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Point source super-resolution via non-convex $$L_1$$ based methods. (English) Zbl 1354.65125
In the super-resolution problem, one has to recover sparse signals $$x\in {\mathbb R}^N$$ consisting only of separated peaks from frequency measurements up to a low frequency cutoff $$f_c$$. Then $$\lambda_c = 1/f_c$$ is called Rayleigh length of $$x$$. If the peak separation $$\Delta$$ of $$x$$ fulfills $$\Delta \geq 2\lambda_c N$$, then such sparse signal $$x$$ can be recovered by $$\ell_1$$ minimization (see [E. J. Candès and C. Fernandez-Granda, Commun. Pure Appl. Math. 67, No. 6, 906–956 (2014; Zbl 1350.94011)]).
In this paper, the authors analyze the so-called minimum separation factor MSF = $$\Delta/(\lambda_c\, N)$$. Thus MSF $$\geq 2$$ guarantees the exact recovery by $$\ell_1$$ minimization. Numerical tests show that $$\ell_1$$ minimization often fails when MSF $$< 1$$. In this case, the authors investigate the recovery of $$x =(x_j)_{j=1}^N$$ by non-convex $$\ell_1$$ based minimization with the regularization term $$R(x) = \| x\|_1 - \| x\|_2$$ or $R(x) = \sum_{j=1}^N \min \{|x_j|, \,\alpha\}$ with some $$\alpha>0$$. The non-convex $$\ell_1$$ based minimization problem is solved via a difference of convex algorithms such that local minimizers are obtained. The authors show several properties of the local minimizers. Numerical experiments for signals and images are examined.

##### MSC:
 65K05 Numerical mathematical programming methods 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 65J22 Numerical solution to inverse problems in abstract spaces 65T40 Numerical methods for trigonometric approximation and interpolation 90C26 Nonconvex programming, global optimization
BLOOMP
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