Point Source Super-resolution Via Non-convex L1 Based Methods

• Published in: Journal of scientific computing Vol. 68; no. 3; pp. 1082 - 1100
• Main Authors: Lou, Yifei, Yin, Penghang, Xin, Jack
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2016; Springer Nature B.V
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Fourier transforms; Frequency measurement; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Optimization; Sparsity; Theoretical; Super-resolution; Rayleigh length; Difference of convex algorithm (DCA); Capped
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-016-0169-x

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 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the L 1 certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two L 1 based nonconvex penalties, the difference of L 1 and L 2 norms ( L 1 - 2 ) and capped L 1 (C L 1 ), 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 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.