A Maximum-Likelihood Formalism for Sub-Resolution Axial Localization of Fluorescent Nanoparticles

One of the ongoing challenges in single particle fluorescence microscopy resides in estimating the axial position of particles with sub-resolution precision. Due to the complexity of the diffraction patterns generated by such particles, the standard fitting methods used to estimate a particle's lateral position are not applicable. A new approach for axial localization is proposed: it consists of a maximum-likelihood estimator based on a theoretical image formation model that incorporates noise. The fundamental theoretical limits on localization are studied, using Cramér-Rao bounds. These indicate that the proposed approach can be used to localize particles with nanometer-scale precision. Using phantom data generated according to the image formation model, it is then shown that the precision of the proposed estimator reaches the fundamental limits. Moreover, the approach is tested on experimental data, and sub-resolution localization at the 10 nm scale is demonstrated.