Correspondence principle between spherical and Euclidean wavelets

Details Correspondence principle between spherical and Euclidean wavelets Correspondence principle between spherical and Euclidean wavelets

2005-02-23

arxiv.org/abs/astro-ph/0502486v2

Astrophys.J.632:15,2005

Astronomy and Astrophysics

Astrophysics

Version accepted in ApJ. Clarified discussion on the unicity of the wavelet formalism on the sphere and on the related corresp

Scientific paper

10.1086/432926

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 the 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 to transfer onto the sphere 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. But the generic results exposed may find numerous applications beyond cosmology and astrophysics.

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