Orthogonal matrix retrieval in cryo-electron microscopy

Tejal Bhamre, Teng Zhang, Amit Singer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

14 Scopus citations


In single particle reconstruction (SPR) from cryo-electron microscopy (EM), the 3D structure of a molecule needs to be determined from its 2D projection images taken at unknown viewing directions. Zvi Kam showed already in 1980 that the autocorrelation function of the 3D molecule over the rotation group SO(3) can be estimated from 2D projection images whose viewing directions are uniformly distributed over the sphere. The autocorrelation function determines the expansion coefficients of the 3D molecule in spherical harmonics up to an orthogonal matrix of size (2l+1)×(2l +1) for each l = 0, 1, 2, · · ·. In this paper we show how techniques for solving the phase retrieval problem in X-ray crystallography can be modified for the cryo-EM setup for retrieving the missing orthogonal matrices. Specifically, we present two new approaches that we term Orthogonal Extension and Orthogonal Replacement, in which the main algorithmic components are the singular value decomposition and semidefinite programming. We demonstrate the utility of these approaches through numerical experiments on simulated data.

Original languageEnglish (US)
Title of host publication2015 IEEE 12th International Symposium on Biomedical Imaging, ISBI 2015
PublisherIEEE Computer Society
Number of pages5
ISBN (Electronic)9781479923748
StatePublished - Jul 21 2015
Event12th IEEE International Symposium on Biomedical Imaging, ISBI 2015 - Brooklyn, United States
Duration: Apr 16 2015Apr 19 2015

Publication series

NameProceedings - International Symposium on Biomedical Imaging
ISSN (Print)1945-7928
ISSN (Electronic)1945-8452


Other12th IEEE International Symposium on Biomedical Imaging, ISBI 2015
Country/TerritoryUnited States

All Science Journal Classification (ASJC) codes

  • Radiology Nuclear Medicine and imaging
  • Biomedical Engineering


  • 3D reconstruction
  • Cryo-electron microscopy
  • ab-initio modelling
  • autocorrelation
  • convex relaxation
  • polar decomposition
  • semidefinite programming
  • single particle analysis
  • spherical harmonics


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