### Abstract

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).

Original language | English (US) |
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Pages (from-to) | 589-616 |

Number of pages | 28 |

Journal | Foundations of Computational Mathematics |

Volume | 11 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2011 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics

### Keywords

- 3D cryo-electron microscopy
- Differential geometry
- Mathematical biology
- Optimization theory
- Representation theory
- Spectral theory