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One of the most demanding tasks in astronomical image processing---in termsof precision---is the centroiding of stars. Upcoming large surveys are going totake images of billions of point sources, including many faint stars, withshort exposure times. Real-time estimation of the centroids of stars is crucialfor real-time PSF estimation, and maximal precision is required formeasurements of proper motion.  The fundamental Cram\'{e}r-Rao lower bound sets a limit on theroot-mean-squared-error achievable by optimal estimators. In this work, we aimto compare the performance of various centroiding methods, in terms ofsaturating the bound, when they are applied to relatively low signal-to-noiseratio unsaturated stars assuming zero-mean constant Gaussian noise. In order tomake this comparison, we present the ratio of the root-mean-squared-errors ofthese estimators to their corresponding Cram\'{e}r-Rao bound as a function ofthe signal-to-noise ratio and the full-width at half-maximum of faint stars.  We discuss two general circumstances in centroiding of faint stars: (i) whenwe have a good estimate of the PSF, (ii) when we do not know the PSF. In thecase that we know the PSF, we show that a fast polynomial centroiding aftersmoothing the image by the PSF can be as efficient as the maximum-likelihoodestimator at saturating the bound. In the case that we do not know the PSF, wedemonstrate that although polynomial centroiding is not as optimal as PSFprofile fitting, it comes very close to saturating the Cram\'{e}r-Rao lowerbound in a wide range of conditions. We also show that the moment-based methodof center-of-light never comes close to saturating the bound, and thus it doesnot deliver reliable estimates of centroids.

Do fast stellar centroiding methods saturate the Cram\'{e}r-Rao lower bound?