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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 3/2016

28.01.2016

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

verfasst von: Yifei Lou, Penghang Yin, Jack Xin

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2016

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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.
Vorheriger Artikel Jacobian-Free Implicit Inner-Iteration Preconditioner for Nonlinear Least Squares Problems
Nächster Artikel Notes on RKDG Methods for Shallow-Water Equations in Canal Networks

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Fußnoten
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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Metadaten
Titel
Point Source Super-resolution Via Non-convex Based Methods
verfasst von
Yifei Lou
Penghang Yin
Jack Xin
Publikationsdatum
28.01.2016
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 3/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-016-0169-x

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