Abstract
Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density ? = ?(x, y, z) generates, and is to be reconstructed from, the number of counts n*(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate ? of ? which maximizes the probability p(n*|?) of observing the actual detector count data n* over all possible densities ?. Let independent Poisson variables n(b) with unknown means ?(b), b = 1, ···, B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ···, D with p(b, d) a one-step transition matrix, assumed known. We observe the total number n* = n*(d) of emissions in each detector unit d and want to estimate the unknown ? = ?(b), b = 1, ···, B. For each ?, the observed data n* has probability or likelihood p(n*|?). The EM algorithm of mathematical statistics starts with an initial estimate ?0 and gives the following simple iterative procedure for obtaining a new estimate ?new, from an old estimate ?old, to obtain ?k, k = 1, 2, ···, ?new(b)= ?old(b) ?Dd=1 n*(d)p(b,d)/??old(b?)p(b?,d),b=1,···B.
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