Classical mechanics is the starting point of physics. Over the centuries, via Newton’s three fundamental laws formulated around 1687, we have built a solid framework describing the material world around us. This textbook includes animations, demos and exercises for studying introductory classical mechanics. Moreover, we will consider the first steps of Einstein’s Special Theory of Relativity published 1905.

This book starts from the three laws of Newton and builds up classical mechanics from scratch. Its content carefully constructs the classical mechanics from Newtons laws, providing a rigorous, fundamental framework. Emphasis is placed on working with the basic concepts of momentum, energy and angular momentum. Special attention is paid to oscillations, collisions including non-ideal ones and the Kepler problem. Rigid body dynamics is extensively discussed. The use of polar coordinates is extensively discussed.

Non-inertial frames of reference are treated starting from linearly accelerating ones and moving to the rotating frames of reference. The centrifugal and Coriolis forces are introduced and explained.

In the second part, the switch to special relativity is made. The problems coming out of Maxwells equations for electro-magnetic waves are used as the step towards modern mechanics. The Lorentz Transformation, where events in 4-dimensional space, time dilation and length contraction are discussed. Furthermore, the book introduces the world of four-vectors, deriving the four-velocity and four-momentum. Velocity transformation, Doppler shift and relativistic collision form the final parts of the book.

Structured illumination microscopy (SIM) is a powerful method for high-resolution 3D-imaging that is compatible with standard fluorescence labeling techniques, as it provides optical sectioning as well as an up to twofold improvement of lateral resolution over widefield microscopy by combining illumination pattern diversity with computational reconstruction. We present a quantitative analysis of the image quality of 3D-SIM using the spectral signal-to-noise ratio (SSNR). In particular, we compare conventional woodpile illumination pattern based 3D-SIM, where the pattern is rotated and translated to acquire the set of raw images that is fed into the reconstruction algorithm, to (square or hexagonal) lattice 3D-SIM, where the pattern is only translated to assemble the input set of raw images. It appears that conventional 3D-SIM has better SSNR than the considered cases of lattice 3D-SIM. In addition, we have also analyzed the impact of the relative amplitude, angle of incidence and polarization of the set of illumination plane waves on image quality, and show how two SSNR derived metrics, SSNR volume and SSNR entropy, can be used to optimize these illumination pattern parameters.

Image quality in single-molecule localization microscopy depends largely on the accuracy and precision of the localizations. While under ideal imaging conditions, the theoretically obtainable precision and accuracy are achieved; in practice, this changes if (field-dependent) aberrations are present. Currently, there is no simple way to measure and incorporate these aberrations into the point-spread function (PSF) fitting; therefore, the aberrations are often taken as constant or neglected altogether. Here we introduce a model-based approach to estimate the field-dependent aberration directly from single-molecule data without a calibration step. This is made possible by using nodal aberration theory to incorporate the field dependency of aberrations into our fully vectorial PSF model. This results in a limited set of aberration fit parameters that can be extracted from the raw frames without a bead calibration measurement, also in retrospect. The software implementation is computationally efficient, enabling the fitting of a full 2D or 3D dataset within a few minutes. We demonstrate our method on 2D and 3D localization data of microtubuli, nuclear pore complexes, and nuclear lamina over fields of view of up to 180 µm and compare it with Gaussian fitting, spline-based fitting, and a deep-learning-based approach.

We address resolution assessment for (light super-resolution) microscopy imaging. In modalities where imaging is not diffraction limited, correlation between two noise independent images is the standard way to infer the resolution. Here we take away the need for two noise independent images by computationally splitting one image acquisition into two noise independent realizations. This procedure generates two Poisson noise distributed images if the input is Poissonian distributed. As most modern cameras are shot-noise limited this procedure is directly applicable. However, also in the presence of readout noise we can compute the resolution faithfully via a correction factor. We evaluate our method on simulations and experimental data of widefield microscopy, STED microscopy, rescan confocal microscopy, image scanning microscopy, conventional confocal microscopy, and transmission electron microscopy. In all situations we find that using one image instead of two results in the same computed image resolution.

DNA-origami nanostructures have shown promising applications in single molecule localization microscopy. They have become a reference standard for benchmarking and for developing new techniques for nanoscopy. Here, we present a pipeline for quantifying the quality of these nano-structures when imaging multiple instances of them using DNA-PAINT technique. We show on several experimental datasets that these structures can have deformations and that the designed binding sites are not equally accessible for the labelled imager strands during the image acquisition process. These limitations result in non-uniform activation of the sites over the origami pattern when fusing the instances into a single reconstruction.

We quantify the precision and bias of dynamic light scattering optical coherence tomography (DLS-OCT) measurements of the diffusion coefficient and flow speed for first and second-order normalized autocovariance functions. For both diffusion and flow, the measurement precision and accuracy are severely limited by correlations between the errors in the normalized autocovariance function. We demonstrate a method of mixing statistically independent normalized autocovariance functions at every time delay for removing these correlations. The mixing method reduces the uncertainty in the obtained parameters by a factor of two but has no effect on the standard error of the mean. We find that the precision in DLS-OCT is identical for different averaging techniques but that the lowest bias is obtained by averaging the measured correlation functions before fitting the model parameters. With our correlation mixing method, it is possible to quantify the precision in DLS-OCT and verify whether the Cramer-Rao bound is reached.

We address (super)resolution assessment of light microscopy via Fourier Ring Correlation (FRC), based on a single camera image. Based on Poisson statistics we can split an image into two noise independent halves, and use this to compute the FRC. The technique is demonstrated on widefield, STED, ISM, and RCM modalities.

[This corrects the article DOI: 10.1117/1.JBO.27.10.106001.].

Correction to: Scientific Reports, published online 16 August 2023 The original version of this Article contained an error in the upper inset of Figure 4, where the atomic model was missing. The original Figure 4 and accompanying legend appear below. (Figure presented.) Overlay of the fluorophore positions from the SMLM particle fusion data (pink) and the SNAP-tag derived from the cryo-EM data (purple). For our overall SMLM emitters (pink), the lateral distance between a unit are 9.1 nm for NR and 10.0 nm for CR. The axial distances between a unit are 2.4 nm for NR and 1.2 nm for CR. The SNAP tags (purple) have lateral distances between a unit of 11.6 nm for NR and 11.5 nm for CR as well as axial distances of 2.5 nm for NR and 2.9 nm for CR. The original Article has been corrected.

Three dimensional modulation-enhanced single-molecule localization techniques, such as ModLoc, offer advancements in axial localization precision across the entire field of view and axial capture range, by applying phase shifting to the illumination pattern. However, this improvement is limited by the pitch of the illumination pattern that can be used and requires registration between separate regions of the camera. To overcome these limitations, we present ZIMFLUX, a method that combines astigmatic point-spread-function (PSF) engineering with a structured illumination pattern in all three spatial dimensions. In order to achieve this we address challenges such as optical aberrations, refractive index mismatch, supercritical angle fluorescence (SAF), and imaging at varying depths within a sample, by implementing a vectorial PSF model. In scenarios involving refractive index mismatch between the sample and immersion medium, the astigmatic PSF loses its ellipticity at greater imaging depths, leading to a deterioration in axial localization precision. In contrast, our simulations demonstrate that ZIMFLUX maintains high axial localization precision even when imaging deeper into the sample. Experimental results show unbiased localization of 3D 80 nm DNA-origami nanostructures in SAF conditions with a 1.5-fold improvement in axial localization precision when comparing ZIMFLUX to conventional SMLM methods that rely solely on astigmatic PSF engineering.

Objectives: Digital endoscopes are connected to a video processor that applies various operations to process the image. One of those operations is edge enhancement that sharpens the image. The purpose of this study was to (1) quantify the level of edge enhancement, (2) measure the effect on sharpness and image noise, and (3) study the influence of edge enhancement on image quality perceived by ENT professionals. Methods: Three digital flexible endoscopic systems were included. The level of edge enhancement and the influence on sharpness and noise were measured in vitro, while systematically varying the levels of edge enhancement. In vivo images were captured at identical levels of one healthy larynx. Each series of in vivo images was presented to 39 ENT professionals according to a forced pairwise comparison test, to select the image with the best image quality for diagnostic purposes. The numbers of votes were converted to a psychometric scale of just noticeable differences (JND) according to the Thurstone V model. Results: The maximum level of edge enhancement varied per endoscopic system and ranged from 0.8 to 1.2. Edge enhancement increased sharpness and noise. Images with edge enhancement were unanimously preferred to images without edge enhancement. The quality difference with respect to zero edge enhancement reaches an optimum at levels between 0.7 and 0.9. Conclusion: Edge enhancement has a major impact on sharpness, noise, and the resulting perceived image quality. We conclude that ENT professionals benefit from this video processing and should verify if their equipment is optimally configured. Level of Evidence: N/A Laryngoscope, 2023.

Single molecule localization microscopy offers resolution nearly down to the molecular level with specific molecular labelling, and is thereby a promising tool for structural biology. In practice, however, the actual value to this field is limited primarily by incomplete fluorescent labelling of the structure. This missing information can be completed by merging information from many structurally identical particles in a particle fusion approach similar to cryo-EM single-particle analysis. In this paper, we present a data analysis of particle fusion results of fluorescently labelled Nup96 nucleoporins in the Nuclear Pore Complex to show that Nup96 occurs in a spatial arrangement of two rings of 8 units with two Nup96 copies per unit giving a total of 32 Nup96 copies per pore. We use Artificial Intelligence assisted modeling in Alphafold to extend the existing cryo-EM model of Nup96 to accurately pinpoint the positions of the fluorescent labels and show the accuracy of the match between fluorescent and cryo-EM data to be better than 3 nm in-plane and 5 nm out-of-plane.

Fusion of multiple chemically identical complexes, so-called particles, in localization microscopy, can improve the signal-to-noise ratio and overcome under-labeling. To this end, structural homogeneity of the data must be assumed. Biological heterogeneity, however, could be present in the data originating from distinct conformational variations or (continuous) variations in particle shapes. We present a prior-knowledge-free method for detecting continuous structural variations with localization microscopy. Detecting this heterogeneity leads to more faithful fusions and reconstructions of the localization microscopy data as their heterogeneity is taken into account. In experimental datasets, we show the continuous variation of the height of DNA origami tetrahedrons imaged with 3D PAINT and of the radius of Nuclear Pore Complexes imaged in 2D with STORM. In simulation, we study the impact on the heterogeneity detection pipeline of Degree Of Labeling and of structural variations in the form of two independent modes.

Summary: We present a fast particle fusion method for particles imaged with single-molecule localization microscopy. The state-of-the-art approach based on all-to-all registration has proven to work well but its computational cost scales unfavorably with the number of particles N, namely as N2. Our method overcomes this problem and achieves a linear scaling of computational cost with N by making use of the Joint Registration of Multiple Point Clouds (JRMPC) method. Straightforward application of JRMPC fails as mostly locally optimal solutions are found. These usually contain several overlapping clusters that each consist of well-aligned particles, but that have different poses. We solve this issue by repeated runs of JRMPC for different initial conditions, followed by a classification step to identify the clusters, and a connection step to link the different clusters obtained for different initializations. In this way a single well-aligned structure is obtained containing the majority of the particles. Results: We achieve reconstructions of experimental DNA-origami datasets consisting of close to 400 particles within only 10 min on a CPU, with an image resolution of 3.2 nm. In addition, we show artifact-free reconstructions of symmetric structures without making any use of the symmetry. We also demonstrate that the method works well for poor data with a low density of labeling and for 3D data.

Single-molecule localization microscopy super-resolution methods rely on stochastic blinking/binding events, which often occur multiple times from each emitter over the course of data acquisition. Typically, the blinking/binding events from each emitter are treated as independent events, without an attempt to assign them to a particular emitter. Here, we describe a Bayesian method of inferring the positions of the tagged molecules by exploring the possible grouping and combination of localizations from multiple blinking/binding events. The results are position estimates of the tagged molecules that have improved localization precision and facilitate nanoscale structural insights. The Bayesian framework uses the localization precisions to learn the statistical distribution of the number of blinking/binding events per emitter and infer the number and position of emitters. We demonstrate the method on a range of synthetic data with various emitter densities, DNA origami constructs and biological structures using DNA-PAINT and dSTORM data. We show that under some experimental conditions it is possible to achieve sub-nanometer precision.

Combining orientation estimation with localization microscopy opens up the possibility to analyze the underlying orientation of biomolecules on the nanometer scale. Inspired by the recent improvement of the localization precision by shifting excitation patterns (MINFLUX, SIMFLUX), we have adapted the idea towards the modulation of excitation polarization to enhance the orientation precision. For this modality two modes are analyzed: i) normally incident excitation with three polarization steps to retrieve the in-plane angle of emitters and ii) obliquely incident excitation with p-polarization with five different azimuthal angles of incidence to retrieve the full orientation. Firstly, we present a theoretical study of the lower precision limit with a Cramér-Rao bound for these modes. For the oblique incidence mode we find a favorable isotropic orientation precision for all molecular orientations if the polar angle of incidence is equal to arccos ^√2/3 ≈ 35 degrees. Secondly, a simulation study is performed to assess the performance for low signal-to-background ratios and how inaccurate illumination polarization angles affect the outcome. We show that a precision, at the Cramér-Rao bound (CRB) limit, of just 2.4 and 1.6 degrees in the azimuthal and polar angles can be achieved with only 1000 detected signal photons and 10 background photons per pixel (about twice better than reported earlier). Lastly, the alignment and calibration of an optical microscope with polarization control is described in detail. With this microscope a proof-of-principle experiment is carried out, demonstrating an experimental in-plane precision close to the CRB limit for signal photon counts ranging from 400 to 10,000.

SignificanceFlexible endoscopes are essential for medical internal examinations. Digital endoscopes are connected to a video processor that can apply various operations to enhance the image. One of those operations is edge enhancement, which has a major impact on the perceived image quality by medical professionals. However, the specific methods and parameters of this operation are undisclosed and the arbitrary units to express the level of edge enhancement differ per video processor.AimObjectively quantify the level of edge enhancement from the recorded images alone, and measure the effect on sharpness and noise.ApproachEdge enhancement was studied in four types of flexible digital ear nose and throat endoscopes. Measurements were performed using slanted edges and gray patches. The level of edge enhancement was determined by subtracting the step response of an image without edge enhancement from images with selected settings of edge enhancement and measuring the resulting peak-to-peak differences. These values were then normalized by the step size. Sharpness was characterized by observing the normalized modulation transfer function (MTF) and computing the spatial frequency at 50% MTF. The noise was measured on the gray patches and computed as a weighted sum of variances from the luminance and two chrominance channels of the pixel values.ResultsThe measured levels were consistent with the level set via the user interface on the video processor and varied typically from 0 to 1.3. Both sharpness and noise increase with larger levels of edge enhancement with factors of 3 and 4 respectively.ConclusionsThe presented method overcomes the issue of vendors expressing the level of edge enhancement each differently in arbitrary units. This allows us to compare the effects, and we can start exploring the relationship with the subjectively perceived image quality by medical professionals to find substantiated optimal settings.

Particle fusion for single molecule localization microscopy improves signal-to-noise ratio and overcomes underlabeling, but ignores structural heterogeneity or conformational variability. We present a-priori knowledge-free unsupervised classification of structurally different particles employing the Bhattacharya cost function as dissimilarity metric. We achieve 96% classification accuracy on mixtures of up to four different DNA-origami structures, detect rare classes of origami occuring at 2% rate, and capture variation in ellipticity of nuclear pore complexes.

Total internal reflection fluorescence (TIRF) microscopy is an important imaging tool for the investigation of biological structures, especially the study on cellular events near the plasma membrane. Imaging at cryogenic temperatures not only enables observing structures in a near-native and fixed state but also suppresses irreversible photo-bleaching rates, resulting in increased photo-stability of fluorophores. Traditional TIRF microscopes produce an evanescent field based on high numerical aperture immersion objective lenses with high magnification, which results in a limited field of view and is incompatible with cryogenic conditions. Here, we present a waveguide-based TIRF microscope, which is able to generate a uniform evanescent field using high refractive index waveguides on photonic chips and to obtain cellular observation at cryogenic temperatures. Our method provides an inexpensive way to achieve total-internal-reflection fluorescence imaging under cryogenic conditions.

With the growing popularity of cryogenic correlative light and electron microscopy, it is becoming increasingly important to bridge the resolution gap between these two modalities. At cryogenic temperatures, the photon yield of fluorophores is a few orders of magnitude higher than at room temperature, enabling localization precisions on the Ångström scale. The current challenge is to induce sparsity at cryogenic temperatures such that individual fluorescent molecules can be localized. In this paper, we demonstrate the progress of using polarized stimulated-emission depletion (STED) to induce sparsity at cryogenic temperatures and in vacuum. We generate linear polarization of arbitrary in-plane orientations to achieve polarized STED with a sparsity of 3.3:1. Furthermore, we have probed the dark-state lifetime of ATTO 647N at cryogenic temperatures and in vacuum at room temperature. This dark state in vacuum is long-lived (τ=38 ms) and could be the cause for reduced photostability of fluorophores under STED illumination in vacuum. The experiments were done on an in-house designed and built liquid nitrogen cryostat, enabling 30 hours of stable cryogenic fluorescence microscopy.