Abstract |
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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 to it through simple quadrature is of order
O(L^5), 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 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 the scalar spherical harmonics transform produces an O(L^3) algorithm
independently of the pixelization. On equi-angular pixelizations, a sampling
theorem introduced by Driscoll and Healy implies the exactness of the
algorithm. The equi-angular 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~10^3) 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 WMAP 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.
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 implementations |