Reconstruction of positron emission tomography images by using MAP estimation

Positron emission tomography (PET) is a technique that has been developed to study the metabolic activity of the human body (Herman 1980) (Deans 1983) (Barrett 1984). In the last years many algorithms have been developed for reconstructing tomography images. The maximum likelihood expectation maximization algorithm (ML-EM) (Vardi & Shepp 1982) is a very stable nonlinear method and was developed by Shepp and Vardi in 1982. For the first time, one was able to model the reconstruction process by taking into account stochastic properties of the underlying physical model. However, the ML-EM algorithm causes some serious problems in the context of the application considered. It is an iterative procedure and seems to converge rapidly to a stationary point, however, the reconstructed image is distorted by superimposed high frequency noise. This poor performance motivated different modifications of the algorithm. These can be divided into two classes. 1. Stopping the algorithm at a step where the reconstructed image seems to have an acceptable quality (Llacer & Veklerov 1989). Obviously; it is a problem to determine a reasonable stop criteria and to explain the gap between this subjective best reconstruction and the optimal one in the maximum likelihood sense. 2. Modification of the nonlinear algorithm by using a priori information in order to improve the algorithm with Bayesian methods (Levitan & Herman 1987). In this chapter the second method is used, based on an analysis of the ML-EM method.