  We propose a sampling scheme that can perfectly reconstruct a collection of
spikes on the sphere from samples of their lowpass-filtered observations.
Central to our algorithm is a generalization of the annihilating filter method,
a tool widely used in array signal processing and finite-rate-of-innovation
(FRI) sampling. The proposed algorithm can reconstruct $K$ spikes from
$(K+\sqrt{K})^2$ spatial samples. This sampling requirement improves over
previously known FRI sampling schemes on the sphere by a factor of four for
large $K$. We showcase the versatility of the proposed algorithm by applying it
to three different problems: 1) sampling diffusion processes induced by
localized sources on the sphere, 2) shot noise removal, and 3) sound source
localization (SSL) by a spherical microphone array. In particular, we show how
SSL can be reformulated as a spherical sparse sampling problem.
