Summary

Based on a realistic physical model of the image-formation process, we have developed a maximum-likelihood estimator for the 3-D position of a fluorescent particle from a focal series of brightfield micrographs. We do also provide theoretical resolution bounds that are well beyond the classical Rayleigh limit.

Introduction

The localization of nanoparticles plays an increasing role in fluorescence microscopy. A key application is particle tracking for the observation of molecular dynamics. In that context, it is highly desirable to be able to determine a the position of a particle with sub-resolution accuracy (at the nanometer scale). A research team has also recently proposed a new microscopy technique (Betzig et al., Science, 2006)—termed photoactivated localization microscopy—whose central component is a least-squares fitting algorithm that allows the precise localization of single fluorescent molecules.

Main Contribution

In many experimental setups involving modern microscope optics, the point-spread function (PSF) of the system is nonstationary due to a mismatch between the refractive indices of the immersion medium and the specimen. This phenomenon needs to be taken into account in order to achieve accurate localization results. Therefore, we proposed an image-formation model that combines a state-of-the-art PSF model with a statistical formulation of acquisition noise. Based on this, we formulated a maximum-likelihood estimator for the 3-D position of a particle. To test the performance of the estimator, we established a theoretical limit on the attainable localization accuracy using Cramér-Rao bounds. These have shown that the localization accuracy along the optical axis is similar to that within the imaging plane, and can be as good as a few nanometers. In practice, using fluorescent beads, we have been able to demonstrate localization accuracies of the order of 10-15 nm, using a conventional widefield fluorescence microscope.