We propose a general framework to perform statistical estimation from low-resolution data, a crucial challenge in applications ranging from microscopy, astronomy and medical imaging to geophysics, signal processing and spectroscopy. First, we show that solving a simple convex program allows to super-resolve a superposition of point sources from bandlimited measurements with infinite precision. This holds as long as the sources are separated by a distance related to the cut-off frequency of the data. The result extends to higher dimensions and to the super-resolution of piecewise-smooth functions. Then, we provide theoretical guarantees that establish the robustness of our methods to noise in a non-asymptotic regime. Finally, we illustrate the flexibility of the framework by discussing extensions to the demixing of sines and spikes and to super-resolution from multiple measurements.
Carlos Fernandez-Granda is a PhD student in Electrical Engineering at Stanford University. Previously, he received an M.Sc. degree from Ecole Normale Superieure de Cachan and engineering degrees from Universidad Politécnica de Madrid and Ecole des Mines in Paris. His research interests are at the intersection of optimization, high-dimensional statistics and harmonic analysis, with emphasis on applications to computer vision, medical imaging and big data.