FFTs for the 2-Sphere-Improvements and Variations

Journal of Fourier Analysis and Applications

Volumn 9, Issue 4, 2003, Pages 341-385

  Healy Jr , D M ^a   Rockmore, D N ^b   Kostelec, P J ^b   Moore, S ^c  

^a Bldg 142   (United States)

^b Dartmouth College   (United States)

^c Cetacean Networks Inc   (United States)

Author keywords

Fast legendre transform; Recurrence relations; Spherical fourier transform; Spherical harmonics

Indexed keywords

EID: 0141888441     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-003-0018-9     Document Type: Article

References (72)

1. Ahmed, N., Natarajan, T., and Rao, K.R. (1974). Discrete cosine transforms, IEEE Transactions on Computers, 23, 90-93.
2. Aho, A., Hopcroft, J., and Ullman, J. (1976). The Design and Analysis of Computer Algorithms, AddisonWesley, Reading.
3. Alpert, B. and Rokhlin, V. (1991). A fast algorithm for the evaluation of Legendre transforms, SIAM J. Sci. Statist. Comput., 12, 158-179.
4. Auslander, L., Johnson, J.R., and Johnson, R.W. (1996). Multidimensional Cooley-Tukey algorithms revisited, Adv. in Appl. Math., 17(4), 477-519.
5. Auslander, L. and Tolmieri, R. (1979). Is computing with the fast Fourier transform pure or applied mathematics?, Bulletin of the American Mathematical Society, (N. S.), 1, 847-897.
6. Barrucand, P. and Dickinson, D. (1968). On the associated Legendre polynomials, in Orthogonal Expansions and their Continuous Analogues, Southern Illinois University Press, Carbondale.
7. Biedenharn, L.C. and Louck, J.D. (1981). Angular Momentum in Quantum Mechanics, Addison-Wesley, Reading.
8. Borodin, A. and Munro, I. (1975). The Computational Complexity of Algebraic and Numeric Problems, Elsevier, New York.
9. Boyd, J. (1989). Chebyshev and Fourier Spectral Methods, Lecture Notes in Engineering, Vol. 49, Springer- Verlag, NY.
10. Bshouty, N., Kaminski, M., and Kirkpatrick, D. (1988). Addition requirements for matrix and transposed matrix products, J. of Algorithms, 9, 354-364.
11. Calvetti, D. (1991). A stochastic roundoff error analysis for the fast Fourier transform, Math. Comput., 56(194), 755-774.
12. Chen, C.W. and Huang, T.S. (1990). Epicardial motion and deformation estimation from coronary artery bifurcation points, in Proc. of Third. Int. Conf. on Comp. Vision, Dec. 4-7, 1990, IEEE Press, 456-460.
13. Chirikjian, G.S. and Kyatkin, A.B. (2000). Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups, CRC Press.
14. Cooley, J.W. and Tukei, J.W. (1965). An algorithm for machine calculation of complex Fourier series, Math. Comput., 19, 297-301.
15. Cormen, T.H., Leiserson, C.E., and Rivest, R.L. (1990). Introduction to Algorithms, MIT Press.
16. Diaconis, P. (1980). Average running time of the fast Fourier transform, J. Algorithms, 1, 187-208.
17. Dilts, G.A.(1985). Computation of spherical harmonic expansion coefficients via FFTs, J. of Computational Physics, 57(3), 439-453.
18. th IEEE FOCS, 344-349; Adv. in Appl. Math., 15, 202-250.
19. th IEEE FOCS, 344-349; Adv. in Appl. Math., 15, 202-250.
20. Driscoll, J.R., Healy, D., and Rockmore, D. (1997). Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs, SIAM J. Comput., 26(4), 1066-1099.
21. Duncan, B.S. and Olson, A.J. (1993). Approximation and characterization of molecular surfaces, Biopolymers, 33, 219-229.
22. Elliot, D.F. and Rao, K.R. (1982). Fast Transforms: Algorithms, Analyses, and Applications, Academic Press, New York.
23. FFTPACK is a freely available collection of FORTRAN programs for computing one-dimensional fast Fourier transforms developed at NCAR by Paul Swarztrauber.
24. FFTW is a C subroutine library for computing the Discrete Fourier Transform (DFT) in one or more dimensions, of both real and complex data, and of arbitrary input size. FFTW is a freely available at http://www.fftw.org.
25. Freeden, W. (1980). On integral formulas of the (unit) sphere and their application to numerical computation of integrals, Computing, 25, 131-146.
26. Gallagher, N.C., Wise, G.L., and Allen, J.W. (1978). A novel approach for the computation of Legendre polynomial expansions, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-26(1), 105-106.
27. Gilbert, F.A. (1971). Inverse problems for the earth's normal modes, in Mathematical Problems in the Geophysical Sciences, Vol. I, American Mathematical Society, Providence.
28. Greengard, L. (1994). Fast algorithms for classical physics, Science, 265, 909-914.
29. Healy, D. and Kim, P. (1996). An empirical Bayes approach to directional data and efficient computation on the sphere, Ann. Stat., 24(1), 232-254.
30. Healy, D., Hendriks, H., and Kim, P. (1993). Spherical deconvolution with application to geometric quality assurance, Technical Report, Department of Mathematics and Computer Science, Dartmouth College.
31. Healy, D., Moore, S., and Rockmore, D. (1994). Efficiency and stability issues in the numerical convolution of Fourier transforms and convolutions on the 2-sphere, Technical Report PCS-TR94-222, Department of Computer Science, Dartmouth College.
32. Healy, D., Olson, T., Rockmore, D., and Mirchandani, G. Wreath products for image processing, in Proceedings of 1996 ICASSP, Vol. 6, 3582-3585.
33. Healy, D., Rockmore, D. and Kostelec, P. Stabilized algebraic FFTs on the 2-sphere, preprint.
34. Healy, D., Rockmore, D., and Moore, S. An FFT for the 2-sphere and applications, in Proceedings of 1996 ICASSP, Vol. 3, 1323-1326.
35. Heideman, M.T., Johnson, D.H., and Burras, C.S. (1985). Gauss and the history of the fast Fourier transform, Arch. for History of Exact Sciences, 34(3), 265-277.
36. Høyer, P. (1997). Efficient quantum transforms, Los Alamos preprint quant-ph/9702028, February.
37. Inda, M.A., Bisseling, R.H., and Maslen, D.K. (2001). On the efficient parallel computation of Legendre transforms, SIAM J. on Scientific Computing, 23(1), 271-303.
38. Kakarala, R. (1992). Triple correlation on groups, Ph.D. thesis, Dept. of Math., University of California, Irivine.
39. Kanatani, K. (1990). Group-Theoretical Methods in Image Understanding, Springer-Verlag, NY.
40. Kay, S. (1993). Fundamentals of Statistical Signal Processing, Prentice-Hall, Englewood Cliffs, NJ.
41. Kobayashi, K. (1985). Solution of multi-dimensional neutron transport equation of the spherical harmonics method using the finite Fourier transformation and quadrature formula, Transport Theory and Stat. Phys., 14, 63-85.
42. Kostelec, P., Rockmore, D., Maslen, D., and Healy Jr., D.M. FFTs on the rotation group, in preparation.
43. Kostelec, P., Maslen, D., Rockmore, D., and Healy Jr., D.M. (2000). Computational harmonic analysis for tensor fields on the two-sphere, J. Comp. Physics, 162, 514-535.
44. -: Many body polarization effects, J. Chem. Phys., 90(7), 3744-3755.
45. Maslen, D. (1993). Fast transforms and sampling for compact roups, Ph.D. thesis, Department of Mathematics, Harvard University.
46. Maslen, D. (1998). Efficient computation of Fourier transforms on compact groups, J. Fourier Anal. Appl., 4(1), 19-52.
47. Maslen, D. and Rockmore, D. (2001). The Cooley-Tukey FFT and group theory, Notices of the AMS, 48(10), 1151-1160.
48. Maslen, D. and Rockmore, D. (1997). Generalized FFTs, in Groups and Computation II, DIMACS Series in Discrete Math. and Computer Science, Vol. 28, Finkelstein, L. and Kantor, W., Eds., 183-237.
49. Miller, M., Joshi, S., Maffitt, D., McNally, J., and Grenander, U. (1994). Membranes, mitochondria and amoebae: shape models, in Advances in Applied Statistics, 137-159.
50. Mohlenkamp, M.P. (1999). A fast transform for spherical harmonics, J. Fourier Anal. Appl., 5(2-3), 159-184.
51. Moore, S. (1994). Efficient stabilization methods for fast polynomial transforms, Ph.D. thesis, Department of Mathematics and Computer Science, Dartmouth College.
52. Moore, S., Healy, D., and Rockmore, D. (1993). Symmetry stabilization for polynomial evaluation and interpolation, Lin. Alg. Appl., 192, 249-299.
53. Peng Oh, S., Spergel, D.N., and Hinshaw, G. (1998). An efficient technique to determine the power spectrum from cosmic microwave background maps, preprint.
54. Oppenheim, A. and Schafer, R. (1989). Discrete-Time Signal Processing, Prentice-Hall, NJ.
55. Orzag, S.A. (1986). Fast eigenfunction transforms, in Science and Computers, Academic Press, Orlando.
56. Potts, D., Steidl, G., and Tasche, M. (1998). Fast and stable algorithms for discrete spherical Fourier transforms, Linear Algebra Appl., 275/276, 433-450.
57. Potts, D., Steidl, G., and Tasche, M. (1998). Fast algorithms for discrete polynomial transforms, Math. Comp., 67, 1577-1590.
58. Ramos, G.U. (1971). Roundoff error analysis of the fast Fourier transform, Math. Comp., 25, 757-768.
59. Rockmore, D. (1997). Some applications of generalized FFTs, (Appendix with D. Healy), in Groups and Computation II, DIMACS Series in Discrete Math and Computer Science, Vol. 28, Finkelstein, L. and Kantor, W., Eds., 329-369.
60. Schwartz, J.T. (1985). Mathematics addresses problems in computer vision for advanced robotics, SIAM News, 18(3).
61. Swarztrauber, P. (1993). The vector harmonic transform method for solving partial differential equations in spherical geometry, Monthly Weather Review, 121(12), 3415-3437.
62. SpharmonicKit is a freely available collection of C programs for doing Legendre and scalar spherical transforms. Developed at Dartmouth College by S. Moore, D. Healy, D. Rockmore and P. Kostelec, it is available at www.cs.dartmouth.edu/̃geelong/sphere/.
63. SPHEREPACK is a freely available collection of FORTRAN programs that facilitates computer modeling of geophysical processes developed at NCAR by John Adams and Paul Swarztrauber.
64. Spotz, W.F. and Swarztrauber, P.N. (2001). A performance comparison of associated Legendre projections, J. Comp. Physics, 168(2), 339-355.
65. Steidl, G. and Tasche, M. (1991). A polynomial approach to fast algorithms for discrete Fourier-cosine and Fourier-sine transforms, Math. Comp., 56, 281-296.
66. Swarztrauber, P.N. and Spotz, W.F. (2000). Generalized discrete spherical harmonic transforms, J. Comp. Physics, 159(2), 213-230.
67. Temperton, C. (1991). On scalar and vector transform methods for global spectral models, Mon. Wea. Rev., 119, 1303-1307.
68. Vaidyanathan, P.P. (1993). Multirate Systems and Filter Banks, Prentice-Hall, Englewood Cliffs, NJ.
69. Van Loan, C. (1992). Computational framework for the fast Fourier transform, SIAM, Philadelphia.
70. Van Trees, H. (1968). Detection, Estimation and Modulation Theory, Vol. I, Wiley, New York.
71. Vilenkin, N.J. (1968). Special Functions and the Theory of Group Representations, American Mathematical Society, Providence.
72. Woodward, P.M. (1980). Probability and Information Theory, with Applications to Radar, Artech House, Dedham, MA.