Optimal measurement constellation of the fluoroBancroft localization algorithm for position estimation in tracking confocal microscopy

The fluoroBancroft algorithm is an analytical approach that converts a collection of fluorescence intensity measurements generated by an isolated sub-diffraction limit source into an estimate with nanometer-scale precision of the source position. Based on this algorithm, we have developed a scheme for tracking single fluorescent particles in a confocal microscope. In this paper, we determine an optimal measurement constellation for the estimation algorithm. The position estimation bias and uncertainty arising from the photon counting statistics are calculated based on the assumption that the natural logarithm of a Poisson random variable with large rate can be approximated as a random variable with a Gaussian distribution. A sufficient condition for an unbiased measurement constellation and the optimal radius of a given constellation geometry with six measurements are then derived. The results are illustrated through both numerical simulation and experiments.