### Abstract

The expectation-maximization (EM) algorithm is an important tool for maximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. The authors present a detailed treatment of the algorithm's statistical properties. The specific application they have in mind is image reconstruction in emission tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistics. They authors show that the probability density function for the grey level at a pixel in the image is well approximated by a log-normal law. An expression is derived for the variance of the grey level and for pixel-to-pixel covariance. The variance increases rapidly with iteration number at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportional to the square of the mean image (though other factors may also depend on the mean image), so a map of the standard deviation resembles the object itself. Thus low-intensity regions of the image tend to have low noise. By contrast, linear reconstruction methods, such as filtered back-projection in tomography, show a much more global noise pattern, with high-intensity regions of the object contributing to noise at rather distant low-intensity regions. The theoretical results of this paper depend on two approximations, but in pt.II the authors demonstrate through Monte Carlo simulation that the approximations are justified over a wide range of conditions in emission tomography. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.

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

Article number | 004 |

Pages (from-to) | 833-846 |

Number of pages | 14 |

Journal | Physics in Medicine and Biology |

Volume | 39 |

Issue number | 5 |

DOIs | |

State | Published - 1994 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Radiological and Ultrasound Technology
- Radiology Nuclear Medicine and imaging
- Physics and Astronomy (miscellaneous)
- Biomedical Engineering

### Cite this

*Physics in Medicine and Biology*,

*39*(5), 833-846. [004]. https://doi.org/10.1088/0031-9155/39/5/004

**Noise properties of the EM algorithm. I. Theory.** / Barrett, H. H.; Wilson, D. W.; Tsui, Benjamin.

Research output: Contribution to journal › Article

*Physics in Medicine and Biology*, vol. 39, no. 5, 004, pp. 833-846. https://doi.org/10.1088/0031-9155/39/5/004

}

TY - JOUR

T1 - Noise properties of the EM algorithm. I. Theory

AU - Barrett, H. H.

AU - Wilson, D. W.

AU - Tsui, Benjamin

PY - 1994

Y1 - 1994

N2 - The expectation-maximization (EM) algorithm is an important tool for maximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. The authors present a detailed treatment of the algorithm's statistical properties. The specific application they have in mind is image reconstruction in emission tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistics. They authors show that the probability density function for the grey level at a pixel in the image is well approximated by a log-normal law. An expression is derived for the variance of the grey level and for pixel-to-pixel covariance. The variance increases rapidly with iteration number at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportional to the square of the mean image (though other factors may also depend on the mean image), so a map of the standard deviation resembles the object itself. Thus low-intensity regions of the image tend to have low noise. By contrast, linear reconstruction methods, such as filtered back-projection in tomography, show a much more global noise pattern, with high-intensity regions of the object contributing to noise at rather distant low-intensity regions. The theoretical results of this paper depend on two approximations, but in pt.II the authors demonstrate through Monte Carlo simulation that the approximations are justified over a wide range of conditions in emission tomography. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.

AB - The expectation-maximization (EM) algorithm is an important tool for maximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. The authors present a detailed treatment of the algorithm's statistical properties. The specific application they have in mind is image reconstruction in emission tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistics. They authors show that the probability density function for the grey level at a pixel in the image is well approximated by a log-normal law. An expression is derived for the variance of the grey level and for pixel-to-pixel covariance. The variance increases rapidly with iteration number at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportional to the square of the mean image (though other factors may also depend on the mean image), so a map of the standard deviation resembles the object itself. Thus low-intensity regions of the image tend to have low noise. By contrast, linear reconstruction methods, such as filtered back-projection in tomography, show a much more global noise pattern, with high-intensity regions of the object contributing to noise at rather distant low-intensity regions. The theoretical results of this paper depend on two approximations, but in pt.II the authors demonstrate through Monte Carlo simulation that the approximations are justified over a wide range of conditions in emission tomography. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.

UR - http://www.scopus.com/inward/record.url?scp=0028361508&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028361508&partnerID=8YFLogxK

U2 - 10.1088/0031-9155/39/5/004

DO - 10.1088/0031-9155/39/5/004

M3 - Article

C2 - 15552088

AN - SCOPUS:0028361508

VL - 39

SP - 833

EP - 846

JO - Physics in Medicine and Biology

JF - Physics in Medicine and Biology

SN - 0031-9155

IS - 5

M1 - 004

ER -