On the computation of directional scale-discretized wavelet transforms on the sphere

We review scale-discretized wavelets on the sphere, which are directional and allow one to probe oriented structure in data defined on the sphere. Furthermore, scale-discretized wavelets allow in practice the exact synthesis of a signal from its wavelet coefficients. We present exact and efficient algorithms to compute the scale-discretized wavelet transform of band-limited signals on the sphere. These algorithms are implemented in the publicly available S2DW code. We release a new version of S2DW that is parallelized and contains additional code optimizations. Note that scale-discretized wavelets can be viewed as a directional generalization of needlets. Finally, we outline future improvements to the algorithms presented, which can be achieved by exploiting a new sampling theorem on the sphere developed recently by some of the authors.

S2LET: A code to perform fast wavelet analysis on the sphere

We describe S2LET, a fast and robust implementation of the scale-discretised wavelet transform on the sphere. Wavelets are constructed through a tiling of the harmonic line and can be used to probe spatially localised, scale-depended features of signals on the sphere. The scale-discretised wavelet transform was developed previously and reduces to the needlet transform in the axisymmetric case. The reconstruction of a signal from its wavelets coefficients is made exact here through the use of a sampling theorem on the sphere. Moreover, a multiresolution algorithm is presented to capture all information of each wavelet scale in the minimal number of samples on the sphere. In addition S2LET supports the HEALPix pixelisation scheme, in which case the transform is not exact but nevertheless achieves good numerical accuracy. The core routines of S2LET are written in C and have interfaces in Matlab, IDL and Java. Real signals can be written to and read from FITS files and plotted as Mollweide projections. The S2LET code is made publicly available, is extensively documented, and ships with several examples in the four languages supported. At present the code is restricted to axisymmetric wavelets but will be extended to directional, steerable wavelets in a future release.

Compressed sensing reconstruction of a string signal from interferometric observations of the cosmic microwave background

We propose an algorithm for the reconstruction of the signal induced by cosmic strings in the cosmic microwave background (CMB), from radio-interferometric data at arcminute resolution. Radio interferometry provides incomplete and noisy Fourier measurements of the string signal, which exhibits sparse or compressible magnitude of the gradient due to the Kaiser-Stebbins (KS) effect. In this context the versatile framework of compressed sensing naturally applies for solving the corresponding inverse problem. Our algorithm notably takes advantage of a model of the prior statistical distribution of the signal fitted on the basis of realistic simulations. Enhanced performance relative to the standard CLEAN algorithm is demonstrated by simulated observations under noise conditions including primary and secondary CMB anisotropies.

Wavelet domain Bayesian denoising of string signal in the cosmic microwave background

An algorithm is proposed for denoising the signal induced by cosmic strings in the cosmic microwave background (CMB). A Bayesian approach is taken, based on modeling the string signal in the wavelet domain with generalized Gaussian distributions. Good performance of the algorithm is demonstrated by simulated experiments at arcminute resolution under noise conditions including primary and secondary CMB anisotropies, as well as instrumental noise.

Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005). The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation is firstly identified. A scale discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background (CMB) data, such as the imprint of topological defects, in particular cosmic strings, and for their reconstruction after separation from the other signal components.