Computing Fourier Transforms and Convolutions on the 2-Sphere

This paper considers the problem of efficient computation of the spherical harmonic expansion, or Fourier transform, of functions defined on the two dimensional sphere, S^2. The resulting algorithms are applied to the efficient computation of convolutions of functions on the sphere. We begin by proving convolution theorems generalizing well known and useful results from the abelian case. These convolution theorems are then used to develop a sampling theorem on the sphere. which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations. We show how to perform these efficiently, starting with an O(n(log n)^2) time algorithm for computing the Legendre transform of a function defined on the interval [-1,1] sampled at n points there. Theoretical and experimental results on the effects of finite precision arithmetic are presented. The Legendre transform algorithm is then generalized to obtain an algorithm for the Fourier transform, requiring O(n(log n)^2) time, and an algorithm for its inverse in O(n^1^.^5) time, where n is the number of points on the sphere at which the function is sampled. This improves the naive O(n^2) bound, which is the best previously known. These transforms give an O(n^1^.^5) algorithm for convolving two functions on the sphere.

[1]  H. Goldstein,et al.  Classical Mechanics , 1951, Mathematical Gazette.

[2]  A. R. Edmonds.  Angular Momentum in Quantum Mechanics , 1957 .

[3]  W. Scott,et al.  Group Theory. , 1964 .

[4]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[5]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[6]  L. Schwartz.  Théorie des distributions , 1966 .

[7]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[8]  G. Ramos.  Roundoff error analysis of the fast Fourier transform , 1970 .

[9]  Bede Liu,et al.  Accumulation of Round-Off Error in Fast Fourier Transforms , 1970, JACM.

[10]  A. Oppenheim,et al.  Effects of finite register length in digital filtering and the fast Fourier transform , 1972 .

[11]  Jacques Morgenstern,et al.  Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform , 1973, JACM.

[12]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[13]  D. James.  Quantization errors in the fast Fourier transform , 1975 .

[14]  S. Winograd.  On computing the Discrete Fourier Transform. , 1976, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Tran-Thong,et al.  Fixed-point fast Fourier transform error analysis , 1976 .

[16]  Gary L. Wise,et al.  A novel approach for the computation of Legendre polynomial expansions , 1978 .

[17]  W. Knight,et al.  A simple fixed-point error bound for the fast Fourier transform , 1979 .

[18]  R. Tolimieri,et al.  Is computing with the finite Fourier transform pure or applied mathematics , 1979 .

[19]  Christos H. Papadimitriou,et al.  Optimality of the Fast Fourier transform , 1979, JACM.

[20]  Gabor T. Herman.  Image Reconstruction from Projections , 1979 .

[21]  Salvatore D. Morgera,et al.  Efficient Synthesis and Implementation of Large Discrete Fourier Transformations , 1980, SIAM J. Comput..

[22]  Persi Diaconis,et al.  Average Running Time of the Fast Fourier Transform , 1980, J. Algorithms.

[23]  David C. Munson,et al.  Floating point roundoff error in the prime factor FFT , 1981 .

[24]  V. Rao,et al.  Vector radix FFT error analysis , 1982 .

[25]  Optimal spherical designs and numerical integration on the sphere , 1983 .

[26]  Ganapati Panda,et al.  Error analysis of Good-Winograd algorithm assuming correlated truncation errors , 1983 .

[27]  S. Helgason.  Groups and geometric analysis , 1984 .

[28]  Keisuke Kobayashi.  Solution of multi-dimensional neutron transport equation of the spherical harmonics method using the finite Fourier transformation and quadrature formula , 1985 .

[29]  A. Terras.  Harmonic Analysis on Symmetric Spaces and Applications I , 1985 .

[30]  Gary A Dilts,et al.  Computation of spherical harmonic expansion coefficients via FFT's , 1985 .

[31]  D. V. Satish Chandra.  Accumulation of coefficient roundoff error in fast Fourier transforms implemented with logarithmic number system , 1987, IEEE Trans. Acoust. Speech Signal Process..

[32]  Thomas Beth,et al.  On the Computational Complexity of the General Discrete Fourier Transform , 1987, Theor. Comput. Sci..

[33]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[34]  Daniel N. Rockmore.  Computation of Fourier transforms on the symmetric group , 1989 .

[35]  P. Diaconis,et al.  Efficient computation of the Fourier transform on finite groups , 1990 .

[36]  Bradley K. Alpert,et al.  A Fast Algorithm for the Evaluation of Legendre Expansions , 1991, SIAM J. Sci. Comput..

[37]  D. Calvetti.  A Stochastic Roundoff Error Analysis for the Fast Fourier Transform , 1991 .