can also be complex)
xp = rand(T, Np) .* T(2π)          # non-uniform points in [0, 2π]
ûs = randn(Complex{T}, N ÷ 2 + 1)  # random values at points (we need to store roughly half the Fourier modes for complex-to-real transform)

# Create plan for data of type T
plan_nufft = PlanNUFFT(T, N; m = HalfSupport(4))

# Set non-uniform points
set_points!(plan_nufft, xp)

# Perform type-2 NUFFT on preallocated output
vp = Array{T}(undef, Np)
exec_type2!(vp, plan_nufft, ûs)

More examples

Multidimensional transforms

using NonuniformFFTs
using StaticArrays: SVector  # for convenience

Ns = (256, 256)  # number of Fourier modes in each direction
Np = 1000        # number of non-uniform points

# Generate some non-uniform random data
T = Float64                                      # non-uniform data is real (can also be complex)
d = length(Ns)                                   # number of dimensions (d = 2 here)
xp = [T(2π) * rand(SVector{d, T}) for _ ∈ 1:Np]  # non-uniform points in [0, 2π]ᵈ
vp = randn(T, Np)                                # random values at points

# Create plan for data of type T
plan_nufft = PlanNUFFT(T, Ns; m = HalfSupport(4))

# Set non-uniform points
set_points!(plan_nufft, xp)

# Perform type-1 NUFFT on preallocated output
ûs = Array{Complex{T}}(undef, size(plan_nufft))
exec_type1!(ûs, plan_nufft, vp)

# Perform type-2 NUFFT on preallocated output
exec_type2!(vp, plan_nufft, ûs)

Multiple transforms on the same non-uniform points

using NonuniformFFTs

N = 256   # number of Fourier modes
Np = 100  # number of non-uniform points
ntrans = Val(3)  # number of simultaneous transforms

# Generate some non-uniform random data
T = Float64                # non-uniform data is real (can also be complex)
xp = rand(T, Np) .* T(2π)  # non-uniform points in [0, 2π]
vp = ntuple(_ -> randn(T, Np), ntrans)  # random values at points (one vector per transformed quantity)

# Create plan for data of type T
plan_nufft = PlanNUFFT(T, N; ntransforms = ntrans)

# Set non-uniform points
set_points!(plan_nufft, xp)

# Perform type-1 NUFFT on preallocated output (one array per transformed quantity)
ûs = ntuple(_ -> Array{Complex{T}}(undef, size(plan_nufft)), ntrans)
exec_type1!(ûs, plan_nufft, vp)

# Perform type-2 NUFFT on preallocated output (one vector per transformed quantity)
vp_interp = map(similar, vp)
exec_type2!(vp, plan_nufft, ûs)

Transforms on the GPU

Below is a GPU version of the multidimensional transform example above. The only differences are:

• we import CUDA.jl and Adapt.jl (optional)
• we pass backend = CUDABackend() to PlanNUFFT (CUDABackend is a KernelAbstractions backend and is exported by CUDA.jl). The default is backend = CPU().
• we copy input arrays to the GPU before calling any NUFFT-related functions (set_points!, exec_type1!, exec_type2!)

The example is for an Nvidia GPU (using CUDA.jl), but should also work with e.g. AMDGPU.jl by simply choosing backend = ROCBackend().

using NonuniformFFTs
using StaticArrays: SVector  # for convenience
using CUDA
using Adapt: adapt  # optional (see below)

backend = CUDABackend()  # other options are CPU() or ROCBackend() (untested)

Ns = (256, 256)  # number of Fourier modes in each direction
Np = 1000        # number of non-uniform points

# Generate some non-uniform random data
T = Float64                                          # non-uniform data is real (can also be complex)
d = length(Ns)                                       # number of dimensions (d = 2 here)
xp_cpu = [T(2π) * rand(SVector{d, T}) for _ ∈ 1:Np]  # non-uniform points in [0, 2π]ᵈ
vp_cpu = randn(T, Np)                                # random values at points

# Copy data to the GPU (using Adapt is optional but it makes code more generic).
# Note that all data needs to be on the GPU before setting points or executing transforms.
# We could have also generated the data directly on the GPU.
xp = adapt(backend, xp_cpu)  # returns a CuArray if backend = CUDABackend
vp = adapt(backend, vp_cpu)

# Create plan for data of type T
plan_nufft = PlanNUFFT(T, Ns; m = HalfSupport(4), backend)

# Set non-uniform points
set_points!(plan_nufft, xp)

# Perform type-1 NUFFT on preallocated output
ûs = similar(vp, Complex{T}, size(plan_nufft))  # initialises a GPU array for the output
exec_type1!(ûs, plan_nufft, vp)

# Perform type-2 NUFFT on preallocated output
exec_type2!(vp, plan_nufft, ûs)

Using the AbstractNFFTs.jl interface

This package also implements the AbstractNFFTs.jl interface as an alternative API for constructing plans and evaluating transforms. This can be useful for comparing with similar packages such as NFFT.jl.

using NonuniformFFTs
using AbstractNFFTs: AbstractNFFTs, plan_nfft
using LinearAlgebra: mul!

Ns = (256, 256)  # number of Fourier modes in each direction
Np = 1000        # number of non-uniform points

# Generate some non-uniform random data
T = Float64                      # must be a real data type (Float32, Float64)
d = length(Ns)                   # number of dimensions (d = 2 here)
xp = rand(T, (d, Np)) .- T(0.5)  # non-uniform points in [-1/2, 1/2)ᵈ; must be given as a (d, Np) matrix
vp = randn(Complex{T}, Np)       # random values at points (must be complex)

# Create plan for data of type Complex{T}. Note that we pass the points `xp` as
# a first argument, which calls an AbstractNFFTs-compatible constructor.
p = NonuniformFFTs.NFFTPlan(xp, Ns)
# p = plan_nfft(xp, Ns)  # this is also possible

# Getting the expected dimensions of input and output data.
AbstractNFFTs.size_in(p)   # (256, 256)
AbstractNFFTs.size_out(p)  # (1000,)

# Perform adjoint NFFT, a.k.a. type-1 NUFFT (non-uniform to uniform)
us = adjoint(p) * vp      # allocates output array `us`
mul!(us, adjoint(p), vp)  # uses preallocated output array `us`

# Perform forward NFFT, a.k.a. type-2 NUFFT (uniform to non-uniform)
wp = p * us
mul!(wp, p, us)

# Setting a different set of non-uniform points
AbstractNFFTs.nodes!(p, xp)

Note: the AbstractNFFTs.jl interface currently only supports complex-valued non-uniform data. For real-to-complex transforms, the NonuniformFFTs.jl API demonstrated above should be used instead.

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 (see example above). 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 / 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 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.

Performance benchmarks

NonuniformFFTs.jl can be fast:

On the CPU, for complex non-uniform data, it displays comparable performance to the FINUFFT library written in C++. On the GPU it can be faster than the CUDA-based CuFINUFFT, especially for large problem sizes.

And the results above are for complex non-uniform data. One can obtain important additional gains if one is interested in problems where non-uniform data is real-valued.

See the Performance benchmarks section of the documentation for details and more benchmarks.