Background

The expectation maximization maximum likelihood method (EMML) discussed in the previous chapter has been the subject of much attention in the medical-imaging literature over the past decade. Statisticians like it because it is based on the well-studied principle of likelihood maximization for parameter estimation. Physicists like it because, unlike its competition, filtered backprojection, it permits the inclusion of sophisticated models of the physical situation. Mathematicians like it because it can be derived from iterative optimization theory. Physicians like it because the images are often better than those produced by other means. No method is perfect, however, and the EMML suffers from sensitivity to noise and slow rate of convergence. Research is ongoing to find faster and less sensitive versions of this algorithm.

Another class of iterative algorithms was introduced into medical imaging by Gordon et al. in [104]. These include the algebraic reconstruction technique (ART) and its multiplicative version, MART. These methods were derived by viewing image reconstruction as solving systems of linear equations, possibly subject to constraints, such as positivity. The simultaneous MART (SMART) [81, 168] is a variant of MART that uses all the data at each step of the iteration.

Although the EMML and SMART algorithms have quite different histories and are not typically considered together, they are closely related [30, 31]. In this chapter we examine these two algorithms in tandem, following [32]. Forging a link between the EMML and SMART led to a better understanding of both of these algorithms and to...