Exact reconstruction with directional wavelets on the sphere

TL;DR

This paper introduces a new formalism for analyzing and exactly reconstructing band-limited signals on the sphere using directional wavelets, with applications in astrophysics, notably for cosmic microwave background data analysis.

Contribution

It develops a novel harmonic-space dilation method and a scale discretized wavelet formalism enabling exact reconstruction of signals on the sphere.

Findings

Exact reconstruction of band-limited signals achieved.

Implementation of an efficient multi-resolution algorithm.

Potential applications in astrophysics for detecting localized features.

Abstract

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.