Fast Directional Correlation on the Sphere with Steerable Filters

A fast algorithm is developed for the directional correlation of scalarband-limited signals and band-limited steerable filters on the sphere. Theasymptotic complexity associated to it through simple quadrature is of orderO(L^5), where 2L stands for the square-root of the number of sampling points onthe sphere, also setting a band limit L for the signals and filters considered.The filter steerability allows to compute the directional correlation uniquelyin terms of direct and inverse scalar spherical harmonics transforms, whichdrive the overall asymptotic complexity. The separation of variables techniquefor the scalar spherical harmonics transform produces an O(L^3) algorithmindependently of the pixelization. On equi-angular pixelizations, a samplingtheorem introduced by Driscoll and Healy implies the exactness of thealgorithm. The equi-angular and HEALPix implementations are compared in termsof memory requirements, computation times, and numerical stability. Thecomputation times for the scalar transform, and hence for the directionalcorrelation, of maps of several megapixels on the sphere (L~10^3) are reducedfrom years to tens of seconds in both implementations on a single standardcomputer. These generic results for the scale-space signal processing on thesphere are specifically developed in the perspective of the wavelet analysis ofthe cosmic microwave background (CMB) temperature (T) and polarization (E andB) maps of the WMAP and Planck experiments. As an illustration, we consider thecomputation of the wavelet coefficients of a simulated temperature map ofseveral megapixels with the second Gaussian derivative wavelet.Comment: Version accepted in APJ. 14 pages, 2 figures, Revtex4 (emulateapj). Changes include (a) a presentation of the algorithm as directly built on blocks of standard spherical harmonics transforms, (b) a comparison between the HEALPix and equi-angular implementation