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Related: About this forumTowards a Mathematical Theory of Super-Resolution
http://arxiv.org/abs/1203.5871
Computer Science > Information Theory
Towards a Mathematical Theory of Super-Resolution
Emmanuel Candes, Carlos Fernandez-Granda
(Submitted on 27 Mar 2012 (v1), last revised 9 Jun 2012 (this version, v2))
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 until a frequency cut-off $f_c$. 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 $2/f_c$. 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 {\em super-resolution factor} vary.
Computer Science > Information Theory
Towards a Mathematical Theory of Super-Resolution
Emmanuel Candes, Carlos Fernandez-Granda
(Submitted on 27 Mar 2012 (v1), last revised 9 Jun 2012 (this version, v2))
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 until a frequency cut-off $f_c$. 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 $2/f_c$. 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 {\em super-resolution factor} vary.
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Towards a Mathematical Theory of Super-Resolution (Original Post)
bananas
Sep 2012
OP
struggle4progress
(118,281 posts)1. Thanks! I gotta read this!
TrogL
(32,822 posts)2. I looked at something similar to this in high school
(yes, I was a bit of a nerd)
The accumulated errors became larger than the object I was attempting to resolve.
phantom power
(25,966 posts)3. Possibly wrong, but this seems like a restatement of sampling theorem
In order for a band-limited (i.e., one with a zero power spectrum for frequencies nu>B) baseband (nu>0) signal to be reconstructed fully, it must be sampled at a rate nu>=2B. A signal sampled at nu=2B is said to be Nyquist sampled, and nu=2B is called the Nyquist frequency. No information is lost if a signal is sampled at the Nyquist frequency, and no additional information is gained by sampling faster than this rate.
http://mathworld.wolfram.com/SamplingTheorem.html
http://mathworld.wolfram.com/SamplingTheorem.html