Super-resolution of point sources via convex programming

Research output: Contribution to journalArticle

Abstract

We consider the problem of recovering a signal consisting of a superposition of point sources from lowresolution data with a cutoff frequency fc. If the distance between the sources is under 1/fc, this problem is not well posed in the sense that the low-pass data corresponding to two different signals may be practically the same. We show that minimizing a continuous version of the 1-norm achieves exact recovery as long as the sources are separated by at least 1.26/fc. The proof is based on the construction of a dual certificate for the optimization problem, which can be used to establish that the procedure is stable to noise. Finally, we illustrate the flexibility of our optimization-based framework by describing extensions to the demixing of sines and spikes and to the estimation of point sources that share a common support.

Original languageEnglish (US)
Pages (from-to)251-303
Number of pages53
JournalInformation and Inference
Volume5
Issue number3
DOIs
StatePublished - Jan 1 2016

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Convex optimization
Super-resolution
Convex Programming
Point Source
Cutoff frequency
Certificate
Spike
Superposition
Recovery
Flexibility
Optimization Problem
Norm
Optimization
Framework

Keywords

  • Convex optimization
  • Dual certificates
  • Group sparsity
  • Line-spectra estimation
  • Multiple measurements
  • Overcomplete dictionaries
  • Sparse recovery
  • Super-resolution

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Analysis
  • Applied Mathematics
  • Statistics and Probability
  • Numerical Analysis

Cite this

Super-resolution of point sources via convex programming. / Fernandez-Granda, Carlos.

In: Information and Inference, Vol. 5, No. 3, 01.01.2016, p. 251-303.

Research output: Contribution to journalArticle

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