### Abstract

In cryo-electron microscopy, the data is comprised of noisy 2-D projection images of the 3-D electron scattering intensity of the object where the orientation of the projections is unknown. Often, the images show randomly selected objects from a mixture of different types of objects. Objects of different type may be unrelated, e.g., different species of virus, or related, e.g., different conformations of the same species of virus. Due to the low SNR and the 2-D nature of the data, it is challenging to determine the type of the object shown in an individual image. A statistical model and maximum likelihood estimator that computes simultaneous 3-D reconstruction and labels using an expectation maximization algorithm exists but requires extensive computation due to the numerical evaluation of 3-D or 5-D integrations of a square matrix of dimension equal to the number of degrees of freedom in the 3-D reconstruction. By exploiting the geometry of rotations in 3-D, the estimation problem can be transformed so that the inner-most numerical integral has a scalar rather than a matrix integrand. This leads to a dramatic reduction in computation, especially as the number of degrees of freedom in the 3-D reconstruction increases. Numerical examples of the 3-D reconstructions are provided based on synthetic and experimental images where the objects are small spherical viruses.

Original language | English (US) |
---|---|

Pages (from-to) | 2865-2878 |

Number of pages | 14 |

Journal | IEEE Transactions on Image Processing |

Volume | 16 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2007 |

Externally published | Yes |

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### Keywords

- 3-D signal reconstruction
- Cryo-electron microscopy (cryo EM)
- Expectation maximization algorithm
- Maximum likelihood estimation
- Structural biology
- Tomography
- Virology
- Virus

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Computer Graphics and Computer-Aided Design
- Software
- Theoretical Computer Science
- Computational Theory and Mathematics
- Computer Vision and Pattern Recognition

### Cite this

*IEEE Transactions on Image Processing*,

*16*(12), 2865-2878. https://doi.org/10.1109/TIP.2007.908298

**Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses.** / Lee, Junghoon; Doerschuk, Peter C.; Johnson, John E.

Research output: Contribution to journal › Article

*IEEE Transactions on Image Processing*, vol. 16, no. 12, pp. 2865-2878. https://doi.org/10.1109/TIP.2007.908298

}

TY - JOUR

T1 - Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses

AU - Lee, Junghoon

AU - Doerschuk, Peter C.

AU - Johnson, John E.

PY - 2007/12

Y1 - 2007/12

N2 - In cryo-electron microscopy, the data is comprised of noisy 2-D projection images of the 3-D electron scattering intensity of the object where the orientation of the projections is unknown. Often, the images show randomly selected objects from a mixture of different types of objects. Objects of different type may be unrelated, e.g., different species of virus, or related, e.g., different conformations of the same species of virus. Due to the low SNR and the 2-D nature of the data, it is challenging to determine the type of the object shown in an individual image. A statistical model and maximum likelihood estimator that computes simultaneous 3-D reconstruction and labels using an expectation maximization algorithm exists but requires extensive computation due to the numerical evaluation of 3-D or 5-D integrations of a square matrix of dimension equal to the number of degrees of freedom in the 3-D reconstruction. By exploiting the geometry of rotations in 3-D, the estimation problem can be transformed so that the inner-most numerical integral has a scalar rather than a matrix integrand. This leads to a dramatic reduction in computation, especially as the number of degrees of freedom in the 3-D reconstruction increases. Numerical examples of the 3-D reconstructions are provided based on synthetic and experimental images where the objects are small spherical viruses.

AB - In cryo-electron microscopy, the data is comprised of noisy 2-D projection images of the 3-D electron scattering intensity of the object where the orientation of the projections is unknown. Often, the images show randomly selected objects from a mixture of different types of objects. Objects of different type may be unrelated, e.g., different species of virus, or related, e.g., different conformations of the same species of virus. Due to the low SNR and the 2-D nature of the data, it is challenging to determine the type of the object shown in an individual image. A statistical model and maximum likelihood estimator that computes simultaneous 3-D reconstruction and labels using an expectation maximization algorithm exists but requires extensive computation due to the numerical evaluation of 3-D or 5-D integrations of a square matrix of dimension equal to the number of degrees of freedom in the 3-D reconstruction. By exploiting the geometry of rotations in 3-D, the estimation problem can be transformed so that the inner-most numerical integral has a scalar rather than a matrix integrand. This leads to a dramatic reduction in computation, especially as the number of degrees of freedom in the 3-D reconstruction increases. Numerical examples of the 3-D reconstructions are provided based on synthetic and experimental images where the objects are small spherical viruses.

KW - 3-D signal reconstruction

KW - Cryo-electron microscopy (cryo EM)

KW - Expectation maximization algorithm

KW - Maximum likelihood estimation

KW - Structural biology

KW - Tomography

KW - Virology

KW - Virus

UR - http://www.scopus.com/inward/record.url?scp=36749099441&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36749099441&partnerID=8YFLogxK

U2 - 10.1109/TIP.2007.908298

DO - 10.1109/TIP.2007.908298

M3 - Article

VL - 16

SP - 2865

EP - 2878

JO - IEEE Transactions on Image Processing

JF - IEEE Transactions on Image Processing

SN - 1057-7149

IS - 12

ER -