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Point Source Super-resolution Via Non-convex \(L_1\) Based Methods

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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), \(L_1\) 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.

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Notes

  1. Denote \(.*\) be entry-wise multiplication, and \(|\cdot |\) be entry-wise absolute value (Note \(\Vert \cdot \Vert _1\) is the standard \(L_1\) norm).

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Acknowledgments

The authors would like to thank C. Fernandez-Granda at New York University, V. Morgenshtern at Stanford University, J.-S. Pang at University of Southern California, and M. Yan at Michigan State University for helpful discussions.

Funding Yifei Lou was partially supported by NSF Grant DMS-1522786. Penghang Yin and Jack Xin were partially supported by NSF Grants DMS-1222507 and DMS-1522383.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, 75080, USA

    Yifei Lou

  2. Department of Mathematics, UC Irvine, Irvine, CA, 92697, USA

    Penghang Yin & Jack Xin

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  1. Yifei Lou
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  2. Penghang Yin
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  3. Jack Xin
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Correspondence to Yifei Lou.

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Lou, Y., Yin, P. & Xin, J. Point Source Super-resolution Via Non-convex \(L_1\) Based Methods. J Sci Comput 68, 1082–1100 (2016). https://doi.org/10.1007/s10915-016-0169-x

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