## Ffts on the rotation group (2003)

Venue: | Santa Fe Institute Working Papers Series Paper |

Citations: | 30 - 0 self |

### BibTeX

@INPROCEEDINGS{Kostelec03fftson,

author = {Peter J. Kostelec and Daniel N. Rockmore},

title = {Ffts on the rotation group},

booktitle = {Santa Fe Institute Working Papers Series Paper},

year = {2003},

pages = {03--11}

}

### Years of Citing Articles

### OpenURL

### Abstract

Earlier work by Driscoll and Healy [4] has produced an efficient O(B 2 log 2 B) algorithm for computing the Fourier transform of band-limited functions on the 2-sphere. In this paper, we discuss an implementation of an O(B 4) algorithm for the numerical computation of Fourier transforms of functions defined on the rotation group, SO(3). This compares with the direct O(B 6) approach. The algorithm we implemented is based on the “Separation of Variables ” technique, e.g. as presented by Maslen and Rockmore [19]. In conjunction with the techniques developed in [4], the SO(3) algorithm we implemented may be made truly fast, O(B 3 log 2 B). Basic results will be presented establishing the algorithm’s numerical stability, and examples of applications will be presented. 1

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Citation Context ...′ =−J J� J� J≥0 M=−J M ′ =−J we find the Fourier coefficients of their convolution to be c J MM ′ = J� k=−J a J MM ′DJ MM ′(g) b J MM ′DJ MM ′(g), a J Mkb J kM ′ (10) Details may be found in Vilenkin =-=[31]-=-. One may obtain the Fourier coefficients of their convolution in a very elegant manner. Note that corresponding to each integer J is an irreducible representation of SO(3). Arrange the coefficients o... |

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Citation Context ...s: � C(g) = f(ω) Λ(g)h(ω) dω S 2 and finding the g that maximizes the above integral. This has a number of useful applications, in such diverse areas as molecular biology [16], and 3-D shape-matching =-=[14]-=-. Now, instead of undertaking the time-consuming task of evaluating C(g) for all possible rotations, we may efficiently determine the maximum g by means of the FFT on SO(3). We develop this as follows... |

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Citation Context ...ng all the Fourier coefficients of f as the SO(3) Fast Fourier transform of f, SOF F T (f). In evaluating that last sum, (19), it is possible to apply Risbo’s technique [24], using the identity (from =-=[23]-=-) d l ′ MM ′(β) = ıM−M j� M ′′ =−j d j M ′′ M (π/2)e−ıM ′′ β d j M ′′ M ′(π/2), and so rewrite the entire computation in terms of three Cooley-Tukey FFTs. However, this algorithm is O(B 4 ), and its s... |

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Citation Context ...rect O(B 6 ) approach. The algorithm we implemented is based on the “Separation of Variables” technique, e.g. as presented by Maslen and Rockmore [19]. In conjunction with the techniques developed in =-=[4]-=-, the SO(3) algorithm we implemented may be made truly fast, O(B 3 log 2 B). Basic results will be presented establishing the algorithm’s numerical stability, and examples of applications will be pres... |

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Citation Context ...d, if one is willing to keep the “natural” structure of the algorithm. The algorithm we present preserves this structure, as well as the freedom of applying a Driscoll-Healy based algorithm (e.g. see =-=[12]-=-), which has the potential for an algorithm faster than O(B 4 ). This paper is organized as follows. We begin with a brief introduction to SO(3), including some discussion of harmonic analysis. This i... |

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Citation Context ..., as is observed in [19], one could take advantage of the three-term recurrence the Wigner d-functions satisfy, and hence employ the techniques of Driscoll and Healy [4], Driscoll, Healy and Rockmore =-=[5]-=-, or their more numerically stable variations [12]. The rationale for our “naive” decision will be given in following section. 10sCPU seconds 10 −1 10 −2 10 −3 10 −4 10 −5 10 −6 Alpha Athlon P−III Xeo... |

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Citation Context ... a function defined on the rotation group, SO(3). This software, which we christened “The SOFT Package,” is freely available on the web [27]. This implementation differs from that as derived by Risbo =-=[24]-=-, which uses properties of the Wigner-d function to rewrite the transform over the full group in such a way as to allow the use of three Cooley-Tukey FFTs. However, this introduces overhead which can ... |

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Citation Context ...form. Now the astute reader may ask why the naive method was used in computing the Wigner-d coefficients. Since the d-functions are trigonometric polynomials, certainly methods such as those of Dilts =-=[3]-=- (also called “semi-naive” in [12]) could be used and the discrete sums evaluated in the “discrete cosine transform” domain. Indeed, as is observed in [19], one could take advantage of the three-term ... |

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Generalized FFTs
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(Show Context)
Citation Context ...n the rotation group, SO(3). This compares with the direct O(B 6 ) approach. The algorithm we implemented is based on the “Separation of Variables” technique, e.g. as presented by Maslen and Rockmore =-=[19]-=-. In conjunction with the techniques developed in [4], the SO(3) algorithm we implemented may be made truly fast, O(B 3 log 2 B). Basic results will be presented establishing the algorithm’s numerical... |

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Citation Context ...defined on SO(3): correlating two functions defined on S 2 . This is useful in a variety of applications, including searchable 3-D databases [25], molecular biology [16], and industrial manufacturing =-=[11]-=-. We mention that another application of Fourier transforms functions defined on SO(3) concerns spherical near-field antenna measurements [9]. We conclude with a brief recap and discussion. 2 Life in ... |

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6 |
SpharmonicKit is a freely available collection of C programs for doing Legendre and scalar spherical transforms. Developed at Dartmouth College by
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Citation Context ...lem sizes. We will also present some timing results, and discuss issues of computational efficiency. 8 10 3sAll code was written in C, with some of the more basic routines borrowed from SpharmonicKit =-=[26]-=-. In certain instances, which will be made clear, we availed ourselves to the Fourier transform routines provided by FFTW [8]. Experiments were performed on a variety of mostly GNU/Linux platforms run... |

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3 |
Computation of spherical harmonic expansion coecients via FFTs
- Dilts
- 1985
(Show Context)
Citation Context ...nsform. Now the astute reader may ask why the naive method was used in computing the Wigner-d coecients. Since the d-functions are trigonometric polynomials, certainly methods such as those of Dilts =-=[3]-=- (also called \semi-naive" in [12]) could be used and the discrete sums evaluated in the \discrete cosine transform" domain. Indeed, as is observed in [19], one could take advantage of the three-term ... |

2 |
Computational Harmonic Analysis for Tensor Fields on the Two-sphere
- Rockmore
(Show Context)
Citation Context ...f ∈ L 2 (SO(3)). Furthermore, the Fourier coefficients in Step 2 are randomly generated without the constraint of ensuring that the resulting sample values should be strictly real, e.g. as is done in =-=[15]-=-. We implemented four fundamentally different versions of the full SO(3) Fourier transform, with a variant or two thrown in. The full transform may be treated as two components: the “regular” FFT port... |

2 | An empirical Bayes approach to directional data and e#cient computation on the sphere, Ann - Healy, Kim - 1996 |