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

A new sampling theorem on the sphere has been developed recently, reducing the number of samples required to represent a band-limited signal by a factor of two for equiangular sampling schemes. For signals sparse in a spatially localised measure, such as in a wavelet basis, overcomplete dictionary, or in the magnitude of their gradient, for example, a reduction in the number of samples required to represent a band-limited signal has important implications for sparse image reconstruction on the sphere. A more efficient sampling of the sphere improves the fidelity of sparse image reconstruction through both the dimensionality and spatial 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, which relies on a sampling theorem to define a discrete TV norm on the sphere. Solving these problems is computationally challenging; hence we develop fast methods for this purpose. Numerical simulations are performed, verifying the enhanced fidelity of sparse image reconstruction due to the more efficient sampling of the sphere provided by the new sampling theorem.