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.

Super-Resolution Radar

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.