## Accurate and Fast Discrete Polar Fourier Transform (2003)

Venue: | in Proc. 37th Asilomar Conf. Signals, Systems & Computers |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Averbuch03accurateand,

author = {A. Averbuch and R. R. Coifman and D.L. Donoho and M. Elad and M. Israeli},

title = {Accurate and Fast Discrete Polar Fourier Transform},

booktitle = {in Proc. 37th Asilomar Conf. Signals, Systems & Computers},

year = {2003},

pages = {1933--1937}

}

### OpenURL

### Abstract

In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N N , the proposed algorithm's complexity is O(N log N), just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1-D equispaced FFT's and 1D interpolations. A central tool in our approach is the pseudo-polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of non-angularly equispaced points. The pseudo-polar FFT plays the role of a halfway point -- a nearly-polar system from which conversion to Polar Coordinates uses processes relying purely on interpolation operations. We describe the conversion process, and compare accuracy results obtained by unequallysampled FFT methods to ours and show marked advantage to our approach.

### Citations

145 |
Fast Fourier transforms for nonequispaced data
- Dutt, Rokhlin
- 1993
(Show Context)
Citation Context ...ced non Cartesian set of frequencies, a body of literature we refer to as USFFT (short for Unequally Spaced Frequency Fourier Transform) proposes the use of the regular FFT, followed by interpolation =-=[4, 5, 6, 7, 8, 9, 10, 11, 12, 13]-=-. The existing state-of-the-art expressed by this literature can be adapted to produce an approximate Polar FFT (PFFT) and its inverse. While we are unaware of any publication which does so, we descri... |

96 |
On the fast Fourier transform of functions with singularities
- Beylkin
- 1995
(Show Context)
Citation Context ...ced non Cartesian set of frequencies, a body of literature we refer to as USFFT (short for Unequally Spaced Frequency Fourier Transform) proposes the use of the regular FFT, followed by interpolation =-=[4, 5, 6, 7, 8, 9, 10, 11, 12, 13]-=-. The existing state-of-the-art expressed by this literature can be adapted to produce an approximate Polar FFT (PFFT) and its inverse. While we are unaware of any publication which does so, we descri... |

85 | Nonuniform fast Fourier transforms using min-max interpolation
- Fessler, Sutton
- 2003
(Show Context)
Citation Context ...ced non Cartesian set of frequencies, a body of literature we refer to as USFFT (short for Unequally Spaced Frequency Fourier Transform) proposes the use of the regular FFT, followed by interpolation =-=[4, 5, 6, 7, 8, 9, 10, 11, 12, 13]-=-. The existing state-of-the-art expressed by this literature can be adapted to produce an approximate Polar FFT (PFFT) and its inverse. While we are unaware of any publication which does so, we descri... |

49 | Fast slant stack: A notion of radon transform for data in a cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible
- Averbuch, Coifman, et al.
- 2001
(Show Context)
Citation Context ...aphy. Recently, The pseudo-polar grid (essentially the Linogram with a proper choice of oversampling) was proposed as the base for a stable forward and inverse Radon transform called Fast Slant-Stack =-=[17]. -=-For this grid we have the following fundamental result [17, 3]: Theorem 1 Given the signal f[i1, i2], 0 ≤ i1, i2 < N, the EXACT evaluation of the FT on the oversampled pseudo-polar grid with NS conc... |

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

44 | Nonuniform fast Fourier transform
- Duijndam, Schonewille
- 1999
(Show Context)
Citation Context |

35 |
Rapid computation of the discrete Fourier transform
- Anderson, Dahleh
- 1996
(Show Context)
Citation Context |

33 |
The regular Fourier matrices and nonuniform fast Fourier transforms
- Nguyen, Liu
- 1999
(Show Context)
Citation Context |

33 |
Digital reconstruction of multidimensional signals from their projections
- Mersereau, Oppenheim
- 1974
(Show Context)
Citation Context ...ates (N = 8). This polar-like 2D grid enables fast Fourier computation. This grid has been explored by many since the 1970-s and until recently. The pioneers in this field are Mersereau and Oppenheim =-=[14] -=-who proposed the Linogram grid as an alternative grid to the polar one. Later work by Munson and others [15, 16] have shown ξ xshow these ideas can be extended and used for tomography. Recently, The ... |

23 |
Introduction to the top 10 algorithms
- Dongarra, Sullivan
(Show Context)
Citation Context ... – has had far reaching implications in science and technology in recent decades. The scientific computing community regards the FFT as one of the leading algorithmic achievements of the 20th centur=-=y [1]. In f-=-act, even ordinary consumer-level applications now involve FFT’s – think of web browser decoding a JPEG images – so that development of new tools for Fourier analysis of digital data may be of p... |

18 | A fast algorithm for Chebyshev, Fourier, and sinc interpolation onto an irregular grid
- Boyd
- 1992
(Show Context)
Citation Context |

16 |
A spectral method of characteristics for hyperbolic problems
- Süli, Ware
- 1991
(Show Context)
Citation Context |

9 | Direct-Fourier reconstruction in tomography and synthetic aperture radar, Int
- Choi, Jr, et al.
- 1998
(Show Context)
Citation Context ...ince the 1970-s and until recently. The pioneers in this field are Mersereau and Oppenheim [14] who proposed the Linogram grid as an alternative grid to the polar one. Later work by Munson and others =-=[15, 16] -=-have shown ξ xshow these ideas can be extended and used for tomography. Recently, The pseudo-polar grid (essentially the Linogram with a proper choice of oversampling) was proposed as the base for a ... |

5 |
An owner’s manual for the discrete Fourier transform
- Briggs, Henson
- 1995
(Show Context)
Citation Context ...to be a fast one, similar to the digital transform that produces the Cartesian grid in the frequency domain. Unfortunately, the prevailing belief seems to be that there cannot exist such an algorithm =-=[2]-=-. This is intimately related to the strong reliance of the FFT on separability of the axes, and the equispaced samples on both grids - properties lost when dealing with polar coordinates. In this pape... |

4 |
Fast tomographic reconstruction using chirp-z interpolation
- Moraski, Jr, et al.
- 1991
(Show Context)
Citation Context ...ince the 1970-s and until recently. The pioneers in this field are Mersereau and Oppenheim [14] who proposed the Linogram grid as an alternative grid to the polar one. Later work by Munson and others =-=[15, 16] -=-have shown ξ xshow these ideas can be extended and used for tomography. Recently, The pseudo-polar grid (essentially the Linogram with a proper choice of oversampling) was proposed as the base for a ... |

1 |
Accurate and Fast Discerete Polar Fourier Transform
- Averbuch, Coifman, et al.
- 2003
(Show Context)
Citation Context ...algorithms, and discuss its features such as accuracy and computational complexity. In this paper we concentrate on the forward transform. We refer the reader to a more detailed report of our work in =-=[3]-=-. 2 Current State of the Art For evaluating the Fourier transform of a equally spaced Cartesian grid image over non-equispaced non Cartesian set of frequencies, a body of literature we refer to as USF... |