FFTs for the 2-Sphere-Improvements and Variations

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 article 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 log² 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 article concludes with a brief discussion of a few potential applications.