New Insights into Emission Tomography Via Linear Programming

Abstract

Suppose that each detector unit, d, of an emission scanner measures a count n*(d) which represents the number of emissions into d of an unknown emission density λ.The likelihood, P(n*|λ), is the (Poisson) probability of observing n* under λ.The well-known EM algorithm starts with an estimate λ° of λ and produces a sequence λ^k, k=l,2,…, of estimators having increasing likelihood and which converge to an estimate λ^x with maximum likelihood as k→∞. It is well-known that for large k, λ^k becomes snowy or noisy in appearance, and various methods have been proposed to effectively smooth λ^k, but these all give up likelihood to get smoothness. We give a method using linear programming of smoothing λ^k which produces a smoother estimate λ^k which nevertheless has the same likelihood as λ^k.

If λ^k is nearly unique among estimators with its likelihood then λ^k cannot differ much from λ^k, while of there are many λ with the likelihood of λ^k, then λ^k should be smooth. Experiments described here indicate that for large k, λ^k is not much smoother than λ^k, so that λ^k seems to be nearly unique. This is surprising in view of the fact the problem considered is severely undetermined in the sense that there are typically 3 or more times as many unknowns as equations (measurements), but of course it is the inequality constraints that cause the uniqueness.

We survey the maximum likelihood approach to emission tomography using the iterative EM algorithm and discuss the known difficulty with the algorithm at high iteration numbers.