Bayesian statistics, factor analysis, and PET images I. Mathematical background

The problem of image reconstruction in positron emission tomography (PET) is examined, although the approach is quite general and may have other applications. The approach is based on the maximum-likelihood method L.A. Shepp and Y. Vardi (1982), with their assumption that the number of image pixels is greater than the number of data points. In this situation a (nonunique) solution can be written down directly, although it is not guaranteed to be positive definite. The arbitrariness in this solution can be precisely characterized by a geometric argument. A unique solution can be obtained only by introducing prior information. It is suggested that factor analysis is an efficient way to do this. In the simplest application of the method, the solution is written as the sum of two parts, r_α +t_α, where r_α is determined solely by the data and T_α is determined by r_α and the prior information