Noise analysis of MAP - EM algorithms for emission tomography

Abstract

The ability to theoretically model the propagation of photon noise through PET and SPECT tomographic reconstruction algorithms is crucial in evaluating the reconstructed image quality as a function of parameters of the algorithm. In a previous approach for the important case of the iterative ML-EM (maximum-likelihood-expectation-maximization) algorithm, judicious linearizations were used to model theoretically the propagation of a mean image and a covariance matrix from one iteration to the next. Our analysis extends this approach to the case of MAP (maximum a posteriori)-EM algorithms, where the EM approach incorporates prior terms. We analyse in detail two cases: a MAP-EM algorithm incorporating an independent gamma prior, and a one-step-late (OSL) version of a MAP-EM algorithm incorporating a multivariate Gaussian prior, for which familiar smoothing priors are special cases. To validate our theoretical analyses, we use a Monte Carlo methodology to compare, at each iteration, theoretical estimates of mean and covariance with sample estimates, and show that the theory works well in practical situations where the noise and bias in the reconstructed images do not assume extreme values.

A unified noise analysis for iterative image estimation

“…Similar to , Wang and Gindi, 1997, Soares et al, 2000, we assume that the noise level inx k is low and hence the first-order Taylor series approximation can be used. Thus, we have…”

“…They are valid for a wide range of preconditioned gradient-type algorithms. Using the proper likelihood and prior functions, most results in , Wang and Gindi, 1997, Soares et al, 2000 can be directly obtained from these expressions (see Section 4). One advantage of our method is that the theoretical analysis does not require an explicit expression of the preconditioner and hence is applicable to algorithms that use line searches.…”

“…Barrett et al derived approximate formulae for the mean and covariance of the maximum likelihood (ML) expectation maximization (EM) [Wang and Gindi, 1997] and is examined. An application to transmission tomography is shown in Section 5.…”

Theoretical Evaluation of the Detectability of Random Lesions in Bayesian Emission Reconstruction

“…Barrett et al [6] derived approximate formulae for the mean and covariance of the maximum likelihood (ML) expectation maximization (EM) reconstruction as a function of the iteration number. The same approach was extended to the maximum a posteriori (MAP) EM algorithms by Wang and Gindi [7] and most recently to block iterative algorithms by Soares et al [8]. Using the results in [6] with numerical observer models, Abbey et al [9] have studied lesion detectability in ML EM reconstruction.…”

Optimization of Bayesian emission tomographic reconstruction for region-of-interest quantitation

“…Recently, progress has been made in understanding the nonlinear properties of statistical reconstruction methods. [5][6][7][8][9][10][11] Barrett et al 5 derived approximate formulae for the mean and covariance of the maximum likelihood (ML) expectation maximization (EM) reconstruction as a function of the iteration number. The same approach was extended to maximum a posteriori (MAP) EM algorithms by Wang and Gindi 8 and most recently to block iterative algorithms by Soares et al 12 This iteration-based approach is attractive for methods that are terminated before convergence, as is common practice for the EM algorithm and its ordered-subsets variants.…”