1205.1013 (J. D. McEwen et al.)

Sparse signal reconstruction on the sphere: implications of a new sampling theorem    [PDF]

J. D. McEwen, G. Puy, J. -Ph. Thiran, P. Vandergheynst, D. Van De Ville, Y. Wiaux

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 signal reconstruction on the sphere. A more efficient sampling of the sphere improves the fidelity of sparse signal 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 signals sparse in the magnitude of their gradient. We develop a framework for total variation (TV) inpainting on the sphere by making a connection to the underlying continuous signal via a sampling theorem. Numerical simulations are performed, verifying the enhanced fidelity of sparse signal reconstruction due to the more efficient sampling of the sphere provided by the new sampling theorem.

View original: http://arxiv.org/abs/1205.1013