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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 [Peer Reviewed Journal]

2014 Wiley Periodicals, Inc. ;ISSN: 0010-3640 ;EISSN: 1097-0312 ;DOI: 10.1002/cpa.21455 ;CODEN: CPMAMV

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  • Title:
    Towards a Mathematical Theory of Super-resolution
  • Author: Candès, Emmanuel J. ; Fernandez-Granda, Carlos
  • Subjects: Mathematical problems ; Optimization ; Theory
  • Is Part Of: Communications on pure and applied mathematics, 2014-06, Vol.67 (6), p.906-956
  • Description: 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.
  • Publisher: New York: Blackwell Publishing Ltd
  • Language: English
  • Identifier: ISSN: 0010-3640
    EISSN: 1097-0312
    DOI: 10.1002/cpa.21455
    CODEN: CPMAMV
  • Source: Alma/SFX Local Collection
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