Fast and Precise 3D Fluorophore Localization based on Gradient Fitting.

Localization microscopy, known as different names including (fluorescence) photo-activated localization microscopy [(f) PALM]1,2 and (direct) stochastic optical reconstruction microscopy [(d) STORM]3,4, has become a powerful imaging tool to reveal the ultra-structures and understand the complicated mechanisms behind cellular function. The principle of localization microscopy is straightforward: a small subset of densely labeled fluorophores is sequentially switched “on” to obtain the sparsely distributed individual fluorescent emitters in a single frame, and the position of each emitter is determined by localization algorithm at a nanometer precision; after accumulating the localized positions from thousands of imaging frames, the spatial resolution of the final reconstructed image can be improved by ~10 times. By further combining the point spread function (PSF) engineering methods5,6,7, the capabilities of localization microscopy have been extended to resolve biological structures in all three dimensions. Various PSF engineering methods share a similar underlying principle that the axial position is encoded as the shape of the PSF in the lateral plane, which can be later decoded through image analysis. Among them, astigmatism approach has gained popularity because of its simple experimental configuration. By introducing astigmatism to the optical system (using a cylindrical lens5,8,9 or deformable mirror10,11), the axial position of the fluorophore is encoded as the ellipticity of the PSF. Generally, by employing a 2D elliptical Gaussian function to fit the elliptical PSF, a resolution of ~20 nm in the lateral dimension and ~50 nm in the axial dimension have been achieved5. The spatial resolution of localization microscopy is directly affected by the precision of the localization algorithm. For the best spatial resolution, iterative Gaussian function fitting (GF) based algorithms are usually employed12,13. But the slow execution speed of such algorithm that often takes several hours to reconstruct a standard super-resolution image is an intrinsic disadvantage of the GF based methods. Hence, they do not apply to the cases when fast image reconstruction and online data analysis are needed, such as real-time optimization of imaging parameters. For this purpose, several single-iteration algorithms have been developed in the past few years to accelerate the execution speed while providing comparable precision to the GF based algorithm14,15,16,17,18. Unfortunately, these algorithms are mainly designed for 2D fitting of a circular PSF, and their precision for retrieving the 3D position is significantly compromised when the spatial distribution of fluorescent emission is not isotropic, such as astigmatism-based imaging with elliptical PSF. Hence, it is important to develop a highly efficient 3D localization algorithm for astigmatism-based single particle tracking or super-resolution localization microscopy with both satisfactory localization precision and execution speed. In this paper, we present an algebraic algorithm based on gradient fitting for fast 3D fluorophore localization in astigmatism-based microscopy. We utilize the relationship of the gradient direction distribution and the position of the fluorescent emitter to determine the x–y position and ellipticity of the PSF by finding the best-fit gradient direction distribution (Fig. 1a). Then, this algorithm estimates the position of PSF in all three dimensions by looking up the z-ellipticity calibration curve (Fig. 1b). Through numerical simulation and experiments with fluorescent nanospheres and mammalian cells, we demonstrate that the proposed single-iteration algorithm can achieve localization precision close to multiple iterative GF based algorithm in all three dimensions, while yielding over 100 times faster computation speed. Figure 1 The principle of the gradient fitting based algorithm.