Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

Abstract

We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in (1, 2) of the Rayleigh length (physical resolution limit), $$L_1$$L1 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the $$L_1$$L1 certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two $$L_1$$L1 based nonconvex penalties, the difference of $$L_1$$L1 and $$L_2$$L2 norms ($$L_{1-2}$$L1-2) and capped $$L_1$$L1 (C$$L_1$$L1), subject to the measurement constraints. In one and two dimensional numerical SR examples, the local optimal solutions from difference of convex function algorithms outperform the global $$L_1$$L1 solutions near or below Rayleigh length scales either in the accuracy of ground truth recovery or in finding a sparse solution satisfying the constraints more accurately.