Implications for compressed sensing of a new sampling theorem on the sphere

Sampling theorems on the sphere state that all the information of a continuous band-limited signal on the sphere may be contained in a discrete set of samples. For an equiangular sampling of the sphere, the Driscoll & Healy (DH) [1] sampling theorem has become the standard, requiring ∼ 4L samples on the sphere to represent exactly a signal band-limited in its spherical harmonic decomposition at L. Recently, a new sampling theorem on an equiangular grid has been developed by McEwen & Wiaux (MW) [2], requiring only ∼ 2L samples to represent exactly a band-limited signal, thereby redefining Nyquist rate sampling on the sphere. No sampling theorem on the sphere reaches the optimal number of samples suggested by the L dimension of a band-limited signal in harmonic space (although the MW sampling theorem comes closest to this bound). A reduction by a factor of two in the number of samples required to represent a band-limited signal on the sphere between the DH and MW sampling theorems has important implications for compressed sensing. Compressed sensing on the sphere has been studied recently for signals sparse in harmonic space [3], where a discrete grid on the sphere is not required. However, for signals sparse in the spatial domain (or in its gradient) a discrete grid on the sphere is essential. A reduction in the number of samples of the grid required to represent a band-limited signal improves both the dimensionality and sparsity of the signal, which in turn affects the quality of reconstruction. We illustrate the impact of the number of samples of the DH and MW sampling theorems with an inpainting problem, where measurements are made in the spatial domain (as dictated by many applications). A test signal sparse in its gradient is constructed from a binary Earth map, smoothed to give a signal band-limited at L = 32. We first solve the total variation (TV) inpainting problem directly on the sphere: