Invariant Theory and Imaging

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

This project aims to investigate the mathematical foundations of four important methods in imaging and optics: X-ray crystallography, Cryogenic electron microscopy (Cryo-EM), ptychography, and ultra-short pulse detection. Although X-ray crystallography is the prevalent method for determining the 3-dimensional atomic structure of molecules, the mathematical foundations of this theory are not fully developed. Cryo-EM is a recent technique in biological imaging where a sample (typically a protein) is flash-frozen in a liquid solution and then directed with a low-intensity electron beam. Ptychography is a method of obtaining a high-resolution image by scanning across a sample with a moving mask. It can be applied in a number of contexts, including the imaging of live cells. Ultra-short pulse detection is a key ingredient in time-resolved ultrafast phenomena such as chemical reactions. This research will create a unified set of mathematical techniques to understand these methods. The development of these techniques will have the potential for a myriad of applications in areas as diverse as biochemistry, medicine, and engineering. This research will also provide a training opportunity for graduate students. This research will exploit techniques from invariant theory and algebraic geometry to build a mathematical framework for four methods in imaging and optics. X-ray crystallography corresponds to recovering a signal from its power spectrum. This is arguably the most challenging phase-retrieval problem. The first project aims to develop rigorous mathematical criteria to determine when a discrete periodic signal can be recovered from its power spectrum. Because cryo-EM measurements have a very high noise level, constructing a high-resolution image requires massive data. The second project will use techniques from invariant theory to estimate the sample complexity of cryo-EM and related experiments. Ptychographic measurements can be modeled using the short-time Fourier transform (STFT). The third project will obtain information-theoretic bounds on the number of STFT measurements needed for signal recovery. The last project will focus on the mathematical foundations of a novel technique for pulse characterization using multi-mode fibers. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.