Cramer-Rao Lower Bounds on the Performance of Charge-Coupled-Device Optical Position Estimators

摘要

The problem of optically estimating an object's position by using a charge coupled device (CCD) array composed of square pixels Delta x 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 (sigma subs) (sq rt. of 2), from the peak, and the CCD noise is modeled as Poisson-distributed, dark-current shot noise. A two-dimensional Cramer Rao bound is developed and used to determine a lower limit for the mean squared error of any unbiased position estimator, and the maximum likelihood estimator is also derived. For the one dimensional position estimation problem the lower bound is shown to be minimum for a pixel-to-image size ratio (delta x)/(sigma sub s), of between 1 and 2 over a wide range of signal to noise ratios. Similarily for the two-dimensional 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 an noise power separately and not just of their ratio. Finally, the maximum likelihood estimator is shown to be independent of the signal and noise powers at high signal to noise ratios.