Emission computed tomography is widely applied in medical diagnostic imaging, especially to determine physiological function. The available set of measurements is, however, often incomplete and corrupted, and the quality of image reconstruction is enhanced by the computation of a statistically optimal estimate. Most formulations of the estimation problem use the Poisson model to measure fidelity to data. The intuitive appeal and operational simplicity of quadratic approximations to the Poisson log likelihood make them an attractive alternative, but they imply a potential loss of reconstruction quality which has not often been studied. This paper presents quantitative comparisons between the two models and shows that a judiciously chosen quadratic, as part of a short series of Newton-style steps, yields reconstructions nearly indistinguishable from those under the exact Poisson model.