Noise analysis of regularized EM for SPECT reconstruction

The ability to theoretically model the propagation of photon noise through tomographic reconstruction algorithms is crucial in evaluating reconstructed image quality as a function of parameters of the algorithm. Here, the authors show the theoretical expressions for the propagation of Poisson noise through tomographic SPECT reconstructions using regularized EM algorithms with independent Gamma and multivariate Gaussian priors. The authors' analysis extends the work in H.H. Barrett et al., Phys. Med. Biol., vol. 39, p. 833-46 (1994), in which judicious linearizations were used to enable the propagation of a mean image and covariance matrix from one iteration to the next for the (unregularized) EM algorithm. To validate their theoretical analyses, the authors use a methodology in D.W. Wilson et al., Phys. Med. Biol., vol. 39, p. 847-71 (1994) to compare the results of theoretical calculations to Monte Carlo simulations. The authors also demonstrate an application of the theory to the calculation of an optimal smoothing parameter for a regularized reconstruction. The smoothing parameter is optimal in the context of a quantitation task, defined as the minimization of the expected mean-square error of an estimated number of counts in a hot lesion region. The authors' results thus demonstrate how the theory can be applied to a problem of potential practical use.