Sampling Signals With a Finite Rate of Innovation on the Sphere
The state of the art in sampling theory now contains several theorems for signals that are non-bandlimited. For signals on the sphere however, most theorems still require the assumptions of bandlimitedness. In this work we show that a particular class of non-bandlimited signals, which have a finite rate of innovation, can be exactly recovered using a finite number of samples. We consider a sampling scheme where K weighted Diracs are convolved with a kernel on the rotation group. We prove that if the sampling kernel has a bandlimit L = 2K then (2K - 1)(4K - 1) + 1 equiangular samples are sufficient for exact reconstruction. If the samples are uniformly distributed on the sphere, we argue that the signal can be accurately reconstructed using 4K2 samples and validate our claim through numerical simulations. To further reduce the number of samples required, we design an optimal sampling kernel that achieves accurate reconstruction of the signal using only 3K samples, the number of parameters of the weighted Diracs. In addition to weighted Diracs, we show that our results can be extended to sample Diracs integrated along the azimuth. Finally, we consider kernels with antipodal symmetry which are common in applications such as diffusion magnetic resonance imaging.