Sub-resolution Axial Localization of Nanoparticles in Fluorescence Microscopy

In recent years single particle tracking has significantly contributed to the study of intracellular molecular dynamics [2]. Particle tracking primarily consists of localizing particles in three dimensions (3-D) from a number of images acquired by fluorescence microscopy. While many approaches have been proposed for the lateral (i.e., x-y) localization of particles from acquisitions that are (almost) in focus [3], far fewer methods currently exist for the precise determination of a particle’s axial (i.e., z) position. Since the acquisition of high-resolution z-stacks is inefficient and impractical, axial localization methods try to estimate the position using few acquisitions of the particle. The difficulty of this task comes from the complexity of the diffraction patterns that appear as the particle moves out of focus. To avoid complex models, current axial localization methods often rely on empirical approaches [4]. Under optimal conditions, the spot of light that one observes when viewing a sub-resolution fluorescent particle with a microscope corresponds to the diffraction-limited point spread function (PSF) of the microscope. While most methods for tracking and localizing single particles try to exploit this property, they often make use of a simplified diffraction model. The principal difficulty with respect to axial localization is the non-stationarity of the PSF along the optical axis; if there is a difference in the refractive indices of the specimen and immersion medium, the PSF may vary depending on the particle’s depth within the specimen, which is a frequently occurring situation in practice. In this paper we propose an algorithm based on maximum-likelihood estimation that employs an accurate PSF model to iteratively determine a particle’s axial position. We describe an image formation model, propose a theoretical lower bound on the attainable localization precision, and show how our algorithm is derived. Finally, we demonstrate its efficiency by showing the correspondence with the theoretical lower bound.