A Convex-programming Framework for Super-resolution a Dissertation Submitted to the Department of Electrical Engineering and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

This thesis proposes a general framework to extract information from low-resolution data, a crucial challenge in applications ranging from microscopy, astronomy and medical imaging to geophysics, signal processing and spectroscopy. Assume that we only have information about the spectrum of a superposition of point sources in the low-frequency band [−flo, flo]. We show that as long as the sources are separated by 2/flo, solving a simple convex program achieves exact recovery, in the sense that the original signal is the unique solution to the optimization problem. This is established by leveraging a novel proof technique which constructs a dual certificate by interpolation. In addition, we describe how to implement the recovery algorithm by solving a finite-dimensional semidefinite program, which locates the support of the signal with infinite precision. The guarantees extend to higher dimensions and other models. They imply, for instance, that we can recover a piecewise-smooth function from bandlimited data by super-resolving the discontinuity points. We also provide an analysis of the stability of our method in the presence of noise. On the one hand, we prove that it is possible to extrapolate the spectrum of the original signal up to a frequency fhi > flo to obtain an approximation error between the higher-resolution reconstruction and the truth that is proportional to the noise level times the square of the super-resolution factor (SRF) fhi/flo. On the other hand, we derive supportdetection guarantees that quantify the precision with which we can determine the location of each individual point source in the signal.