Super-Resolution Off the Grid

Super-resolution is the problem of recovering a superposition of point sources using bandlimited measurements, which may be corrupted with noise. This signal processing problem arises in numerous imaging problems, ranging from astronomy to biology to spectroscopy, where it is common to take (coarse) Fourier measurements of an object. Of particular interest is in obtaining estimation procedures which are robust to noise, with the following desirable statistical and computational properties: we seek to use coarse Fourier measurements (bounded by some \emph{cutoff frequency}); we hope to take a (quantifiably) small number of measurements; we desire our algorithm to run quickly. Suppose we have $k$ point sources in $d$ dimensions, where the points are separated by at least $\Delta$ from each other (in Euclidean distance). This work provides an algorithm with the following favorable guarantees:1. The algorithm uses Fourier measurements, whose frequencies are bounded by $O(1/\Delta)$ (up to log factors). Previous algorithms require a \emph{cutoff frequency} which may be as large as $\Omega(\sqrt{d}/\Delta)$.2. The number of measurements taken by and the computational complexity of our algorithm are bounded by a polynomial in both the number of points $k$ and the dimension $d$, with \emph{no} dependence on the separation $\Delta$. In contrast, previous algorithms depended inverse polynomially on the minimal separation and exponentially on the dimension for both of these quantities.Our estimation procedure itself is simple: we take random bandlimited measurements (as opposed to taking an exponential number of measurements on the hyper-grid). Furthermore, our analysis and algorithm are elementary (based on concentration bounds of sampling and singular value decomposition).