Towards a mathematical theory of super-resolution
Super-resolution is the problem of recovering the fine details of an object from coarse scale information. Suppose we have many point sources at unknown locations and with unknown complex-valued amplitudes and that we only observe Fourier samples of this object up until a frequency cut-off F. In this talk we discuss an algorithm that allows to super-resolve the point sources with infinite precision, i.e. recover the exact locations and amplitudes. The method consists of 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. In addition, we introduce a framework for understanding the stability of our theory and methods, which establishes that super-resolution via convex optimization is robust to noise.
This is joint work with Emmanuel Candes from the Department of Mathematics and Statistics.