Photo-activation localization microscopy is a far-field superresolution imaging technique based on the localization of single molecules with subdiffraction limit precision. Known under acronyms such as PALM (photo-activated localization microscopy) or STORM (stochastic optical reconstruction microscopy), these techniques achieve superresolution by allowing only a sparse, random set of molecules to emit light at any given time and subsequently localizing each molecule with great precision. Recently, such techniques have been extended to three dimensions, opening up unprecedented possibilities to explore the structure and function of cells. Interestingly, proper engineering of the three-dimensional (3D) point spread function (PSF) through additional optics has been demonstrated to theoretically improve 3D position estimation and ultimately resolution. In this paper, an optimal 3D single-molecule localization estimator is presented in a general framework for noisy, aberrated and/or engineered PSF imaging. To find the position of each molecule, a phase-retrieval enabled maximum-likelihood estimator is implemented. This estimator is shown to be efficient, meaning it reaches the fundamental Cramer-Rao lower bound of x, y, and z localization precision. Experimental application of the phase-retrieval enabled maximum-likelihood estimator using a particular engineered PSF microscope demonstrates unmatched low-photon-count 3D wide-field single-molecule localization performance.
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