Spherical 3D Isotropic Wavelets
  Future cosmological surveys will provide 3D large scale structure maps with
large sky coverage, for which a 3D Spherical Fourier-Bessel (SFB) analysis in
spherical coordinates is natural. Wavelets are particularly well-suited to the
analysis and denoising of cosmological data, but a spherical 3D isotropic
wavelet transform does not currently exist to analyse spherical 3D data. The
aim of this paper is to present a new formalism for a spherical 3D isotropic
wavelet, i.e. one based on the SFB decomposition of a 3D field and accompany
the formalism with a public code to perform wavelet transforms. We describe a
new 3D isotropic spherical wavelet decomposition based on the undecimated
wavelet transform (UWT) described in Starck et al. 2006. We also present a new
fast Discrete Spherical Fourier-Bessel Transform (DSFBT) based on both a
discrete Bessel Transform and the HEALPIX angular pixelisation scheme. We test
the 3D wavelet transform and as a toy-application, apply a denoising algorithm
in wavelet space to the Virgo large box cosmological simulations and find we
can successfully remove noise without much loss to the large scale structure.
We have described a new spherical 3D isotropic wavelet transform, ideally
suited to analyse and denoise future 3D spherical cosmological surveys, which
uses a novel Discrete Spherical Fourier-Bessel Transform. We illustrate its
potential use for denoising using a toy model. All the algorithms presented in
this paper are available for download as a public code called MRS3D at
http://jstarck.free.fr/mrs3d.html
