## Fast and accurate Polar Fourier transform (2006)

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Venue: | Appl. Comput. Harmon. Anal. |

Citations: | 17 - 1 self |

### BibTeX

@MISC{Averbuch06fastand,

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

title = {Fast and accurate Polar Fourier transform},

year = {2006}

}

### OpenURL

### Abstract

In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. 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^2 log N), just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, 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 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid.

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Citation Context ...equispaced rays. The literature of tomography concerns reconstruction of an image from a collection of its onedimensional integrals. An important approach to this problem is the direct Fourier method =-=[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]-=-. The direct Fourier method uses the projection-slice theorem, which suggests that one can convert a collection of projection data (Rf)(·, θ) into a two-dimensional Fourier data ˆ f(ξ) on a Polar grid... |

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Citation Context ...that one can convert a collection of projection data (Rf)(·, θ) into a two-dimensional Fourier data ˆ f(ξ) on a Polar grid, and then reconstruct by Fourier inversion. In the second body of literature =-=[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]-=- one has data at an equally spaced Cartesian grid, but wishes to evaluate its discrete Fourier transform as a non-equispaced non Cartesian set of frequencies. This problem is known as either the USFFT... |

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Citation Context ...that one can convert a collection of projection data (Rf)(·, θ) into a two-dimensional Fourier data ˆ f(ξ) on a Polar grid, and then reconstruct by Fourier inversion. In the second body of literature =-=[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]-=- one has data at an equally spaced Cartesian grid, but wishes to evaluate its discrete Fourier transform as a non-equispaced non Cartesian set of frequencies. This problem is known as either the USFFT... |

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Citation Context ...any since the 1970-s. The pioneers in this field are Mersereau and Oppenheim [31] who proposed the concentric squares grid as an alternative to the Polar grid. Work by Pasciak [32], Edholm and Herman =-=[33]-=-, and Lawton [34] showed that fast exact evaluation on such grids is possible. Later work by Munson and others [8, 10] have shown how these ideas can be extended and used for tomography. Recently, the... |

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Citation Context ...that one can convert a collection of projection data (Rf)(·, θ) into a two-dimensional Fourier data ˆ f(ξ) on a Polar grid, and then reconstruct by Fourier inversion. In the second body of literature =-=[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]-=- one has data at an equally spaced Cartesian grid, but wishes to evaluate its discrete Fourier transform as a non-equispaced non Cartesian set of frequencies. This problem is known as either the USFFT... |

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Citation Context ...is viewed a source of inspiration only. Rather than hewing to some general process deriving from polar FT, each relevant application area has custom built tools specialized to the application at hand =-=[4, 5, 6, 7]-=-. It is perhaps also not a coincidence that in these areas the methods most directly inspired by continuum polar Ft ideas are not the ones in the most widespread use. For example, in Radon inversion, ... |

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Citation Context ...es lie in an oversampled set of non-angularly equispaced points (see Figure 2). The Pseudo-Polar FFT offers us a near-Polar frequency coordinate system for which exact rapid FT evaluation is possible =-=[30]-=-. Whereas the Polar grid points sit at the intersection between linearly growing concentric circles and angularly equispaced rays, the Pseudo-Polar points sit at the intersection between linearly grow... |

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Citation Context ...equispaced rays. The literature of tomography concerns reconstruction of an image from a collection of its onedimensional integrals. An important approach to this problem is the direct Fourier method =-=[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]-=-. The direct Fourier method uses the projection-slice theorem, which suggests that one can convert a collection of projection data (Rf)(·, θ) into a two-dimensional Fourier data ˆ f(ξ) on a Polar grid... |

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Citation Context ...equispaced rays. The literature of tomography concerns reconstruction of an image from a collection of its onedimensional integrals. An important approach to this problem is the direct Fourier method =-=[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]-=-. The direct Fourier method uses the projection-slice theorem, which suggests that one can convert a collection of projection data (Rf)(·, θ) into a two-dimensional Fourier data ˆ f(ξ) on a Polar grid... |

19 |
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Citation Context ...] have shown how these ideas can be extended and used for tomography. Recently, the Pseudo-Polar grid was proposed as the base for a stable forward and inverse Radon transform called Fast Slant-Stack =-=[30, 35]-=-. It has also been used for image registration [36]. 2 We should note that Cartesian-based USFFT methods may also be made cache-efficient in various ways [25]. However, this property is far more natur... |

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Citation Context ...in the exponent turns this expression into a regular 1D-FFT, this time applied on the rows of the array ˆ f1. With this factor α in the above summation, the required operation is known as the Chirp-Z =-=[37]-=-, or the Fractional Fourier Transform (FRFT) [38]. In Appendix A we show how this transform can be computed efficiently for any α, with 30N log N operations, based on 1D-FFT use. As before, due to a s... |

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Citation Context ...0-s. The pioneers in this field are Mersereau and Oppenheim [31] who proposed the concentric squares grid as an alternative to the Polar grid. Work by Pasciak [32], Edholm and Herman [33], and Lawton =-=[34]-=- showed that fast exact evaluation on such grids is possible. Later work by Munson and others [8, 10] have shown how these ideas can be extended and used for tomography. Recently, the Pseudo-Polar gri... |

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Citation Context ...d has been explored by many since the 1970-s. The pioneers in this field are Mersereau and Oppenheim [30] who proposed the concentric squares grid as an alternative to the polar grid. Work by Pasciak =-=[31]-=-, Edholm and Herman [32], and Lawton [33] showed that fast exact evaluation on such grids is possible. Later work by Munson and others [12, 13] have shown how these ideas can be extended and used 9fo... |

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Citation Context ...iquitous equispaced Cartesian format. The prevailing belief seems to be that there is no such algorithm. For example, in Briggs’ treatise The FFT: an Owner’s Manual for the Discrete Fourier Transform =-=[2]-=-, which is widely considered comprehensive and authoritative, the index contains the entry “Polar FFT”, continuing with “no FFT for, 284”! Indeed, several difficulties stand in our way to construct su... |

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2 | About BeamLab a Toolbox for New Multiscale Methodologies
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Citation Context ...oject has been development of a software tool that performs the computations given here. We intend to make this toolbox available to the public, in a fashion parallel to Wavelab and Beamlab libraries =-=[40, 41]-=-. Our implementation uses Matlab code to perform various tasks around the idea of Polar FFT. The PFFT toolbox is freely available in http://www.cs.technion.ac.il/~elad/PolarFFT.zip. As part of the fre... |

1 | Accurate and fast Polar Fourier transform - Averbuch, Coifman, et al. - 2003 |

1 |
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1 |
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Citation Context ...or tomography. Recently, the Pseudo-Polar grid was proposed as the base for a stable forward and inverse Radon transform called Fast Slant-Stack [30, 35]. It has also been used for image registration =-=[36]-=-. 2 We should note that Cartesian-based USFFT methods may also be made cache-efficient in various ways [25]. However, this property is far more natural when the data is structured as in the Pseudo-Pol... |

1 |
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Citation Context ...the proposed PFFT, speed, accuracy, stability, vectorizability and nonexpansivity. These terms will be given meaning as we deepen our description. A brief and partial version of this work appeared in =-=[3]-=-, and here we give an expanded and more detailed description of this work. 1.4 Relation to State of the Art Two existing bodies of literature contain ideas relevant to the above definition of Polar FT... |

1 |
Introduction the top 10 algorithms, Special issue of Computing in
- Dongarra, Sullivan
- 2000
(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 century =-=[1]-=-. In fact, even ordinary consumer-level applications now involve FFT’s – think of web browser decoding JPEG images – so that development of new tools for Fourier analysis of digital data may be of pot... |