Super-Resolution and Subwavelength Imaging

The investigator aims to put the super-resolution phenomenon of signal estimation on a quantitative footing, and to explore the consequences of reliable super-resolution in tomographic imaging. The estimation of a signal from the knowledge of its Fourier transform in a band qualifies as super-resolved when the fine structure of the signal is recovered on a length scale smaller than the inverse of the band's width --- the so-called Shannon-Nyquist scaling. It is a poorly understood, yet real and decidedly nonlinear numerical phenomenon. Sparsity has been suggested as a principle according to which super-resolution reliably occurs. However, recent insights from analysis of convex programs indicate that a second principle, sign compatibility, is needed in tandem with sparsity to unlock positive results. Building on this, the investigator seeks quantitative guarantees of a new kind for successful signal recovery up to the ideal resolution level that no method can beat. Such results would generalize the theory of compressed sensing in the regime of very coherent dictionaries, where no restricted isometry-type property can be expected to hold. The investigator also studies algorithms based on translation-invariance such as the matrix pencil method, modified matching pursuits, precorrective filtering, and generalizations to higher dimensions. He explores implications, both positive and negative, for tomographic imaging modalities such as seismic and synthetic aperture radar in the subwavelength regime.

The project aims to design advanced, nonlinear processing methods to deliver higher image resolution and interpretation levels in situations where the data come with poor frequency content, such as in the case of a blurring. A methodical understanding of the possibilities offered by super-resolution could eventually transform the methods by which information is extracted from data in a host of inverse problems, ranging from medical imaging (MRI, fluorescence microscopy) to geophysical imaging (exploration seismology) and defense applications. The project also offers an opportunity for students to acquire strong interdisciplinary training at the intersection of computational mathematics, modern data processing, and tomographic imaging. Graduate curricula typically fall short of covering this evolving material in a satisfactory way, even though there is a great demand in industry for quantitative-minded specialists who can operate as experts on all these fronts. The investigator is involved in several initiatives to introduce this new research area widely, through the organization of graduate summer schools, the administration of an internship program geared toward minorities, and remote teaching via MIT's online educational platforms OCW and EdX.