Sparse image reconstruction on the sphere: implications of a new sampling theorem

Abstract

We study the impact of sampling theorems on the fidelity of sparse image reconstruction on the sphere. We discuss how a reduction in the number of samples required to represent all information content of a band-limited signal acts to improve the fidelity of sparse image reconstruction, through both the dimensionality and sparsity of signals. To demonstrate this result we consider a simple inpainting problem on the sphere and consider images sparse in the magnitude of their gradient. We develop a framework for total variation (TV) inpainting on the sphere, including fast methods to render the inpainting problem computationally feasible at high-resolution. Recently a new sampling theorem on the sphere was developed, reducing the required number of samples by a factor of two for equiangular sampling schemes. Through numerical simulations we verify the enhanced fidelity of sparse image reconstruction due to the more efficient sampling of the sphere provided by the new sampling theorem.

11 pages, 5 figures

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• Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results
• Spatiospectral concentration on a sphere
• A novel sampling theorem on the sphere
• Sparse recovery for spherical harmonic expansions
• Compressed sensing for wide-field radio interferometric imaging
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• Differentiable and accelerated spherical harmonic and Wigner transforms
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• The Stochastic Properties of ${\ell }^1$-Regularized Spherical Gaussian Fields
• Sparse Inpainting and Isotropy
• Slepian Scale-Discretised Wavelets on the Sphere
• Sparse image reconstruction on the sphere: a general approach with uncertainty quantification
• Accelerating spherical harmonic transforms for a large number of sky maps
• Bayesian variational regularization on the ball
• Super-Resolution on the Two-Dimensional Unit Sphere
• Exact recovery of Dirac ensembles from the projection onto spaces of spherical harmonics