## Accelerating the nonuniform Fast Fourier Transform (2004)

Venue: | SIAM REVIEW |

Citations: | 35 - 2 self |

### BibTeX

@ARTICLE{Greengard04acceleratingthe,

author = {Leslie Greengard and June-yub Lee},

title = {Accelerating the nonuniform Fast Fourier Transform},

journal = {SIAM REVIEW},

year = {2004},

volume = {46},

number = {3},

pages = {443--454}

}

### Years of Citing Articles

### OpenURL

### Abstract

The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N) operations rather than O(N 2) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid [A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1383]. In this paper, we observe that one of the standard interpolation or “gridding ” schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two- and threedimensional settings, saving either 10dN in storage in d dimensions or a factor of about 5–10 in CPUtime (independent of dimension).

### Citations

145 |
Fast Fourier transforms for nonequispaced data
- Dutt, Rokhlin
- 1993
(Show Context)
Citation Context ...g some interpolation scheme with the standard FFT. Oddly enough, it was a number of years after their use in applications before a rigorous analysis of such schemes was introduced by Dutt and Rokhlin =-=[5]-=-. Subsequent papers, such as [1, 3, 9, 10], described variants based on alternative interpolation/approximation approaches. Before discussing the algorithm itself, we would like to comment briefly on ... |

112 | Fast fourier transforms for nonequispaced data: A tutorial
- Potts, Steidl, et al.
- 2001
(Show Context)
Citation Context ...cribe an extremely simple and efficient implementation of the nonuniform fast Fourier transform (NUFFT). There are a host of applications of such algorithms, and we refer the reader to the references =-=[2, 6, 8, 11, 13, 14, 17]-=- for examples. We restrict our attention here to one: function (or image) reconstruction from Fourier data as discussed in [6, 8, 11, 14]. Let us begin, however, with a more precise description of the... |

96 |
On the fast Fourier transform of functions with singularities
- Beylkin
- 1995
(Show Context)
Citation Context ... the standard FFT. Oddly enough, it was a number of years after their use in applications before a rigorous analysis of such schemes was introduced by Dutt and Rokhlin [5]. Subsequent papers, such as =-=[1, 3, 9, 10]-=-, described variants based on alternative interpolation/approximation approaches. Before discussing the algorithm itself, we would like to comment briefly on applications that involve evaluation of (3... |

85 | Selection of a convolution function for Fourier inversion using gridding
- Jackson, Meyer, et al.
- 1991
(Show Context)
Citation Context ...cribe an extremely simple and efficient implementation of the nonuniform fast Fourier transform (NUFFT). There are a host of applications of such algorithms, and we refer the reader to the references =-=[2, 6, 8, 11, 13, 14, 17]-=- for examples. We restrict our attention here to one: function (or image) reconstruction from Fourier data as discussed in [6, 8, 11, 14]. Let us begin, however, with a more precise description of the... |

84 | Nonuniform fast Fourier transforms using min-max interpolation
- Fessler, Sutton
- 2003
(Show Context)
Citation Context ...cribe an extremely simple and efficient implementation of the nonuniform fast Fourier transform (NUFFT). There are a host of applications of such algorithms, and we refer the reader to the references =-=[2, 6, 8, 11, 13, 14, 17]-=- for examples. We restrict our attention here to one: function (or image) reconstruction from Fourier data as discussed in [6, 8, 11, 14]. Let us begin, however, with a more precise description of the... |

50 |
A fast sinc function gridding algorithm for Fourier invenion in computer tomogrnphy. lEEE Tmns. Med. Imaging hCI-4
- O'Sullivan
(Show Context)
Citation Context |

46 | Fast approximate Fourier transforms for irregularly spaced data
- Ware
- 1998
(Show Context)
Citation Context |

35 |
Rapid computation of the discrete Fourier transform
- Anderson, Dahleh
- 1996
(Show Context)
Citation Context ... the standard FFT. Oddly enough, it was a number of years after their use in applications before a rigorous analysis of such schemes was introduced by Dutt and Rokhlin [5]. Subsequent papers, such as =-=[1, 3, 9, 10]-=-, described variants based on alternative interpolation/approximation approaches. Before discussing the algorithm itself, we would like to comment briefly on applications that involve evaluation of (3... |

33 |
The regular Fourier matrices and nonuniform fast Fourier transforms
- Nguyen, Liu
- 1999
(Show Context)
Citation Context ... the standard FFT. Oddly enough, it was a number of years after their use in applications before a rigorous analysis of such schemes was introduced by Dutt and Rokhlin [5]. Subsequent papers, such as =-=[1, 3, 9, 10]-=-, described variants based on alternative interpolation/approximation approaches. Before discussing the algorithm itself, we would like to comment briefly on applications that involve evaluation of (3... |

30 |
The nonuniform discrete Fourier transform and its applications in filter design
- Bagchi, Mitra
- 1996
(Show Context)
Citation Context |

18 | Fast potential theory. II. Layer potentials and discrete sums
- Strain
- 1991
(Show Context)
Citation Context ...entrated on the one-dimensional case, higher dimensional versions have been considered by a variety of authors [3, 6, 14]. The first rigorous two-dimensional version can be found in a paper by Strain =-=[15]-=-, which uses the NUFFT to solve a class of elliptic partial differential equations. Remark 1. Not all schemes for reconstructing Fourier integrals of the type (3) can be represented formally as a quad... |

14 |
An Accurate Algorithm for Nonuniform Fast Fourier Transforms (NUFFT’s
- Liu, Nguyen
- 1998
(Show Context)
Citation Context |

12 |
Direct reconstruction of non-cartesian k-space data using a nonuniform fast Fourier transform
- Sarty, Bennett, et al.
- 2001
(Show Context)
Citation Context |

10 | Spectral approximation of the free-space heat kernel - Greengard, Lin |

7 |
Reconstruction of MRI images from non-uniform sampling and its application to intrascan motion correction
- Bourgeois, Wajer, et al.
- 2001
(Show Context)
Citation Context ...used for image reconstruction. For a variety of technical reasons, however, nonuniform data sampling techniques are much better suited for fast data acquisition, motion correction, and functional MRI =-=[4]-=-. In this example, we create simulated MRI452 LESLIE GREENGARD AND JUNE-YUB LEE data by using a type-2 transformation in two dimensions: F (s k x,s k y)= ∑ ∑ (17) f(j1,j2) e −i(j1,j2)·(sk x ,sk y ) ,... |

7 | Interpolation and Fourier transformation of fringe visibilities - Thompson, Bracewell - 1974 |

1 |
Multidimensional Fourier transformation in magnetic resonance imaging, in The Fourier Transformation
- Pike
- 1998
(Show Context)
Citation Context ...tor of 21 from gridding and a factor of 4 from the oversampled FFT. Example 4 (MRIImage Reconstruction). One of the important applications of the nonuniform FFT is to magnetic resonance imaging (MRI) =-=[6, 8, 11, 12, 13, 14]-=-. The MRI hardware is able to acquire the Fourier transform of a particular tissue property at selected points in the frequency domain. In most clinical systems, the device is designed to acquire data... |