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Cramer-Rao lower bounds on the performance of CCD optical position estimators

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The problem of optically estimating an object’s position using a charge coupled device (CCD) array composed of square pixels Δx on a side is analyzed. The object’s image spot at the CCD is assumed to have a Gaussian intensity profile with a 1/e point at a radial distance of 2 σs from the peak, and the CCD noise is modeled as Poisson distributed dark current shot noise. A 2-D Cramer-Rao bound is developed and used to determine a lower limit for the mean-square error of any unbiased position estimator, and the maximum likelihood estimator is also derived. For the 1-D position estimation problem the lower bound is shown to be minimum for a pixel-to-image size ratio Δx/σs of between 1 and 2 over a wide range of signal-to-noise ratios. Similarly for the 2-D problem, the optimum ratio is shown to lie between 1.5 and 2.5. As is customary in direct detection systems, it is also observed that the lower bound is a function of both the signal power and noise power separately and not just their ratio. Finally at high signal-to-noise ratios, the maximum likelihood estimator is shown to be independent of the signal and noise powers.

© 1986 Optical Society of America

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