February Fourier Talks 2016

Single particle cryo-electron microscopy (EM) recently joined X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy as a high-resolution structural method for biological macromolecules. In single particle cryo-EM, the 3-D structure needs to be determined from many noisy 2-D projection images of individual, ideally identical frozen-hydrated macromolecules whose orientations and positions are random and unknown (i.e., random X-ray transform). This lecture will explore algorithms for estimating the unknown pose parameters. The main focus will be on semidefinite programming relaxations that are based on the Fourier transform over the group SO(3). Such semidefinite programs can be viewed as extensions to existing approximation algorithms to max-cut and unique games, two fundamental problems in theoretical computer science. The approach is quite general and can be used to handle other groups of transformations that arise in other applications in signal processing, image analysis, computer vision, and computer graphics.