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.
