An alternating minimization weighted least squares reconstruction algorithm

In this paper, the authors introduce a new iterative algorithm for reconstructing positron emission tomography (PET) images. This algorithm seeks to minimize an objective function of weighted least squares (WLS) type. However, unlike conventional WLS methods, the weights do not need to be estimated from the data, but are incorporated in the objective function and relies heavily on the Poisson nature of the data. As a result, the objective function is not quadratic, but is convex. The iterative algorithm is obtained in a manner similar to an analytic derivation of the ML-EM (maximum likelihood-expectation maximization) algorithm which employs an alternating minimization procedure between two convex sets of matrices. However, the distance metric is quite different in the authors' case, and much more difficult to analyze. This algorithm is similar in form to, and shares many properties in common with, the ML-EM algorithm. The mathematical proof of the global convergence of the algorithm remains an open problem.