Maximum Likelihood Reconstruction for Emission Tomography

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 λ = λ(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 λ of λ which maximizes the probability p(n*|λ) of observing the actual detector count data n* over all possible densities λ. Let independent Poisson variables n(b) with unknown means λ(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 λ = λ(b), b = 1, ···, B. For each λ, the observed data n* has probability or likelihood p(n*|λ). The EM algorithm of mathematical statistics starts with an initial estimate λ0 and gives the following simple iterative procedure for obtaining a new estimate λnew, from an old estimate λold, to obtain Σk, k = 1, 2, ···, λnew(b)= λold(b) λDd=1 n*(d)p(b,d)/Σλold(bΣ)p(bλ,d),b=1,···B.