NFFT is a software library, written in C, for computing non-equispaced fast Fourier transforms and related variations. It implements the following transforms:

1. Non-equispaced fast Fourier transform (NFFT)

   • forward transform (NFFT), i.e. frequency to time/space domain
   • adjoint transform (adjoint NFFT), i.e. time/space to frequency domain

2. Generalisations

   • to arbitrary nodes in time and frequency domain (NNFFT)
   • to real-valued data, i.e. (co)sine transforms, (NFCT, NFST)
   • to the sphere S^2 (NFSFT)
   • to the rotation group (NFSOFT)
   • to the hyperbolic cross (NSFFT)

3. Generalised inverse transformations based on iterative methods, e.g. CGNR/CGNE

Some examples for application of these transforms are provided:

1. Medical imaging

   • magnetic resonance imaging (mri)
   • computerised tomography (radon)

2. Summation schemes

   • fast summation (fastsum)
   • fast Gauss transform (FGT)
   • singular kernels
   • zonal kernels

3. polar FFT, discrete Radon transform, ridgelet transform

Detailed API documentation in HTML format can be found in doc/html/index.html, if you are working from a release tarball. When working from a source repository, the documentation can be generated with Doxygen (which requires the doxygen-latex and perl packages):

make doc

The NFFT depends on the FFTW library, which is available for many Linux distros, Homebrew on macOS and MSYS2 on Windows. If you compile the FFTW yourself, it should be configured with the flag --enable-shared (and --enable-threads for the multi-threaded version). Building the NFFT requires make and a C compiler such as gcc.

When working from a source repository, you need to run libtoolize and autoreconf first. A bash script to do this is provided. This step requries the tools autoconf, automake and libtool.

./bootstrap.sh

The rest of the build process is standard.

./configure --enable-all --enable-openmp [add options as necessary, see below]

Alternatively, you might run the configure script for Matlab.

./configure --enable-all --enable-openmp --with-matlab=/path/to/matlab

Here are some useful optional flags for ./configure:

• --enable-all specifies that all modules should be compiled,
• --enable-openmp enables the multicore support and
• --enable-julia specifies that the julia interface will be compiled.
• --with-matlab=/path/to/matlab specifies the path of a Matlab installation, and
• --with-octave=/path/to/octave does the same for GNU Octave.
• For a list of all available options, run ./configure --help.

Build the software.

make

Optionally, unit tests may be run. Some of the unit tests require an installation of cunit.

make check

Optionally, install NFFT on your system.

make install

The current general paper, the one that we recommend if you wish to cite NFFT, is Keiner, J., Kunis, S., and Potts, D. ''Using NFFT 3 - a software library for various nonequispaced fast Fourier transforms'' ACM Trans. Math. Software 36, Article 19, 1-30, 2009. BibTeX entry:

@article{KeKuPo09,
 author = {Jens Keiner and Stefan Kunis and Daniel Potts},
 title = {Using {NFFT3} - a Software Library for Various Nonequispaced Fast {Fourier} Transforms},
 journal = {{ACM} Trans. Math. Software},
 year = {2009},
 volume = {36},
 pages = {Article 19, 1--30},
 doi = {10.1145/1555386.1555388}}

Your comments are welcome! This is the third version of the library and may not be as robust or well documented as it should be. Please keep track of bugs or missing/confusing instructions and report them in our issue tracker or directly to Daniel Potts. The postal address is

  Prof. Dr. Daniel Potts
  TU Chemnitz, Fakultaet fuer Mathematik
  Reichenhainer Str. 39
  09107 Chemnitz
  GERMANY

Alternatively, you might contact Stefan Kunis or Jens Keiner.

If you find NFFT useful, we would be delighted to hear about what application you are using NFFT for!

Copyright (c) 2002, 2017 Jens Keiner, Stefan Kunis, Daniel Potts

This software was written by Jens Keiner, Stefan Kunis and Daniel Potts. It was developed at the Mathematical Institute, University of Luebeck, and at the Faculty of Mathematics, Chemnitz University of Technology.

NFFT3 is free software. You can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. If not stated otherwise, this applies to all files contained in this package and its sub-directories.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA