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Earlier work by Driscoll and Healy has produced an efficient algorithm for computing the Fourier transform of band-limited functions on the 2-sphere. In this paper we present a reformulation and variation of the original algorithm which results in a greatly improved inverse transform, and consequent improved convolution algorithm for such functions. All require at most $O(N\log^2 N)$ operations where $N$ is the number of sample points. We also address implementation considerations and give heuristics for allowing reliable and computationally efficient floating point implementations of slightly modified algorithms.
These claims are supported by extensive numerical experiments from our implementation in C on DEC, HP, SGI and Linux Pentium platforms. These results indicate that variations of the algorithm are both reliable and efficient for a large range of useful problem sizes. Performance appears to be architecture-dependent. The paper concludes with a brief discussion of a few potential applications.
Preliminary versions of some of these results have appeared in the Dartmouth College Department of Computer Science Technical Report PCS-TR94-222 and ``An FFT for the 2-sphere and Applications'', Proc. of ICASSP-96, Volume 3, pp. 1323--1326.
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Dennis M. Healy, Daniel N. Rockmore, Peter J. Kostelec, and Sean S. B. Moore, "FFTs for the 2-Sphere - Improvements and Variations." Dartmouth Computer Science Technical Report TR2002-419, March 2002.
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