Representation Theoretic Patterns in Three-Dimensional Cryo-Electron Microscopy II—The Class Averaging Problem

• Published in: Foundations of computational mathematics Vol. 11; no. 5; pp. 589 - 616
• Main Authors: Hadani, Ronny, Singer, Amit
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.10.2011; Springer; Springer Nature B.V
• Subjects: Algebra; Algorithms; Applications of Mathematics; Biomathematics; Classification; Computational mathematics; Computer Science; Economics; Electron microscopy; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Microscopy; Numerical Analysis; Proving; Representations; Three dimensional; United States; Optimization theory; Spectral theory; Differential geometry; 3D cryo-electron microscopy; Representation theory; 20G05; Mathematical biology
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-011-9095-3

In this paper we study the formal algebraic structure underlying the intrinsic classification algorithm, recently introduced in Singer et al. (SIAM J. Imaging Sci. 2011 , accepted), for classifying noisy projection images of similar viewing directions in three-dimensional cryo-electron microscopy (cryo-EM). This preliminary classification is of fundamental importance in determining the three-dimensional structure of macromolecules from cryo-EM images. Inspecting this algebraic structure we obtain a conceptual explanation for the admissibility (correctness) of the algorithm and a proof of its numerical stability. The proof relies on studying the spectral properties of an integral operator of geometric origin on the two-dimensional sphere, called the localized parallel transport operator. Along the way, we continue to develop the representation theoretic set-up for three-dimensional cryo-EM that was initiated in Hadani and Singer (Ann. Math. 2010 , accepted).