Towards a Mathematical Theory of Super-Resolution

Emmanuel Candes
Stanford University
Applied and Computational Mathematics

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 and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-off F. 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. We also show that the theory and methods are robust to noise. We introduce a framework for understanding stability and present 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. We shall present numerical examples and discuss ongoing work applying these ideas in the area of single molecule imaging.

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