Thu, 17.01.2019, 16:15

Thu, 17.01.2019, 17:15
Super-Resolution of Positive Sources
Kolloquium Angewandte Mathematik

Referent: Veniamin Morgenshtern (FAU, Lehrstuhl für Multimediakommunikation und Signalverarbeitung)
Veranstalter: Frauke Liers
Raum: H13

Abstract: The resolution of all microscopes is limited by diffraction. The observed data is a convolution of the emitted signal with a low-pass kernel, the point-spread function (PSF) of the microscope. The frequency cut-off of the PSF is inversely proportional to the wavelength of light. Hence, the features of the object that are smaller than the wavelength of light are difficult to observe. In single-molecule microscopy the emitted signal is a collection of point sources, produced by blinking molecules. The goal is to recover the location of these sources with precision that is much higher than the wavelength of light. This leads to the problem of super-resolution of positive sources in the presence of noise. I will show that the problem can be solved by using convex optimization in a stable fashion. The stability of reconstruction depends on Rayleigh-regularity of the signal support, i.e., on how many point sources can occur within an interval of one wavelength. The stability estimate is complemented by a converse result: the performance of the convex algorithm is nearly optimal. I will describe an application, developed in collaboration with the group of Prof. W.E. Moerner, where we use these theoretical ideas to improve data processing in modern microscopes.