Sparse image reconstruction on the sphere: analysis and synthesis

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.

11 pages, 6 Figures

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Cited by in corpus (9)

• Spherical Bayesian mass-mapping with uncertainties: full sky observations on the celestial sphere
• Spatial field reconstruction with INLA: Application to IFU galaxy data
• Differentiable and accelerated spherical harmonic and Wigner transforms
• Wavelet-Based Segmentation on the Sphere
• Posterior sampling for inverse imaging problems on the sphere in seismology and cosmology
• Slepian Scale-Discretised Wavelets on the Sphere
• Sparse image reconstruction on the sphere: a general approach with uncertainty quantification
• Bayesian variational regularization on the ball
• Edge detection with polynomial frames on the sphere