Towards a mathematical theory of super-resolution

Emmanuel J. Candès, Carlos Fernandez-Granda

Research output: Contribution to journalArticle

Abstract

This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object-the high end of its spectrum-from coarse scale information only-from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff fc. We show that one can super-resolve these point sources with infinite precision-i.e., recover the exact locations and amplitudes-by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/fc. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary.

Original languageEnglish (US)
Pages (from-to)906-956
Number of pages51
JournalCommunications on Pure and Applied Mathematics
Volume67
Issue number6
DOIs
StatePublished - 2014

Fingerprint

Super-resolution
Point Source
Convex optimization
Cutoff frequency
Piecewise Smooth Functions
Unknown
Semidefinite Program
Convex Optimization
One Dimension
Higher Dimensions
Discontinuity
Resolve
Vary
Optimization Problem
Object
Model

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Towards a mathematical theory of super-resolution. / Candès, Emmanuel J.; Fernandez-Granda, Carlos.

In: Communications on Pure and Applied Mathematics, Vol. 67, No. 6, 2014, p. 906-956.

Research output: Contribution to journalArticle

@article{1704514564014c418bc934842f9d71f3,
title = "Towards a mathematical theory of super-resolution",
abstract = "This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object-the high end of its spectrum-from coarse scale information only-from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff fc. We show that one can super-resolve these point sources with infinite precision-i.e., recover the exact locations and amplitudes-by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/fc. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary.",
author = "Cand{\`e}s, {Emmanuel J.} and Carlos Fernandez-Granda",
year = "2014",
doi = "10.1002/cpa.21455",
language = "English (US)",
volume = "67",
pages = "906--956",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "6",

}

TY - JOUR

T1 - Towards a mathematical theory of super-resolution

AU - Candès, Emmanuel J.

AU - Fernandez-Granda, Carlos

PY - 2014

Y1 - 2014

N2 - This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object-the high end of its spectrum-from coarse scale information only-from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff fc. We show that one can super-resolve these point sources with infinite precision-i.e., recover the exact locations and amplitudes-by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/fc. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary.

AB - This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object-the high end of its spectrum-from coarse scale information only-from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff fc. We show that one can super-resolve these point sources with infinite precision-i.e., recover the exact locations and amplitudes-by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/fc. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary.

UR - http://www.scopus.com/inward/record.url?scp=84896651071&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896651071&partnerID=8YFLogxK

U2 - 10.1002/cpa.21455

DO - 10.1002/cpa.21455

M3 - Article

VL - 67

SP - 906

EP - 956

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 6

ER -