Correspondence principle between spherical and euclidean wavelets

Yves Wiaux, L. Jacques, Pierre Vandergheynst

Research output: Contribution to journalArticlepeer-review

83 Citations (Scopus)

Abstract

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 stereo-graphic 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.

Original languageEnglish
Pages (from-to)15-28
Number of pages14
JournalAstrophysical Journal
Volume632
Issue number1
DOIs
Publication statusPublished - 10 Oct 2005

Keywords

  • Cosmic microwave background
  • Methods: data analysis

ASJC Scopus subject areas

  • Space and Planetary Science

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