### Abstract

We develop an approximate theoretical formula for fast computation of the covariance of PET images reconstructed using maximum a posteriori (MAP) estimation. The results assume a Poisson likelihood for the data and a quadratic prior on the image. The covariance for each voxel is computed using 2D FFTs and is a function of a single data dependent parameter. This parameter is computed using a modified backprojection. For a small region of interest (ROI), the correlation can be assumed to be locally stationary so that computation of the variance of an ROI can be performed very rapidly. Previous approximate formulae for the variance of MAP estimators have performed poorly in areas of low activity since they do not account for the non-negativity constraints that are routinely used in MAP algorithms. Here a `truncated Gaussian' model is used to compensate for the effect of the non-negativity constraints. Accuracy of the theoretical expressions is evaluated using both Monte Carlo simulations and a multiple-frame
^{15}O-water brain study. The Monte Carlo studies show that the truncated Gaussian model is effective in compensating for the effect of the non-negativity constraint. These results also show good agreement between Monte Carlo covariances and the theoretical approximations. The
^{15}O-water brain study further confirms the accuracy of the theoretical approximations.

Original language | English (US) |
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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Publisher | Society of Photo-Optical Instrumentation Engineers |

Pages | 344-355 |

Number of pages | 12 |

Volume | 3661 |

Edition | I |

State | Published - 1999 |

Externally published | Yes |

Event | Proceedings of the 1999 Medical Imaging - Image Processing - San Diego, CA, USA Duration: Feb 22 1999 → Feb 25 1999 |

### Other

Other | Proceedings of the 1999 Medical Imaging - Image Processing |
---|---|

City | San Diego, CA, USA |

Period | 2/22/99 → 2/25/99 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(I ed., Vol. 3661, pp. 344-355). Society of Photo-Optical Instrumentation Engineers.

**Fast computation of the covariance of MAP reconstructions of PET images.** / Qi, Jinyi; Leahy, Richard M.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Proceedings of SPIE - The International Society for Optical Engineering.*I edn, vol. 3661, Society of Photo-Optical Instrumentation Engineers, pp. 344-355, Proceedings of the 1999 Medical Imaging - Image Processing, San Diego, CA, USA, 2/22/99.

}

TY - CHAP

T1 - Fast computation of the covariance of MAP reconstructions of PET images

AU - Qi, Jinyi

AU - Leahy, Richard M.

PY - 1999

Y1 - 1999

N2 - We develop an approximate theoretical formula for fast computation of the covariance of PET images reconstructed using maximum a posteriori (MAP) estimation. The results assume a Poisson likelihood for the data and a quadratic prior on the image. The covariance for each voxel is computed using 2D FFTs and is a function of a single data dependent parameter. This parameter is computed using a modified backprojection. For a small region of interest (ROI), the correlation can be assumed to be locally stationary so that computation of the variance of an ROI can be performed very rapidly. Previous approximate formulae for the variance of MAP estimators have performed poorly in areas of low activity since they do not account for the non-negativity constraints that are routinely used in MAP algorithms. Here a `truncated Gaussian' model is used to compensate for the effect of the non-negativity constraints. Accuracy of the theoretical expressions is evaluated using both Monte Carlo simulations and a multiple-frame 15O-water brain study. The Monte Carlo studies show that the truncated Gaussian model is effective in compensating for the effect of the non-negativity constraint. These results also show good agreement between Monte Carlo covariances and the theoretical approximations. The 15O-water brain study further confirms the accuracy of the theoretical approximations.

AB - We develop an approximate theoretical formula for fast computation of the covariance of PET images reconstructed using maximum a posteriori (MAP) estimation. The results assume a Poisson likelihood for the data and a quadratic prior on the image. The covariance for each voxel is computed using 2D FFTs and is a function of a single data dependent parameter. This parameter is computed using a modified backprojection. For a small region of interest (ROI), the correlation can be assumed to be locally stationary so that computation of the variance of an ROI can be performed very rapidly. Previous approximate formulae for the variance of MAP estimators have performed poorly in areas of low activity since they do not account for the non-negativity constraints that are routinely used in MAP algorithms. Here a `truncated Gaussian' model is used to compensate for the effect of the non-negativity constraints. Accuracy of the theoretical expressions is evaluated using both Monte Carlo simulations and a multiple-frame 15O-water brain study. The Monte Carlo studies show that the truncated Gaussian model is effective in compensating for the effect of the non-negativity constraint. These results also show good agreement between Monte Carlo covariances and the theoretical approximations. The 15O-water brain study further confirms the accuracy of the theoretical approximations.

UR - http://www.scopus.com/inward/record.url?scp=0032593853&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032593853&partnerID=8YFLogxK

M3 - Chapter

AN - SCOPUS:0032593853

VL - 3661

SP - 344

EP - 355

BT - Proceedings of SPIE - The International Society for Optical Engineering

PB - Society of Photo-Optical Instrumentation Engineers

ER -