## Super-Resolution of Positive SourcesVeniamin Morgenshtern ## AbstractThe resolution of all microscopes is limited by diffraction. The observed signal 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. We show that the problem can be solved using convex optimization in a stable fashion. The stability of reconstruction depends on Rayleigh-regularity of the support of the signal, 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 also give a brief summary on the ongoing project, developed in collaboration with the group of Prof. W.E. Moerner, where we use the theoretical ideas to improve microscopes. ## BioVeniamin Morgenshtern is a Postdoctoral researcher in the Statistics Department at Stanford University, where he works with Emmanuel Candès. His research interests are in Mathematical Signal Processing and in Information Theory. Veniamin studied mathematics and computer science at Saint-Petersburg State University, Russia, graduating with the Dipl. Math. degree with honors in 2004. He then joined the Communication Technology Laboratory at ETH Zurich, Switzerland, as a research assistant. His advisor at ETH was Helmut Bölcskei. In 2007 Veniamin was a visiting researcher at the University of Illinois at Urbana-Champaign, US. He graduated from ETH Zurich in 2010, receiving the Dr. Sc. degree. Veniamin's Ph.D. thesis was awarded with an ETH Medal. From 2010 to 2012, Veniamin was a postdoctoral researcher at ETH Zurich. In 2012, he received a two-year Fellowship for Advanced Researchers by the Swiss National Science Foundation. In 2015, the team that Veniamin led received a Second Prize in Thomson Reuters Eikon Text Tagging Challenge (Innocentive 9933333), a machine learning and natural language processing competition. |