# tknopp/NFFT.jl

Julia implementation of the Non-equidistant Fast Fourier Transform (NFFT)
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# NFFT.jl

This package provides a Julia implementation of the Non-equidistant Fast Fourier Transform (NFFT). This algorithm is also referred as Gridding in the literature (e.g. in MRI literature). For a detailed introduction into the NFFT and its application please have a look at www.nfft.org.

The NFFT is a fast implementation of the Non-equidistant Discrete Fourier Transform (NDFT) that is basically a DFT with non-equidistant sampling nodes in either Fourier or time/space domain. In contrast to the FFT, the NFFT is an approximative algorithm whereas the accuracy can be controlled by two parameters: the window width m and the oversampling factor sigma.

In Julia, run

## Basic usage

Basic usage of NFFT.jl is shown in the following example for 1D:

using NFFT

M, N = 1024, 512
x = range(-0.4, stop=0.4, length=M)  # nodes at which the NFFT is evaluated
fHat = randn(ComplexF64,M)           # data to be transformed
p = NFFTPlan(x, N)                   # create plan. m and sigma are optional parameters
g = nfft(p, f)                       # calculate forward NFFT

In 2D:

M, N = 1024, 16
x = rand(2, M) .- 0.5
fHat = randn(ComplexF64,M)
p = NFFTPlan(x, (N,N))
g = nfft(p, f)

### Directional NFFT

There are special methods for computing 1D NFFT's for each 1D slice along a particular dimension of a higher dimensional array.

M = 11
y = rand(M) .- 0.5
N = (16,20)
P1 = NFFTPlan(y, 1, N)
f = randn(ComplexF64,N)
fHat = nfft(P1, f)

Here size(f) = (16,20) and size(fHat) = (11,20) since we compute an NFFT along the first dimension. To compute the NFFT along the second dimension

P2 = NFFTPlan(y, 2, N)
fHat = nfft(P2, f)

Now size(fHat) = (16,11).