Interior-point methodology for 3-D PET reconstruction

Interior-point methods have been successfully applied to a wide variety of linear and nonlinear programming applications. This paper presents a class of algorithms, based on path-following interior-point methodology, for performing regularized maximum-likelihood (ML) reconstructions on three-dimensional (3-D) emission tomography data. The algorithms solve a sequence of subproblems that converge to the regularized maximum likelihood solution from the interior of the feasible region (the nonnegative orthant). We propose two methods, a primal method which updates only the primal image variables and a primal-dual method which simultaneously updates the primal variables and the Lagrange multipliers. A parallel implementation permits the interior-point methods to scale to very large reconstruction problems. Termination is based on well-defined convergence measures, namely, the Karush-Kuhn-Tucker first-order necessary conditions for optimality. We demonstrate the rapid convergence of the path-following interior-point methods using both data from a small animal scanner and Monte Carlo simulated data. The proposed methods can readily be applied to solve the regularized, weighted least squares reconstruction problem.