000087281 001__ 87281 000087281 005__ 20190316233751.0 000087281 0247_ $$2doi$$a10.1086/507692 000087281 02470 $$2DAR$$a9601 000087281 02470 $$2ISI$$a000242125700074 000087281 037__ $$aARTICLE 000087281 245__ $$aFast directional correlation on the sphere with steerable filters 000087281 269__ $$a2006 000087281 260__ $$c2006 000087281 336__ $$aJournal Articles 000087281 520__ $$aA 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. 000087281 6531_ $$acomputational methods 000087281 6531_ $$asphere 000087281 6531_ $$acosmology 000087281 6531_ $$aCMB 000087281 6531_ $$aLTS2 000087281 700__ $$0240427$$g163268$$aWiaux, Y. 000087281 700__ $$0243987$$g182131$$aJacques, L. 000087281 700__ $$aVandergheynst, P.$$g120906$$0240428 000087281 773__ $$j652$$tAstrophys. J.$$q820 000087281 8564_ $$uhttp://www.journals.uchicago.edu/ApJ/$$zURL 000087281 8564_ $$uhttps://infoscience.epfl.ch/record/87281/files/fastcorr-arXiv-astro-ph-0508516v2.pdf$$zn/a$$s250892 000087281 909C0 $$xU10380$$0252392$$pLTS2 000087281 909CO $$qGLOBAL_SET$$pSTI$$ooai:infoscience.tind.io:87281$$particle 000087281 937__ $$aEPFL-ARTICLE-87281 000087281 970__ $$aWiaux2005_1327/LTS 000087281 973__ $$rREVIEWED$$sPUBLISHED$$aEPFL 000087281 980__ $$aARTICLE