## Three-dimensional reconstruction of the statistics of heterogeneous objects from a collection of one projection image of each object |

JOSA A, Vol. 29, Issue 6, pp. 959-970 (2012)

http://dx.doi.org/10.1364/JOSAA.29.000959

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### Abstract

An estimation problem for statistical reconstruction of heterogeneous three-dimensional objects from two-dimensional tomographic data (single-particle cryoelectron microscope images) is posed as the problem of estimating class probabilities, means, and covariances for a Gaussian mixture where both the mean and covariance are stochastically structured. Both discrete (i.e., classes) and continuous heterogeneity is included. A maximum likelihood solution computed by a generalized expectation-maximization algorithm is presented and demonstrated on experimental images of Flock House Virus.

© 2012 Optical Society of America

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.3190) Image processing : Inverse problems

(100.6890) Image processing : Three-dimensional image processing

(100.6950) Image processing : Tomographic image processing

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: August 30, 2011

Manuscript Accepted: December 9, 2011

Published: May 23, 2012

**Virtual Issues**

Vol. 7, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Yili Zheng, Qiu Wang, and Peter C. Doerschuk, "Three-dimensional reconstruction of the statistics of heterogeneous objects from a collection of one projection image of each object," J. Opt. Soc. Am. A **29**, 959-970 (2012)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-6-959

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### References

- J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies (Academic, 1996).
- S. H. W. Scheres, H. Gao, M. Valle, G. T. Herman, P. P. B. Eggermont, J. Frank, and J.-M. Carazo, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization,” Nat. Methods 4, 27–29(2007). [CrossRef]
- J. Lee, P. C. Doerschuk, and J. E. Johnson, “Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses,” IEEE Trans. Image Process 16, 2865–2878 (2007). [CrossRef]
- Z. Yin, Y. Zheng, P. C. Doerschuk, P. Natarajan, and J. E. Johnson, “A statistical approach to computer processing of cryo electron microscope images: Virion classification and 3-D reconstruction,” J. Struct. Biol. 144, 24–50(2003). [CrossRef]
- P. C. Doerschuk and J. E. Johnson, “Ab initio reconstruction and experimental design for cryo electron microscopy,” IEEE Trans. Inf. Theory 46, 1714–1729 (2000). [CrossRef]
- J. Tang, J. M. Johnson, K. A. Dryden, M. J. Young, A. Zlotnick, and J. E. Johnson, “The role of subunit hinges and molecular “switches’’ in the control of viral capsid polymorphism,” J. Struct. Biol. 154, 59–67 (2006). [CrossRef]
- Y. Zheng, “Novel statistical models and a high-performance computing toolkit for the solution of cryo electron microscopy inverse problems in viral structural biology,” Ph.D. dissertation (School of Electrical and Computer Engineering, Purdue University, West Lafayette, Ind., USA, 2008).
- Q. Wang, is preparing a Ph.D. dissertation called “Maximum likelihood reconstruction of heterogeneous 3-D objects from 2-D projections of unknown orientation and application to electron microscope images of viruses” (School of Electrical and Computer Engineering, Cornell University, Ithaca, N.Y., USA; expected completion date 2013).
- A. Berge and A. H. Schistad Solberg, “Structured Gaussian components for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens. 44, 3386–3396 (2006). [CrossRef]
- K. C. Sim and M. J. F. Gales, “Minimum phone error training of precision matrix models,” IEEE Trans. Audio Speech Lang. Process. 14, 882–889 (2006). [CrossRef]
- Y. Tian, J.-L. Zhou, H. Lin, and H. Jiang, “Tree-based covariance modeling of hidden Markov models,” IEEE Trans. Audio Speech Lang. Process. 14, 2134–2146 (2006). [CrossRef]
- S. Dharanipragada and K. Visweswariah, “Gaussian mixture models with covariances or precisions in shared multiple subspaces,” IEEE Trans. Audio Speech Lang. Process. 14, 1255–1266 (2006). [CrossRef]
- H. Snoussi and A. Mohammad-Djafari, “Estimation of structured Gaussian mixtures: the inverse EM algorithm,” IEEE Trans. Signal Process. 55, 3185–3191 (2007). [CrossRef]
- G. Casella and R. L. Berger, Statistical Inference, 2nd ed.(Duxbury2002).
- R. A. Redner, and H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM Rev. 26, 195–239 (1984). [CrossRef]
- G. J. McLachlan and T. Krishnan, The EM Algorithm and Extensions (Wiley-Interscience, 1997).
- J. A. Bilmes, “A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models,” Tech. Rep. TR-97-021 (Department of Electrical Engineering and Computer Science, University of California at Berkeley, 1998).
- B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, 1979).
- M. van Heel, “Similarity measures between images,” Ultramicroscopy 21, 95–100 (1987). [CrossRef]
- G. Harauz and M. van Heel, “Exact filters for general geometry three dimensional reconstruction,” Optik 73, 146–156(1986).
- T. S. Baker, N. H. Olson, and S. D. Fuller, “Adding the third dimension to virus life cycles: three-dimensional reconstruction of icosahedral viruses from cryo-electron micrographs,” Microbiol. Molec. Biol. Rev. 63, 862–922 (1999).
- B. Efron and D. V. Hinkley, “Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information,” Biometrika 65, 457–482 (1978). [CrossRef]
- C. J. Prust, P. C. Doerschuk, G. C. Lander, and J. E. Johnson, “Ab initio maximum likelihood reconstruction from cryo electron microscopy images of an infectious virion of the tailed bacteriophage P22 and maximum likelihood versions of Fourier Shell Correlation appropriate for measuring resolution of spherical or cylindrical objects,” J. Struct. Biol. 167, 185–199 (2009). [CrossRef]
- A. J. Fisher and J. E. Johnson, “Ordered duplex RNA controls capsid architecture in an icosahedral animal virus,” Nature 361, 176–179 (1993). [CrossRef]
- J. Lanman, J. Crum, T. J. Deerinck, G. M. Gaietta, A. Schneemann, G. E. Sosinsky, M. H. Ellisman, and J. E. Johnson, “Visualizing flock house virus infection in Drosophila cells with correlated fluorescence and electron microscopy,” J. Struct. Biol. 161, 439–446 (2008). [CrossRef]
- URL, “Flock House Virus (FHV) web page,” http://viperdb.scripps.edu/info_page.php?VDB=2q25.
- Y. Zheng and P. C. Doerschuk, “Explicit computation of orthonormal symmetrized harmonics with application to the identity representation of the icosahedral group,” SIAM J. Math. Anal. 32, 538–554 (2000). [CrossRef]
- R. H. Cheng, V. S. Reddy, N. H. Olson, A. J. Fisher, T. S. Baker, and J. E. Johnson, “Functional implications of quasi-equivalence in a T=3 icosahedral animal virus established by cryo-electron microscopy and x-ray crystallography,” Structure 2, 271–282 (1994). [CrossRef]
- D. Bubeck, D. J. Filman, and J. M. Hogle, “Cryo-electron microscopy reconstruction of a poliovirus-receptor-membrane complex,” Nat. Struct. Mol. Biol. 12, 615–618 (2005). [CrossRef]
- J. M. Hogle, “Poliovirus cell entry: common structural themes in viral cell entry pathways,” Annu. Rev. Microbiol. 56, 677–702 (2002). [CrossRef]
- M. Tihova, K. A. Dryden, T. L. Le, S. C. Harvey, J. E. Johnson, M. Yeager, and A. Schneemann, “Nodavirus coat protein imposes dodecahedral RNA structure independent of nucleotide sequence and length,” J. Virol. 78, 2897–2905(2004). [CrossRef]
- E. F. Pettersen, T. D. Goddard, C. C. Huang, G. S. Couch, D. M. Greenblatt, E. C. Meng, and T. E. Ferrin, “UCSF Chimera–A visualization system for exploratory research and analysis,” J. Comput. Chem. 25, 1605–1612 (2004). [CrossRef]
- URL http://www.mathworks.com/ .

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