Fast computation of the covariance of MAP reconstructions of PET images

Jinyi Qi; Richard M. Leahy

doi:10.1117/12.348589

21 May 1999 Fast computation of the covariance of MAP reconstructions of PET images

Jinyi Qi, Richard M. Leahy

Proceedings Volume 3661, Medical Imaging 1999: Image Processing; (1999) https://doi.org/10.1117/12.348589
Event: Medical Imaging '99, 1999, San Diego, CA, United States

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 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.

Citation Download Citation

Jinyi Qi and Richard M. Leahy "Fast computation of the covariance of MAP reconstructions of PET images", Proc. SPIE 3661, Medical Imaging 1999: Image Processing, (21 May 1999); https://doi.org/10.1117/12.348589

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