Correspondence Principle between Spherical and Euclidean Wavelets

Wavelets on the sphere are re-introduced and further developed independently of the original group theoretic formalism, in an alternative and completely equivalent approach, as inverse stereographic projection of wavelets on the plane. These developments are motivated by the interest of the scale-space analysis of the cosmic microwave background (CMB) anisotropies on the sky. In addition to offering a practical and intuitive approach to the filtering by spherical wavelets (i.e. wavelets on the sphere), the correspondence principle established here notably allows to transfer onto the sphere properties of Euclidean wavelets (i.e. wavelets on the plane). In that regard, we also define and develop the notions of directionality and steerability of filters on the sphere. In the context of the CMB analysis, these notions are essential 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. But the generic results exposed may find numerous applications beyond cosmology and astrophysics. Subject headings: cosmology: cosmic microwave background — methods: data analysis