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Towards a Mathematical Theory of Super-resolution
Communications on pure and applied mathematics, 2014-06, Vol.67 (6), p.906-956
Candès, Emmanuel J.
Fernandez-Granda, Carlos
2014
Details
Autor(en) / Beteiligte
Candès, Emmanuel J.
Fernandez-Granda, Carlos
Titel
Towards a Mathematical Theory of Super-resolution
Ist Teil von
Communications on pure and applied mathematics, 2014-06, Vol.67 (6), p.906-956
Ort / Verlag
New York: Blackwell Publishing Ltd
Erscheinungsjahr
2014
Link zum Volltext
Quelle
Wiley Blackwell Single Titles
Beschreibungen/Notizen
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. © 2014 Wiley Periodicals, Inc.
Sprache
Englisch
Identifikatoren
ISSN: 0010-3640
eISSN: 1097-0312
DOI: 10.1002/cpa.21455
Titel-ID: cdi_proquest_journals_1516511575
Format
–
Schlagworte
Mathematical problems
,
Optimization
,
Theory
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