Motion Correction and Synchronization

Abstract

Motion Correction of 2D images is a crucial step in the process of reconstructing the shape of 3D molecules from their 2D image projections, which is the aim of Cryo-Electron Microscopy. In this paper, we propose a new method for Motion Correction that involves the use of invariant features of a shifted, noisy signal. We examine alternate methods involving Angular Synchronization and Least-Squares and discuss their relative strengths and weaknesses. The performances of the invariant features and Angular Synchronization Methods on various one-dimensional signals are compared, and we indicate instances in which the former is better. Motivated by the Motion Correction Problem, the last chapter of this paper investigates the Synchronization Problem in more generality. We were interested in the problem of determining the tightness of the performance bound of synchronization on a graph. The tightness of this bound is determined by a property of the adjacency matrix of the graph that we call the Sign Coherence. Investigating the properties of the Sign Coherence leads us to the Gershgorin Circle Theorem and we discover some surprising symmetries between the Gershgorin inequality and the performance bound on synchronization.