  We consider simultaneously identifying the membership and locations of point
sources that are convolved with different band-limited point spread functions,
from the observation of their superpositions. This problem arises in
three-dimensional super-resolution single-molecule imaging, neural spike
sorting, multi-user channel identification, among other applications. We
propose a novel algorithm, based on convex programming, and establish its
near-optimal performance guarantee for exact recovery in the noise-free setting
by exploiting the spectral sparsity of the point source models as well as the
incoherence between point spread functions. Furthermore, robustness of the
recovery algorithm in the presence of bounded noise is also established.
Numerical examples are provided to demonstrate the effectiveness of the
proposed approach.
