The EM algorithm for solving the linear system *Af = g* reads

Multiplications and divisions in this formula are understood component wise. It is derived from a statistical model of image reconstruction, see Shepp and Vardi (1982). The purpose is to compute a minimizer of the likelihood function

The convergence of (4.4) to a minimizer of (4.5) follows from the general EM theory in Dempster et al. (1977). Not that (4.4) preserves positivity if .

The pure EM method (4.4) produces unpleasant images which contain too much
detail. Several remedies have been suggested. An obvious one is to stop the iteration early, typically after 12 steps. One also can perform a smoothing step
after each iteration (EMS-algorithm of Silverman et al. (1990)). A theoretically more
satisfying method is to add a penalty term - *B(f)* to (4.5), i.e. to minimize

*-B(f)* may be interpreted either in a Bayesian setting or simply as smoothing term.
Typically,

with a positive definite matrix *B* and a reference picture . Unfortunately,
minimizing (4.6) is much more difficult than minimizing (4.5) and cannot be done with a simple iteration such as (4.4). For partial solutions to this problem see Levitan and Herman (1987), Green (1990), Setzepfandt (1992).

As in ART, a judicious arrangement of the equations can speed up the convergence significantly. The directions have to be arranged in ``ordered subsets'', see Hudson et al. (1992).

Thu Sep 10 10:51:17 MET DST 1998