On the Feasibility of Axial Tracking of a Fluorescent Nano-Particle Using a Defocusing Model

The image of a sub-resolution nano-particle in fluorescence microscopy corresponds to a slice of the 3D point spread function (PSF). This slice relates to the out-of-focus distance of the nano-particle. In this paper, we investigate to which extent it is possible to estimate the out-of-focus distance of the nano-particle from a 2D image based on the knowledge of the 3D PSF. To this end, we compute the Cramér-Rao bound (CRB) that provides a lower bound on the error of the best estimator of the axial position. The calculation of the CRB involves the specification of a 3D PSF model, the assumption of a signal-dependent Poisson noise, and some acquisition parameters. Our derivation shows that the CRB depends on the defocusing distance. Interestingly, nanometer precision can be attained over a range of defocus distances and for sufficiently high SNR levels. The theoretical results are confirmed with simulated experiments using estimators based on the least-squares (LS) and normalized cross-correlation (NCC) criterion. The results obtained are very close to the theoretical CRB.