Correspondence Principle Between Spherical and Euclidean Wavelets, and Fast Directional Correlation on the Sphere With Steerable Filters

Details Correspondence Principle Between Spherical and Euclidean Wavelets, and Fast Directional Correlation on the Sphere With Steerable Filters Correspondence Principle Between Spherical and Euclidean Wavelets, and Fast Directional Correlation on the Sphere With Steerable Filters

Dec 2005

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2005agufmin23a1201w&link_type=abstract

American Geophysical Union, Fall Meeting 2005, abstract #IN23A-1201

Statistics

Computation

0520 Data Analysis: Algorithms And Implementation, 0540 Image Processing, 3205 Fourier Analysis (3255), 3280 Wavelet Transform (3255, 4455), 7599 General Or Miscellaneous

Scientific paper

Wavelets on the sphere are reintroduced and further developed on both the theoretical and the algorithmic grounds. A specific application to cosmology is also discussed. First, a new practical approach to the wavelet filtering of signals on the sphere is developed. Translations and rotations of the filters are naturally implemented through three-dimensional rotations of the group SO(3), and a unitary, radial, and conformal dilation operator is required. The resulting formalism is unique. A correspondence principle is also established, stating that the inverse stereographic projection of a wavelet on the plane (i.e., Euclidean wavelet) also uniquely leads to a wavelet on the sphere (i.e., spherical wavelet). It simplifies the construction of wavelets on the sphere and allows the transfer onto the sphere of properties of wavelets on the plane, such as directionality and steerability. Second, an exact fast algorithm is developed for the directional correlation on the sphere of band-limited signals of band limit L and steerable (wavelet) filters, on 2L×2L equi-angular grids in the coordinates (θ,φ). On the one hand, the algorithm is based on a technique of separation of variables in the Wigner D-functions, basis functions for the harmonic analysis on the rotation group SO(3). The asymptotic complexity of the algorithm is correspondingly reduced from O(L5) to O(L4). On the other hand, the filter steerability and the use of the Driscoll and Healy fast scalar spherical harmonics transform further reduce the algorithm complexity to a simple O(L2log22L). Finally, we consider the perspective of the wavelet analysis of the cosmic microwave background (CMB) temperature and polarization anisotropies on the sphere of the sky. The notions of directionality and steerability are important tools for the identification of local directional features in the wavelet coefficients of the signal, and for their interpretation in cosmology. In this context, computation times for the exact directional correlation of megapixels all-sky maps from the ongoing WMAP or the forthcoming Planck Surveyor satellite missions are typically reduced from years to seconds on a single standard computer, and thus made easily affordable. But the generic results exposed for the scale-space signal processing on the sphere may find numerous applications beyond cosmology and astrophysics, from geophysics to computer vision.

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