Measurement-based estimation of global pupil functions in 3D localization microscopy

Universal inverse modeling of point spread functions for SMLM localization and microscope characterization

The point spread function (PSF) of a microscope describes the image of a point emitter. Knowing the accurate PSF model is essential for various imaging tasks, including single-molecule localization, aberration correction and deconvolution. Here we present universal inverse modeling of point spread functions (uiPSF), a toolbox to infer accurate PSF models from microscopy data, using either image stacks of fluorescent beads or directly images of blinking fluorophores, the raw data in single-molecule localization microscopy (SMLM). Our modular framework is applicable to a variety of microscope modalities and the PSF model incorporates system- or sample-specific characteristics, for example, the bead size, field- and depth- dependent aberrations, and transformations among channels. We demonstrate its application in single or multiple channels or large field-of-view SMLM systems, 4Pi-SMLM, and lattice light-sheet microscopes using either bead data or single-molecule blinking data.

Large-FOV 3D localization microscopy by spatially variant point spread function generation

Accurate characterization of the microscopic point spread function (PSF) is crucial for achieving high-performance localization microscopy (LM). Traditionally, LM assumes a spatially invariant PSF to simplify the modeling of the imaging system. However, for large fields of view (FOV) imaging, it becomes important to account for the spatially variant nature of the PSF. Here, we propose an accurate and fast principal components analysis-based field-dependent 3D PSF generator (PPG3D) and localizer for LM. Through simulations and experimental three-dimensional (3D) single-molecule localization microscopy (SMLM), we demonstrate the effectiveness of PPG3D, enabling super-resolution imaging of mitochondria and microtubules with high fidelity over a large FOV. A comparison of PPG3D with a shift-variant PSF generator for 3D LM reveals a threefold improvement in accuracy. Moreover, PPG3D is approximately 100 times faster than existing PSF generators, when used in image plane-based interpolation mode. Given its user-friendliness, we believe that PPG3D holds great potential for widespread application in SMLM and other imaging modalities.

Universal inverse modelling of point spread functions for SMLM localization and microscope characterization

The point spread function (PSF) of a microscope describes the image of a point emitter. Knowing the accurate PSF model is essential for various imaging tasks, including single molecule localization, aberration correction and deconvolution. Here we present uiPSF (universal inverse modelling of Point Spread Functions), a toolbox to infer accurate PSF models from microscopy data, using either image stacks of fluorescent beads or directly images of blinking fluorophores, the raw data in single molecule localization microscopy (SMLM). The resulting PSF model enables accurate 3D super-resolution imaging using SMLM. Additionally, uiPSF can be used to characterize and optimize a microscope system by quantifying the aberrations, including field-dependent aberrations, and resolutions. Our modular framework is applicable to a variety of microscope modalities and the PSF model incorporates system or sample specific characteristics, e.g., the bead size, depth dependent aberrations and transformations among channels. We demonstrate its application in single or multiple channels or large field-of-view SMLM systems, 4Pi-SMLM, and lattice light-sheet microscopes using either bead data or single molecule blinking data.

Fluorescence Microscopy: a statistics-optics perspective

Fundamental properties of light unavoidably impose features on images collected using fluorescence microscopes. Modeling these features is ever more important in quantitatively interpreting microscopy images collected at scales on par or smaller than light's wavelength. Here we review the optics responsible for generating fluorescent images, fluorophore properties, microscopy modalities leveraging properties of both light and fluorophores, in addition to the necessarily probabilistic modeling tools imposed by the stochastic nature of light and measurement.

Deep learning using a residual deconvolutional network enables real-time high-density single-molecule localization microscopy

High-density localization based on deep learning is a very effective method to accelerate single molecule localization microscopy (SMLM). Compared with traditional high-density localization methods, deep learning-based methods enable a faster data processing speed and a higher localization accuracy. However, the reported high-density localization methods based on deep learning are still not fast enough to enable real time data processing for large batches of raw images, which is probably due to the heavy computational burden and computation complexity in the U-shape architecture used in these models. Here we propose a high-density localization method called FID-STORM, which is based on an improved residual deconvolutional network for the real-time processing of raw images. In FID-STORM, we use a residual network to extract the features directly from low-resolution raw images rather than the U-shape network from interpolated images. We also use a model fusion from TensorRT to further accelerate the inference of the model. In addition, we process the sum of the localization images directly on GPU to obtain an additional speed gain. Using simulated and experimental data, we verified that the FID-STORM method achieves a processing speed of 7.31 ms/frame at 256 × 256 pixels @ Nvidia RTX 2080 Ti graphic card, which is shorter than the typical exposure time of 10∼30 ms, thus enabling real-time data processing in high-density SMLM. Moreover, compared with a popular interpolated image-based method called Deep-STORM, FID-STORM enables a speed gain of ∼26 times, without loss of reconstruction accuracy. We also provided an ImageJ plugin for our new method.

Using fluorescent beads to emulate single fluorophores

We study the conditions under which fluorescent beads can be used to emulate single fluorescent molecules in the calibration of optical microscopes. Although beads are widely used due to their brightness and easy manipulation, there can be notable differences between the point spread functions (PSFs) they produce and those for single-molecule fluorophores, caused by their different emission patterns and sizes. We study theoretically these differences for various scenarios, e.g., with or without polarization channel splitting, to determine the conditions under which the use of beads as a model for single molecules is valid. We also propose methods to model the blurring due to the size difference and compensate for it to produce PSFs that are more similar to those for single molecules.

Towards optimal point spread function design for resolving closely spaced emitters in three dimensions

The past decade has brought many innovations in optical design for 3D super-resolution imaging of point-like emitters, but these methods often focus on single-emitter localization precision as a performance metric. Here, we propose a simple heuristic for designing a point spread function (PSF) that allows for precise measurement of the distance between two emitters. We discover that there are two types of PSFs that achieve high performance for resolving emitters in 3D, as quantified by the Cramér-Rao bounds for estimating the separation between two closely spaced emitters. One PSF is very similar to the existing Tetrapod PSFs; the other is a rotating single-spot PSF, which we call the crescent PSF. The latter exhibits excellent performance for localizing single emitters throughout a 1-µm focal volume (localization precisions of 7.3 nm in x, 7.7 nm in y, and 18.3 nm in z using 1000 detected photons), and it distinguishes between one and two closely spaced emitters with superior accuracy (25-53% lower error rates than the best-performing Tetrapod PSF, averaged throughout a 1-µm focal volume). Our study provides additional insights into optimal strategies for encoding 3D spatial information into optical PSFs.

Image Reconstruction in Light-Sheet Microscopy: Spatially Varying Deconvolution and Mixed Noise

We study the problem of deconvolution for light-sheet microscopy, where the data is corrupted by spatially varying blur and a combination of Poisson and Gaussian noise. The spatial variation of the point spread function of a light-sheet microscope is determined by the interaction between the excitation sheet and the detection objective PSF. We introduce a model of the image formation process that incorporates this interaction and we formulate a variational model that accounts for the combination of Poisson and Gaussian noise through a data fidelity term consisting of the infimal convolution of the single noise fidelities, first introduced in L. Calatroni et al. (SIAM J Imaging Sci 10(3):1196-1233, 2017). We establish convergence rates and a discrepancy principle for the infimal convolution fidelity and the inverse problem is solved by applying the primal-dual hybrid gradient (PDHG) algorithm in a novel way. Numerical experiments performed on simulated and real data show superior reconstruction results in comparison with other methods.

Phase optimization algorithm for 3D particle localization with large axial depth

We propose an optimization algorithm based on Fresnel approximation (FA) imaging to optimize an extended-axial-depth point spread function (PSF) for 3D particle localization. The transfer function efficiency of the PSF is improved by repeatedly imposing constraints in the object plane, the spatial domain, and the Fourier domain. During the iterative calculation, the effective photon number or Cramer-Rao lower bound is used as the termination condition of the iteration. The algorithm allows flexible adjustment of the peak intensity ratio of the two main lobes. Moreover, the transfer function efficiency can be balanced by increasing the weight of the modulation function of the expected PSF at each axial position. The twin-Airy (TA) PSF optimized by the FA optimization algorithm does not require complex post-processing, whereas post-processing is an essential step for the unoptimized TA-PSF. The optimization algorithm is significant for extended-axial-depth PSFs used for 3D particle localization, as it improves localization precision and temporal resolution.

Accurate localization microscopy by intrinsic aberration calibration

A standard paradigm of localization microscopy involves extension from two to three dimensions by engineering information into emitter images, and approximation of errors resulting from the field dependence of optical aberrations. We invert this standard paradigm, introducing the concept of fully exploiting the latent information of intrinsic aberrations by comprehensive calibration of an ordinary microscope, enabling accurate localization of single emitters in three dimensions throughout an ultrawide and deep field. To complete the extraction of spatial information from microscale bodies ranging from imaging substrates to microsystem technologies, we introduce a synergistic concept of the rigid transformation of the positions of multiple emitters in three dimensions, improving precision, testing accuracy, and yielding measurements in six degrees of freedom. Our study illuminates the challenge of aberration effects in localization microscopy, redefines the challenge as an opportunity for accurate, precise, and complete localization, and elucidates the performance and reliability of a complex microelectromechanical system.

Quantifying accuracy and heterogeneity in single-molecule super-resolution microscopy

The resolution and accuracy of single-molecule localization microscopes (SMLMs) are routinely benchmarked using simulated data, calibration rulers, or comparisons to secondary imaging modalities. However, these methods cannot quantify the nanoscale accuracy of an arbitrary SMLM dataset. Here, we show that by computing localization stability under a well-chosen perturbation with accurate knowledge of the imaging system, we can robustly measure the confidence of individual localizations without ground-truth knowledge of the sample. We demonstrate that our method, termed Wasserstein-induced flux (WIF), measures the accuracy of various reconstruction algorithms directly on experimental 2D and 3D data of microtubules and amyloid fibrils. We further show that WIF confidences can be used to evaluate the mismatch between computational models and imaging data, enhance the accuracy and resolution of reconstructed structures, and discover hidden molecular heterogeneities. As a computational methodology, WIF is broadly applicable to any SMLM dataset, imaging system, and localization algorithm.

Standard benchmarking of single-molecule localization microscopy cannot quantify nanoscale accuracy of arbitrary datasets. Here, the authors present Wasserstein-induced flux, a method using a chosen perturbation and knowledge of the imaging system to measure confidence of individual localizations.

Effect of Pixelation on the Parameter Estimation of Single Molecule Trajectories

The advent of single molecule microscopy has revolutionized biological investigations by providing a powerful tool for the study of intercellular and intracellular trafficking processes of protein molecules which was not available before through conventional microscopy. In practice, pixelated detectors are used to acquire the images of fluorescently labeled objects moving in cellular environments. Then, the acquired fluorescence microscopy images contain the numbers of the photons detected in each pixel, during an exposure time interval. Moreover, instead of having the exact locations of detection of the photons, we only know the pixel areas in which the photons impact the detector. These challenges make the analysis of single molecule trajectories, from pixelated images, a complex problem. Here, we investigate the effect of pixelation on the parameter estimation of single molecule trajectories. In particular, we develop a stochastic framework to calculate the maximum likelihood estimates of the parameters of a stochastic differential equation that describes the motion of the molecule in living cells. We also calculate the Fisher information matrix for this parameter estimation problem. The analytical results are complicated through the fact that the observation process in a microscope prohibits the use of standard Kalman filter type approaches. The analytical framework presented here is illustrated with examples of low photon count scenarios for which we rely on Monte Carlo methods to compute the associated probability distributions.

Recent advances in point spread function engineering and related computational microscopy approaches: from one viewpoint

This personal hybrid review piece, written in light of my recipience of the UIPAB 2020 young investigator award, contains a mixture of my scientific biography and work so far. This paper is not intended to be a comprehensive review, but only to highlight my contributions to computation-related aspects of super-resolution microscopy, as well as their origins and future directions.

Microscopic scan-free surface profiling over extended axial ranges by point-spread-function engineering

The shape of a surface, i.e., its topography, influences many functional properties of a material; hence, characterization is critical in a wide variety of applications. Two notable challenges are profiling temporally changing structures, which requires high-speed acquisition, and capturing geometries with large axial steps. Here, we leverage point-spread-function engineering for scan-free, dynamic, microsurface profiling. The presented method is robust to axial steps and acquires full fields of view at camera-limited framerates. We present two approaches for implementation: fluorescence-based and label-free surface profiling, demonstrating the applicability to a variety of sample geometries and surface types.

Correction of refractive index mismatch-induced aberrations under radially polarized illumination by deep learning

Radially polarized field under strong focusing has emerged as a powerful manner for fluorescence microscopy. However, the refractive index (RI) mismatch-induced aberrations seriously degrade imaging performance, especially under high numerical aperture (NA). Traditional adaptive optics (AO) method is limited by its tedious procedure. Here, we present a computational strategy that uses artificial neural networks to correct the aberrations induced by RI mismatch. There are no requirements for expensive hardware and complicated wavefront sensing in our framework when the deep network training is completed. The structural similarity index (SSIM) criteria and spatial frequency spectrum analysis demonstrate that our deep-learning-based method has a better performance compared to the widely used Richardson-Lucy (RL) deconvolution method at different imaging depth on simulation data. Additionally, the generalization of our trained network model is tested on new types of samples that are not present in the training procedure to further evaluate the utility of the network, and the performance is also superior to RL deconvolution.

Integrating engineered point spread functions into the phasor-based single-molecule localization microscopy framework

In single-molecule localization microscopy (SMLM), the use of engineered point spread functions (PSFs) provides access to three-dimensional localization information. The conventional approach of fitting PSFs with a single 2-dimensional Gaussian profile, however, often falls short in analyzing complex PSFs created by placing phase masks, deformable mirrors or spatial light modulators in the optical detection pathway. Here, we describe the integration of PSF modalities known as double-helix, saddle-point or tetra-pod into the phasor-based SMLM (pSMLM) framework enabling fast CPU based localization of single-molecule emitters with sub-pixel accuracy in three dimensions. For the double-helix PSF, pSMLM identifies the two individual lobes and uses their relative rotation for obtaining z-resolved localizations. For the analysis of saddle-point or tetra-pod PSFs, we present a novel phasor-based deconvolution approach entitled circular-tangent pSMLM. Saddle-point PSFs were experimentally realized by placing a deformable mirror in the Fourier plane and modulating the incoming wavefront with specific Zernike modes. Our pSMLM software package delivers similar precision and recall rates to the best-in-class software package (SMAP) at signal-to-noise ratios typical for organic fluorophores and achieves localization rates of up to 15 kHz (double-helix) and 250 kHz (saddle-point/tetra-pod) on a standard CPU. We further integrated pSMLM into an existing software package (SMALL-LABS) suitable for single-particle imaging and tracking in environments with obscuring backgrounds. Taken together, we provide a powerful hardware and software environment for advanced single-molecule studies.

Addressing systematic errors in axial distance measurements in single-emitter localization microscopy

Nanoscale localization of point emitters is critical to several methods in optical fluorescence microscopy, including single-molecule super-resolution imaging and tracking. While the precision of the localization procedure has been the topic of extensive study, localization accuracy has been less emphasized, in part due to the challenge of producing an experimental sample containing unperturbed point emitters at known three-dimensional positions in a relevant geometry. We report a new experimental system which reproduces a widely-adopted geometry in high-numerical aperture localization microscopy, in which molecules are situated in an aqueous medium above a glass coverslip imaged with an oil-immersion objective. We demonstrate a calibration procedure that enables measurement of the depth-dependent point spread function (PSF) for open aperture imaging as well as imaging with engineered PSFs with index mismatch. We reveal the complicated, depth-varying behavior of the focal plane position in this system and discuss the axial localization biases incurred by common approximations of this behavior. We compare our results to theoretical calculations.

Three-Dimensional Single-Molecule Localization Microscopy in Whole-Cell and Tissue Specimens

Super-resolution microscopy techniques are versatile and powerful tools for visualizing organelle structures, interactions, and protein functions in biomedical research. However, whole-cell and tissue specimens challenge the achievable resolution and depth of nanoscopy methods. We focus on three-dimensional single-molecule localization microscopy and review some of the major roadblocks and developing solutions to resolving thick volumes of cells and tissues at the nanoscale in three dimensions. These challenges include background fluorescence, system- and sample-induced aberrations, and information carried by photons, as well as drift correction, volume reconstruction, and photobleaching mitigation. We also highlight examples of innovations that have demonstrated significant breakthroughs in addressing the abovementioned challenges together with their core concepts as well as their trade-offs.

Three-dimensional nanoscopy of whole cells and tissues with in situ point spread function retrieval

Single-molecule localization microscopy is a powerful tool for visualizing subcellular structures, interactions and protein functions in biological research. However, inhomogeneous refractive indices inside cells and tissues distort the fluorescent signal emitted from single-molecule probes, which rapidly degrades resolution with increasing depth. We propose a method that enables the construction of an in situ 3D response of single emitters directly from single-molecule blinking datasets, and therefore allows their locations to be pinpointed with precision that achieves the Cramér-Rao lower bound and uncompromised fidelity. We demonstrate this method, named in situ PSF retrieval (INSPR), across a range of cellular and tissue architectures, from mitochondrial networks and nuclear pores in mammalian cells to amyloid-β plaques and dendrites in brain tissues and elastic fibers in developing cartilage of mice. This advancement expands the routine applicability of super-resolution microscopy from selected cellular targets near coverslips to intra- and extracellular targets deep inside tissues.

VIPR: vectorial implementation of phase retrieval for fast and accurate microscopic pixel-wise pupil estimation

In microscopy, proper modeling of the image formation has a substantial effect on the precision and accuracy in localization experiments and facilitates the correction of aberrations in adaptive optics experiments. The observed images are subject to polarization effects, refractive index variations, and system specific constraints. Previously reported techniques have addressed these challenges by using complicated calibration samples, computationally heavy numerical algorithms, and various mathematical simplifications. In this work, we present a phase retrieval approach based on an analytical derivation of the vectorial diffraction model. Our method produces an accurate estimate of the system's phase information, without any prior knowledge about the aberrations, in under a minute.

Deep learning in single-molecule microscopy: fundamentals, caveats, and recent developments [Invited]

Deep learning-based data analysis methods have gained considerable attention in all fields of science over the last decade. In recent years, this trend has reached the single-molecule community. In this review, we will survey significant contributions of the application of deep learning in single-molecule imaging experiments. Additionally, we will describe the historical events that led to the development of modern deep learning methods, summarize the fundamental concepts of deep learning, and highlight the importance of proper data composition for accurate, unbiased results.

Accurate and rapid background estimation in single-molecule localization microscopy using the deep neural network BGnet

Background fluorescence, especially when it exhibits undesired spatial features, is a primary factor for reduced image quality in optical microscopy. Structured background is particularly detrimental when analyzing single-molecule images for 3-dimensional localization microscopy or single-molecule tracking. Here, we introduce BGnet, a deep neural network with a U-net-type architecture, as a general method to rapidly estimate the background underlying the image of a point source with excellent accuracy, even when point-spread function (PSF) engineering is in use to create complex PSF shapes. We trained BGnet to extract the background from images of various PSFs and show that the identification is accurate for a wide range of different interfering background structures constructed from many spatial frequencies. Furthermore, we demonstrate that the obtained background-corrected PSF images, for both simulated and experimental data, lead to a substantial improvement in localization precision. Finally, we verify that structured background estimation with BGnet results in higher quality of superresolution reconstructions of biological structures.

Accurate phase retrieval of complex 3D point spread functions with deep residual neural networks

Phase retrieval, i.e., the reconstruction of phase information from intensity information, is a central problem in many optical systems. Imaging the emission from a point source such as a single molecule is one example. Here, we demonstrate that a deep residual neural net is able to quickly and accurately extract the hidden phase for general point spread functions (PSFs) formed by Zernike-type phase modulations. Five slices of the 3D PSF at different focal positions within a two micrometer range around the focus are sufficient to retrieve the first six orders of Zernike coefficients.

Spatio-angular fluorescence microscopy II. Paraxial 4f imaging

We investigate the properties of a single-view fluorescence microscope in a 4f geometry when imaging fluorescent dipoles without using the monopole or scalar approximations. We show that this imaging system has a spatio-angular band limit, and we exploit the band limit to perform efficient simulations. Notably, we show that information about the out-of-plane orientation of ensembles of in-focus fluorophores is recorded by paraxial fluorescence microscopes. Additionally, we show that the monopole approximation may cause biased estimates of fluorophore concentrations, but these biases are small when the sample contains either many randomly oriented fluorophores in each resolvable volume or unconstrained rotating fluorophores.

Computational correction of spatially variant optical aberrations in 3D single-molecule localization microscopy

3D single-molecule localization microscopy relies on fitting the shape of point-spread-functions (PSFs) recorded on a wide-field detector. However, optical aberrations distort those shapes, which compromises the accuracy and precision of single-molecule localization microscopy. Here, we employ a computational phase retrieval based on a vectorial PSF model to quantify the spatial variance of optical aberrations in a two-channel ultrawide-field single-molecule localization microscope. The use of a spatially variant PSF model enables accurate and precise emitter localization in x-, y- and z-directions throughout the entire field of view.

IsoSense: frequency enhanced sensorless adaptive optics through structured illumination

We present IsoSense, a wavefront sensing method that mitigates sample dependency in image-based sensorless adaptive optics applications in microscopy. Our method employs structured illumination to create additional high spatial frequencies in the image through custom illumination patterns. This improves the reliability of image quality metric calculations and enables sensorless wavefront measurement even in samples with sparse spatial frequency content. We demonstrate the feasibility of IsoSense for aberration correction in a deformable-mirror-based structured illumination super-resolution fluorescence microscope.

Multicolor localization microscopy and point-spread-function engineering by deep learning

Deep learning has become an extremely effective tool for image classification and image restoration problems. Here, we apply deep learning to microscopy and demonstrate how neural networks can exploit the chromatic dependence of the point-spread function to classify the colors of single emitters imaged on a grayscale camera. While existing localization microscopy methods for spectral classification require additional optical elements in the emission path, e.g., spectral filters, prisms, or phase masks, our neural net correctly identifies static and mobile emitters with high efficiency using a standard, unmodified single-channel configuration. Furthermore, we show how deep learning can be used to design new phase-modulating elements that, when implemented into the imaging path, result in further improved color differentiation between species, including simultaneously differentiating four species in a single image.

Ring resonator based polarization diversity WDM receiver

A ring resonator based 4 channel wavelength division multiplexing (WDM) receiver with polarization diversity is demonstrated at 10 Gb/s per channel. By forming a waveguide loop between the two output ports of a polarization splitter-rotator (PSR), the input signals in the quasi-transverse-electric (quasi-TE) and the quasi-transverse-magnetic (quasi-TM) polarizations can be demultiplexed by the same set of ring resonator filters, thus reducing the number of required channel control circuits by half compared to methods which process the two polarizations individually. Large signal measurement results indicate that the design can tolerate a signal delay of up to 30% of the unit interval (UI) between the two polarizations, which implies that compensating for manufacturing variability with optical delay lines on chip is not necessary for a robust operation. The inter-channel crosstalk is found negligible down to 0.4nm (50 GHz) spacing, at which point the adjacent channel isolation is 17 dB, proving the design's compatibility for dense WDM application.

Generalized method to design phase masks for 3D super-resolution microscopy

Point spread function (PSF) engineering by phase modulation is a novel approach to three-dimensional (3D) super-resolution microscopy, with different point spread functions being proposed for specific applications. It is often not easy to achieve the desired shape of engineered point spread functions because it is challenging to determine the correct phase mask. Additionally, a phase mask can either encode 3D space information or additional time information, but not both simultaneously. A robust algorithm for recovering a phase mask to generate arbitrary point spread functions is needed. In this work, a generalized phase mask design method is introduced by performing an optimization. A stochastic gradient descent algorithm and a Gauss-Newton algorithm are developed and compared for their ability to recover the phase masks for previously reported point spread functions. The new Gauss-Newton algorithm converges to a minimum at much higher speeds. This algorithm is used to design a novel stretching-lobe phase mask to encode temporal and 3D spatial information simultaneously. The stretching-lobe phase mask and other masks are fabricated in-house for proof-of-concept using multi-level light lithography and an optimized commercially sourced stretching-lobe phase mask (PM) is validated experimentally to encode 3D spatial and temporal information. The algorithms' generalizability is further demonstrated by generating a phase mask that comprises four different letters at different depths.

Three-dimensional localization microscopy using deep learning

Single molecule localization microscopy (SMLM) is one of the fastest evolving and most broadly used super-resolving imaging techniques in the biosciences. While image recordings could take up to hours only ten years ago, scientists are now reaching for real-time imaging in order to follow the dynamics of biology. To this end, it is crucial to have data processing strategies available that are capable of handling the vast amounts of data produced by the microscope. In this article, we report on the use of a deep convolutional neural network (CNN) for localizing particles in three dimensions on the basis of single images. In test experiments conducted on fluorescent microbeads, we show that the precision obtained with a CNN can be comparable to that of maximum likelihood estimation (MLE), which is the accepted gold standard. Regarding speed, the CNN performs with about 22k localizations per second more than three orders of magnitude faster than the MLE algorithm of ThunderSTORM. If only five parameters are estimated (3D position, signal and background), our CNN implementation is currently slower than the fastest, recently published GPU-based MLE algorithm. However, in this comparison the CNN catches up with every additional parameter, with only a few percent extra time required per additional dimension. Thus it may become feasible to estimate further variables such as molecule orientation, aberration functions or color. We experimentally demonstrate that jointly estimating Zernike mode magnitudes for aberration modeling can significantly improve the accuracy of the estimates.

Fundamental Precision Bounds for Three-Dimensional Optical Localization Microscopy with Poisson Statistics

Point source localization is a problem of persistent interest in optical imaging. In particular, a number of widely used biological microscopy techniques rely on precise three-dimensional localization of single fluorophores. As emitter depth localization is more challenging than lateral localization, considerable effort has been spent on engineering the response of the microscope in a way that reveals increased depth information. Here, we prove the (sub)optimality of these approaches by deriving and comparing to the measurement-independent quantum Cramér-Rao bound (QCRB). We show that existing methods for depth localization with single-objective collection exceed the QCRB, and we gain insight into the bound by proposing an interferometer arrangement that approaches it. We also show that for light collection with two opposed objectives, an established interferometric technique globally reaches the QCRB in all three dimensions simultaneously, and so this represents an interesting case study from the point of view of quantum multiparameter estimation.

ZOLA-3D allows flexible 3D localization microscopy over an adjustable axial range

Single molecule localization microscopy can generate 3D super-resolution images without scanning by leveraging the axial variations of normal or engineered point spread functions (PSF). Successful implementation of these approaches for extended axial ranges remains, however, challenging. We present Zernike Optimized Localization Approach in 3D (ZOLA-3D), an easy-to-use computational and optical solution that achieves optimal resolution over a tunable axial range. We use ZOLA-3D to demonstrate 3D super-resolution imaging of mitochondria, nuclear pores and microtubules in entire nuclei or cells up to ~5 μm deep.

3D single-molecule localization is limited in depth and often requires using a wide range of point spread functions (PSFs). Here the authors present an optical solution featuring a deformable mirror to generate different PSFs and easy-to-use software for super-resolution imaging up to 5 µm deep.

Light sheet approaches for improved precision in 3D localization-based super-resolution imaging in mammalian cells [Invited]

The development of imaging techniques beyond the diffraction limit has paved the way for detailed studies of nanostructures and molecular mechanisms in biological systems. Imaging thicker samples, such as mammalian cells and tissue, in all three dimensions, is challenging due to increased background and volumes to image. Light sheet illumination is a method that allows for selective irradiation of the image plane, and its inherent optical sectioning capability allows for imaging of biological samples with reduced background, photobleaching, and photodamage. In this review, we discuss the advantage of combining single-molecule imaging with light sheet illumination. We begin by describing the principles of single-molecule localization microscopy and of light sheet illumination. Finally, we present examples of designs that successfully have married single-molecule super-resolution imaging with light sheet illumination for improved precision in mammalian cells.

Nonlinear Focal Modulation Microscopy

We demonstrate nonlinear focal modulation microscopy (NFOMM) to achieve superresolution imaging. Traditional approaches to superresolution that utilize point scanning often rely on spatially reducing the size of the emission pattern by directly narrowing (e.g., through minimizing the detection pinhole in Airyscan, Zeiss) or indirectly peeling its outer profiles [e.g., through depleting the outer emission region in stimulated emission depletion (STED) microscopy]. We show that an alternative conceptualization that focuses on maximizing the optical system's frequency shifting ability offers advantages in further improving resolution while reducing system complexity. In NFOMM, a spatial light modulator and a suitably intense laser illumination are used to implement nonlinear focal-field modulation to achieve a transverse spatial resolution of ∼60  nm (∼λ/10). We show that NFOMM is comparable with STED microscopy and suitable for fundamental biology studies, as evidenced in imaging nuclear pore complexes, tubulin and vimentin in Vero cells. Since NFOMM is readily implemented as an add-on module to a laser-scanning microscope, we anticipate wide utility of this new imaging technique.

High precision wavefront control in point spread function engineering for single emitter localization

Point spread function (PSF) engineering is used in single emitter localization to measure the emitter position in 3D and possibly other parameters such as the emission color or dipole orientation as well. Advanced PSF models such as spline fits to experimental PSFs or the vectorial PSF model can be used in the corresponding localization algorithms in order to model the intricate spot shape and deformations correctly. The complexity of the optical architecture and fit model makes PSF engineering approaches particularly sensitive to optical aberrations. Here, we present a calibration and alignment protocol for fluorescence microscopes equipped with a spatial light modulator (SLM) with the goal of establishing a wavefront error well below the diffraction limit for optimum application of complex engineered PSFs. We achieve high-precision wavefront control, to a level below 20 mλ wavefront aberration over a 30 minute time window after the calibration procedure, using a separate light path for calibrating the pixel-to-pixel variations of the SLM, and alignment of the SLM with respect to the optical axis and Fourier plane within 3 μm (x/y) and 100 μm (z) error. Aberrations are retrieved from a fit of the vectorial PSF model to a bead z-stack and compensated with a residual wavefront error comparable to the error of the SLM calibration step. This well-calibrated and corrected setup makes it possible to create complex '3D+λ' PSFs that fit very well to the vectorial PSF model. Proof-of-principle bead experiments show precisions below 10 nm in x, y, and λ, and below 20 nm in z over an axial range of 1 μm with 2000 signal photons and 12 background photons.

Tilted Light Sheet Microscopy with 3D Point Spread Functions for Single-Molecule Super-Resolution Imaging in Mammalian Cells

To obtain a complete picture of subcellular nanostructures, cells must be imaged with high resolution in all three dimensions (3D). Here, we present tilted light sheet microscopy with 3D point spread functions (TILT3D), an imaging platform that combines a novel, tilted light sheet illumination strategy with engineered long axial range point spread functions (PSFs) for low-background, 3D super localization of single molecules as well as 3D super-resolution imaging in thick cells. TILT3D is built upon a standard inverted microscope and has minimal custom parts. The axial positions of the single molecules are encoded in the shape of the PSF rather than in the position or thickness of the light sheet, and the light sheet can therefore be formed using simple optics. The result is flexible and user-friendly 3D super-resolution imaging with tens of nm localization precision throughout thick mammalian cells. We validated TILT3D for 3D super-resolution imaging in mammalian cells by imaging mitochondria and the full nuclear lamina using the double-helix PSF for single-molecule detection and the recently developed Tetrapod PSF for fiducial bead tracking and live axial drift correction. We envision TILT3D to become an important tool not only for 3D super-resolution imaging, but also for live whole-cell single-particle and single-molecule tracking.

3D single-molecule super-resolution microscopy with a tilted light sheet

Tilted light sheet microscopy with 3D point spread functions (TILT3D) combines a novel, tilted light sheet illumination strategy with long axial range point spread functions (PSFs) for low-background, 3D super-localization of single molecules as well as 3D super-resolution imaging in thick cells. Because the axial positions of the single emitters are encoded in the shape of each single-molecule image rather than in the position or thickness of the light sheet, the light sheet need not be extremely thin. TILT3D is built upon a standard inverted microscope and has minimal custom parts. The result is simple and flexible 3D super-resolution imaging with tens of nm localization precision throughout thick mammalian cells. We validate TILT3D for 3D super-resolution imaging in mammalian cells by imaging mitochondria and the full nuclear lamina using the double-helix PSF for single-molecule detection and the recently developed tetrapod PSFs for fiducial bead tracking and live axial drift correction.

Observation of live chromatin dynamics in cells via 3D localization microscopy using Tetrapod point spread functions

We report the observation of chromatin dynamics in living budding yeast (Saccharomyces cerevisiae) cells, in three-dimensions (3D). Using dual color localization microscopy and employing a Tetrapod point spread function, we analyze the spatio-temporal dynamics of two fluorescently labeled DNA loci surrounding the GAL locus. From the measured trajectories, we obtain different dynamical characteristics in terms of inter-loci distance and temporal variance; when the GAL locus is activated, the 3D inter-loci distance and temporal variance increase compared to the inactive state. These changes are visible in spite of the large thermally- and biologically-driven heterogeneity in the relative motion of the two loci. Our observations are consistent with current euchromatin vs. heterochromatin models.