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Presents a method for approximating constrained maximum entropy (ME) reconstructions of SPECT data with modifications to a block-iterative maximum a posteriori (MAP) algorithm. Maximum likelihood (ML)-based reconstruction algorithms require some form of noise smoothing. Constrained ME provides a more formal method of noise smoothing without requiring the user to select parameters. In the context of SPECT, constrained ME seeks the minimum-information image estimate among those whose projections are a given distance from the noisy measured data, with that distance determined by the magnitude of the Poisson noise. Images that meet the distance criterion are referred to as feasible images. The authors find that modeling of all principal degrading factors (attenuation, detector response, and scatter) in the reconstruction is critical because feasibility is not meaningful unless the projection model is as accurate as possible. Because the constrained ME solution is the same as a MAP solution for a particular value of the MAP weighting parameter, β, the constrained ME solution can be found with a MAP algorithm if the correct value of β is found. The authors show that the RBI-MAP algorithm, if used with a dynamic scheme for estimating β, can approximate constrained ME solutions in 20 or fewer iterations. The authors compare results for various methods of achieving feasible images on a simulation of Tl-201 cardiac SPECT data. Results show that the RBI-MAP ME approximation provides images and quantitative estimates close to those from a slower algorithm that gives the true ME solution. Also, the authors find that the ME results have higher spatial resolution and greater high-frequency noise content than a feasibility-based stopping rule, feasibility-based low-pass filtering, and a quadratic Gibbs prior with β selected according to the feasibility criterion. The authors conclude that fast ME approximation is possible using either RBI-MAP wit- - h the dynamic procedure or a feasibility-based stopping rule, and that such reconstructions may be particularly useful in applications where resolution is critical.