Blind super-resolution of sparse spike signals

In many applications, the observations can be modeled as a linear combination of a small number of scaled and shifted copies of a bandlimited point spread function, either determined by the nature or designed by the users. Examples include neural spike trains, returns in radar and sonar, images in astronomy and single-molecule microscopy, etc. It is of great interest to resolve the spike signal as accurate as possible from the observation. When the point spread function is assumed unknown, this problem is terribly ill-posed. This paper proposes a convex optimization framework based on minimization of the atomic norm for jointly spectrally-sparse ensembles to simultaneously estimate the point spread function as well as the spike signal with provable performance guarantees, by mildly constraining the point spread function lies in a known low-dimensional subspace with an unknown orientation. Numerical examples are provided to validate the effectiveness of the proposed approach.