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In this paper, we present novel maximum likelihood reconstruction algorithms for positron emission tomography (PET). The key idea behind the algorithms is that the set of maximum likelihood estimates is equivalent to the intersection of certain convex sets. Given this equivalence, we exploit results from set theoretic estimation and develop subgradient projection algorithms to maximize the log likelihood function. From experiments using synthetic data, it was found that the proposed algorithms produced images that were both smoother and more well defined than those obtained using the ML-EM algorithm.