Best linear unbiased axial localization in three-dimensional fluorescent bead tracking with subnanometer resolution using off-focus images.

A three-dimensional particle tracking technique, based on microscope off-focus images, was introduced in Z. Zhang and C.-H. Menq, Appl. Opt.47, 2361 (2008) and applied to bright-field imaging. This paper presents two major improvements to the axial localization algorithm of the 3D particle tracking technique. First, it extends the algorithm to measure fluorescent particles in the presence of photobleaching and excitation variation. Second, it enhances the measurement resolution by achieving the best linear unbiased estimation of the particle's axial position. Similarly to the original algorithm, a radius vector is first converted from the off-focus 2D image of the particle, and the axial position is estimated by comparing the radius vector with an object-specific model, calibrated automatically prior to each experiment. Although it was an intensity-based method, by normalizing the radius vectors the improved algorithm becomes a shape-based method, thus invariant to image intensity change and robust to photobleaching. Moreover, when considering the noise variance of each point in the radius vector and their correlations, the best linear unbiased estimation based on a linearized model is achieved. It is shown that variance equalization and correlation-weighted optimization greatly reduce the estimation variance and lead to near-uniform localization resolution over the entire measurement range. Estimation resolution is theoretically analyzed and validated by experiments. Theoretical analysis enables the prediction of measurement resolution based on calibration data. Finally, experimental results are presented to illustrate the performance of the measurement method in terms of measurement precision and range, as well as its robustness to intensity variation.