Correspondence principle between spherical and Euclidean wavelets
arXiv:astro-ph/0502486 · doi:10.1086/432926
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 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.
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)
References in corpus (19)
- First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters
- Non-Gaussianity from Inflation: Theory and Observations
- Asymmetries in the CMB anisotropy field
- Detection of non-Gaussianity in the WMAP 1-year data using spherical wavelets
- Testing the cosmological principle of isotropy: local power spectrum estimates of the WMAP data
- Multipole Vectors--a new representation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2<=l<=8
- First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Interpretation of the TT and TE Angular Power Spectrum Peaks
- Measuring Statistical isotropy of the CMB anisotropy
- A high-significance detection of non-Gaussianity in the WMAP 1-year data using directional spherical wavelets
- Wavelets and WMAP non-Gaussianity
- Phase Correlations in Cosmic Microwave Background Temperature Maps
- The N-point correlation functions of the first-year Wilkinson Microwave Anisotropy Probe sky maps
- Statistical Isotropy of the Wilkinson Microwave Anisotropy Probe Data: A Bipolar Power Spectrum Analysis
- Point Source Detection using the Spherical Mexican Hat Wavelet on simulated all-sky Planck maps
- Detection and discrimination of cosmological non-Gaussian signatures by multi-scale methods
- Polynomial Interpretation of Multipole Vectors
- Maximum-entropy image reconstruction using wavelets
- Cosmological parameters and the WMAP data revisited
- Cosmic microwave background constraints on the strong equivalence principle
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