Sparse image reconstruction on the sphere: analysis and synthesis

TL;DR

This paper introduces advanced sparse regularisation techniques for solving ill-posed inverse problems on the sphere, improving image reconstruction fidelity by optimizing sampling schemes and wavelet sparsity, with applications to astrophysical data.

Contribution

It develops new methods for sparse regularisation on the sphere, analyzing analysis and synthesis approaches with various sampling schemes, and demonstrates improved reconstruction in astrophysical imaging.

Findings

Efficient sampling schemes enhance reconstruction quality.

Sparse regularisation improves denoising and inpainting results.

Application to Planck data reveals better Galactic dust structure extraction.

Abstract

We develop techniques to solve ill-posed inverse problems on the sphere by sparse regularisation, exploiting sparsity in both axisymmetric and directional scale-discretised wavelet space. Denoising, inpainting, and deconvolution problems, and combinations thereof, are considered as examples. Inverse problems are solved in both the analysis and synthesis settings, with a number of different sampling schemes. The most effective approach is that with the most restricted solution-space, which depends on the interplay between the adopted sampling scheme, the selection of the analysis/synthesis problem, and any weighting of the l1 norm appearing in the regularisation problem. More efficient sampling schemes on the sphere improve reconstruction fidelity by restricting the solution-space and also by improving sparsity in wavelet space. We apply the technique to denoise Planck 353 GHz observations, improving the ability to extract the structure of Galactic dust emission, which is important for studying Galactic magnetism.