2005/02/28 by Yves Wiaux, Y. Wiaux, Laurent Jacques +3 · 5 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #Scientific Research and Discoveries #astro-ph #gr-qc #math-ph #math.MP
paper · pdf · doi:10.1086/432926
published as Astrophys.J.632:15,2005 · Version accepted in ApJ. Clarified discussion on the unicity of the wavelet formalism on the sphere and on the related correspondence principle. 14 pages, 8 figures, Revtex4 (emulateapj)
arxiv created 2005/08/24 · openalex publication_date 2005/10/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
Wavelets on the sphere are reintroduced and further developed independently of the original group theoretic formalism, in an equivalent, but more straightforward approach. These developments are motivated by the interest of the scale-space analysis of the cosmic microwave background (CMB) anisotropies on the sky. A new, self-consistent, and practical approach to wavelet filtering on the sphere is developed. It is also established that the inverse stereographic projection of a wavelet on the plane (i.e., Euclidean wavelet) leads to a wavelet on the sphere (i.e., spherical wavelet). This new correspondence principle simplifies the construction of wavelets on the sphere and allows the transfer onto the sphere of properties of wavelets on the plane. In that regard, we define and develop the notions of directionality and steerability of filters on the sphere. In the context of the CMB analysis, these notions are important tools for the identification of local directional features in the wavelet coefficients of the signal and for their interpretation as possible signatures of non-Gaussianity, statistical anisotropy, or foreground emission. However, the generic results exposed may find numerous applications beyond cosmology and astrophysics.