From dfac6aea53ec6df88bbc4c576deff31a5ee28181 Mon Sep 17 00:00:00 2001
From: Juan Ignacio Polanco <juan-ignacio.polanco@cnrs.fr>
Date: Mon, 11 Dec 2023 17:23:58 +0100
Subject: [PATCH] Add README

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+# NonuniformFFTs.jl
+
+[![Build Status](https://github.com/jipolanco/NonuniformFFTs.jl/workflows/CI/badge.svg)](https://github.com/jipolanco/NonuniformFFTs.jl/actions)
+[![Coverage](https://codecov.io/gh/jipolanco/NonuniformFFTs.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/jipolanco/NonuniformFFTs.jl)
+
+Yet another package for computing multidimensional [non-uniform fast Fourier transforms (NUFFTs)](https://en.wikipedia.org/wiki/NUFFT) in Julia.
+
+## Differences with other packages
+
+This package roughly follows the same notation and conventions of the [FINUFFT library](https://finufft.readthedocs.io/en/latest/)
+and its [Julia interface](https://github.com/ludvigak/FINUFFT.jl), with a few differences detailed below.
+
+For now, parallelism is not supported by this package, but this will come in the near future.
+On a single thread, performance is comparable (and often better) than other libraries, including those mentioned 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`](https://finufft.readthedocs.io/en/latest/opts.html#data-handling-options) 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](http://fftw.org/fftw3_doc/The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html).
+  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`](https://ludvigak.github.io/FINUFFT.jl/latest/#FINUFFT.finufft_makeplan-Tuple{Integer,%20Union{Integer,%20Array{Int64}},%20Integer,%20Integer,%20Real}) 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.
+
+### Differences with [NFFT.jl](https://github.com/JuliaMath/NFFT.jl)
+
+- This package allows changing the non-uniform points associated to a NUFFT plan.
+  In other words, once a plan already exists, computing a NUFFT for a different set of points is efficient and doesn't need to allocate a new plan.
+
+- This package allows NUFFTs of purely real non-uniform data.
+
+- Setting non-uniform points and executing plans is allocation-free.
+
+- 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 allows NUFFTs of purely real non-uniform data.
+  Moreover, transforms can be performed in for an arbitrary number of dimensions.
+
+- A different smoothing kernel function is used (backwards Kaiser–Bessel kernel by default).
+
+- Unlike the FINUFFT.jl interface, this package guarantees zero allocations when setting non-uniform points and executing plans.