Stable Super-Resolution of Positive Sources
In this project, we discovered that an efficient convex optimization algorithm is a near-optimal method for super-resolution of positive sources in the presence of noise. Good algorithms for super-resolution of positive sources are central for future improvements in super-resolved fluorescence microscopy, a method that gives researchers the unique ability to image small structures inside the living cell. The importance of super-resolved fluorescence microscopy is now widely recognized and its inventors were awarded the Nobel Prize in Chemistry 2014. Mathematically, our work relies on a new interpolation construction in Fourier analysis and on convex duality.
The method is based on semidefinite programming and allows to increase the resolution of radar beyond its natural limit. To achieve these gains, it is important to use a random (or pseudorandom) probing signal. Mathematically, this work merges ideas from the theory of super-resolution and the theory of compressed sensing with Gabor dictionary.