Fast Directional Correlation on the Sphere with Steerable Filters

A fast algorithm is developed for the directional correlation of scalar band-limited signals and band-limited steerable filters on the sphere. The asymptotic complexity associated with it through simple quadrature is of order O(L⁵), where 2L stands for the square root of the number of sampling points on the sphere, also setting a band limit L for the signals and filters considered. The filter steerability allows us to compute the directional correlation uniquely in terms of direct and inverse scalar spherical harmonics transforms, which drive the overall asymptotic complexity. The ``separation of variables'' technique for a scalar spherical harmonics transform produces an O(L³) algorithm independent of the pixelization. On equiangular pixelizations, a sampling theorem introduced by Driscoll and Healy implies the exactness of the algorithm. The equiangular and HEALPix implementations are compared in terms of memory requirements, computation times, and numerical stability. The computation times for the scalar transform, and hence for the directional correlation, of maps of several megapixels on the sphere (L~=10³) are reduced from years to tens of seconds in both implementations on a single standard computer. These generic results for the scale-space signal processing on the sphere are specifically developed in the perspective of the wavelet analysis of the cosmic microwave background (CMB) temperature (T) and polarization (E and B) maps of the Wilkinson Microwave Anisotropy Probe and Planck experiments. As an illustration, we consider the computation of the wavelet coefficients of a simulated temperature map of several megapixels with the second Gaussian derivative wavelet.