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Information Center for Mathematical Science

PAC

Towards a Mathematical Theory of Super-Resolution
Author Emmanuel J. Candes (Stanford University)
Homepage Url http://www-stat.stanford.edu/~candes/publications.html
Coauthors Carlos Fernandez-Granday
Abstract This paper develops a mathematical theory of super-resolution. Broadly speaking, superresolution is the problem of recovering the ne 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-o fc. We show that one can super-resolve these point sources with in nite precision|i.e. recover the exact locations and amplitudes|by solving a simple convex optimization problem, which can essentially be reformulated as a semide nite 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 in nite 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.
Abstract Url http://www-stat.stanford.edu/~candes/papers/super-res.pdf