The long-term simulation of planetary systems poses significant challenges due to the inherently chaotic and non-integrable nature of gravitational interactions in the N-body problem. This thesis examines the Wisdom–Holman symplectic integration scheme, a method specifically designed for nearly integrable systems. This scheme separates the dominant Keplerian motion from weaker perturbative forces, enabling stable integration over astronomical timescales. Emphasis is placed on understanding the practical limitations and capabilities of this method when using large time steps, particularly in the presence of mean-motion resonances and step-size resonances. Through extensive numerical experiments using the Rebound simulation package, the scaling behavior of integration errors is characterized, revealing a transition from secondorder to lower-order error regimes at large time steps. This shows that in certain systems, time steps significantly larger than the shortest orbital period can still yield acceptable accuracy. However, in systems with mean-motion resonance, strong sensitivity to step-size resonances is observed, requiring careful step-size selection. A comparison between Jacobi and Democratic Heliocentric coordinates shows that the former performs best when orbits are nested, while the latter is better suited to systems with crossing or unordered orbits. These findings provide practical guidelines for applying Wisdom–Holman integration effectively across a range of dynamical regimes.

Single molecule localization microscopy (SMLM) surpasses the diffraction limit by sequentially imaging individual fluorescent molecules. By incorporating 4Pi detection to capture self-interference from the fluorescence emitters via two objective lenses, 4Pi-SMLM achieves isotropic sub 5 nm resolution in three dimensions (3D). This thesis details the design, construction, and evaluation of a Cryo-4Pi-SMLM system, aiming to enable subcellular 3D fluorescence imaging in native states with isotropic localization precision under cryogenic conditions, providing a 3D correlative light and electron microscopy solution...

The introduction and adoption of seismic full waveform inversion (FWI) revolutionized Earth's subsurface imaging practices. FWI uses all the information in the seismic data (amplitudes and phases) to reconstruct a detailed Earth's subsurface model. However, it does come with limitations. Beyond the reach of refractions and diving waves, FWI cannot effectively reconstruct subsurface layers. This led to the development of reflection waveform inversion (RWI), which exclusively uses the pair of transmission-after-reflection wavepaths to sample deeper compared to FWI. RWI reconstructs the background velocity model of the subsurface by alternating between a migration loop and a tomography loop.

Despite the theoretical appeal, RWI has its own share of limitations. This dissertation investigates the barriers limiting the optimal performance of reflection waveform inversion in the context of one-way RWI (ORWI), a variation of reflection waveform inversion that adopts one-way wavefield propagators to forward model seismic reflection data. After exploring the barriers, the dissertation offers solutions to improve the reliability, accuracy, and convergence of ORWI.

The dissertation acknowledges several barriers that limit the optimal performance of conventional/standard ORWI. First, ORWI relies on accurate subsurface images. However, limited-resolution images with unpreserved amplitudes, resulting from the migration loop, lead to suboptimal background velocity updates. This issue also extends to the tomography loop, where limited-resolution tomographic wavepaths impede optimal background velocity updates. Second, ORWI overlooks the immediate impact that updating the velocity model has on the reflectivity model, as the reflectors' positions in depth remain fixed while the background velocity is updated. This oversight leads to inconsistent reflectivity and velocity models in ORWI, introducing full-wave inconsistencies in the short-offset residual waveforms for tomography. Third, similar to other seismic waveform inversion techniques, ORWI suffers from the detrimental effect of including cycle-skipped data from long offsets.

To mitigate the barriers, the dissertation proposes a range of solutions. Initially, the dissertation introduces a computationally efficient high-resolution migration algorithm called preconditioned least-squares wave-equation migration (PLS-WEM) through depth-dependent gradient preconditioning. PLS-WEM reconstructs high-resolution, amplitude-preserved seismic images in fewer iterations.

Following that, by incorporating PLS-WEM into standard ORWI, the dissertation enhances ORWI, achieving improved reflectivity imaging and thereby reconstructing tomograms that are more representative of the true subsurface layers. The dissertation also proposes the following data solutions: (a) Muting short-offset residual waveforms in the tomography data to reduce the adverse imprint of inconsistencies between the reflectivity and velocity models on the tomographic gradient of ORWI. (b) Building on (a), extending the migration offset to the maximum effective migration offset (MEMO) to enhance both the signal-to-noise ratio and the illumination of the reflectivity model. (c) Introducing a data selection algorithm to minimize the impact of cycle-skipped long-offset data.

The dissertation then presents high-resolution ORWI (HR-ORWI) technology, which leverages depth-dependent gradient preconditioning in both migration and tomography loops to reconstruct optimal tomograms in fewer cycles.

The dissertation next evaluates three approaches to depth-dependent gradient preconditioning: conventional, source-interference-free, and source-interference-inclusive. Numerical results show the superiority of the source-interference-inclusive approach, offering enhanced resolution, reduced computational demands, and resilience to source interference.

Lastly, the dissertation develops a mathematical framework that integrates early-arrival waveform inversion with ORWI through the subspace gradient method, combining the strengths of transmission and transmission-after-reflection wavepaths to enhance tomogram reconstruction.

In conclusion, this dissertation offers a comprehensive set of solutions to overcome the limitations of ORWI, facilitating its broader adoption and application in seismic exploration and velocity model building.

High-resolution microscopy techniques, such as Single-Molecule Localization Microscopy (SMLM) and Cryogenic Electron Microscopy (Cryo-EM), can utilize particle fusion or averaging to reconstruct a macromolecular structure of increased signal-to-noise ratio and of potentially higher resolution. This process assumes that all fused particles are structurally equal. Structural heterogeneity, however, is often present due to biological variations and should not be ignored. In particularly continuous and subtle conformational changes present in the data lead to undesired blurring of the reconstruction. This thesis develops methods to detect continuous structural heterogeneity and to exploit it for more faithful reconstructions, enabling more accurate interpretations and insights into molecular structures.

In Chapter 2, we propose a method to detect continuous structural heterogeneity in SMLM datasets based on an all-to-all pairwise comparison of the found structures. The method is applied to both experimental and simulated data, where continuous variations such as the height of 3D DNA origami tetrahedrons and the radius of 2D Nuclear Pore Complexes (NPCs) are detected. The chapter highlights how accounting for these structural variations leads to more reliable particle fusion and reconstruction.

In Chapter 3, we propose a Point Cloud Variational Auto-Encoder (PCVAE) that operates directly on 2D and 3D localization data to detect structural heterogeneity. Unlike common neural networks that rely on pixelated images, our method utilizes raw localization coordinates. This not only reduces the required memory but also has low computational complexity and thus allows scalability to many structures. In contrast to multi-dimensional scaling approaches, where the computational complexity scales quadratically, here it remains linear with the number of particles. Our method is capable of identifying multiple modes of variation and reveals nanometer-scale changes such as radius and height variations in both simulated and experimental datasets.

In Chapter 4, we propose a method to detect continuous structural heterogeneity in Cryo-EM datasets. Recent approaches rely on machine learning models that often require large training datasets and careful tuning of hyperparameters.
%These machine learning methods are often hindered by a lack of interpretability and consistency due to the non-linear mapping onto a low-dimensional latent space.
In contrast, our method detects underlying continuous variations in 2D projections by pairwise comparison of images within orientation classes. The approach reconstructs intermediate conformational states representing the continuous structural heterogeneity in synthetic SARS-CoV-2 spike protein data, simulated under ideal conditions. More realistic simulations, incorporating varying defocus per particle and radiation damage, do not lead to the same favourable results, still posing a challenge for future research.

For a long time, the resolution of light microscopy was restricted to approximately 200 nm, as described by Abbe’s diffraction limit. Single-Molecule Localization Microscopy (SMLM) overcomes this limit by capturing many frames of a sample labeled with blinking fluorophores, where each frame shows a different subset of molecules. The fluorophores are located by fitting a model of their point spread function (PSF) and these localizations are combined to create an image with 20-50 nm resolution. Localizing a fluorophore is only possible when no other fluorescent molecules are active in the surrounding diffraction-limited area so that its PSF appears as an isolated blob. However, high-density regions unavoidably contain overlapping PSFs, which lead to localization errors as the number of emitters in the region of interest is unknown. To prevent this, the density of active fluorophores is typically kept between 0.01-0.1 μm−2 and data acquisition can take up to days, making dynamic imaging impossible. This theoretical research explores the possibility of enabling high-density SMLM by using Single-Photon Avalanche Diode (SPAD) arrays, detecting every incident photon with picosecond timing precision, instead of capturing the total intensity in a 10-100 ms interval like the conventionally used sCMOS cameras. The photon arrival times from simulated SPAD measurements are used to determine the number of emitters in the field of view, which is directly related to the second-order quantum coherence of the signal. This coherence can be calculated by dividing the arrival times into discrete time bins and convoluting the signal with itself. To correct the number of emitters for the bias that is introduced by the discretization of the data, an analytical expression is derived and validated with simulations. Using this correction, the number of Alexa647 fluorophores can be determined from a 0.1 ms simulated measurement with a relative standard error of 1% independent of emitter count, using a laser intensity of 330 kWcm−2 and 100% detection efficiency. Considering an experimental setting in which 10 kWcm−2 intensity and 10% detection efficiency are more realistic, a 1.5 s measurement is needed to obtain the same accuracy, and a 15 ms interval is required to obtain a standard error of 10%. The newly acquired information about the number of emitters will make it possible to locate fluorophores with overlapping point spread functions for multi-emitter fitting. Consequently, SPAD arrays will provide the ability to image high emitter densities, which enables faster data acquisition and dynamic imaging.

This thesis explores advanced computational techniques in super-resolution microscopy (SRM), with the primary goal of pushing the limits of achievable resolution towards the 1 nm scale. It includes developments in particle fusion algorithms, data analysis of complex biological structures, and exploration of the impact of molecular dipole orientation on MINFLUX localization accuracy and precision.

In the first part, we present a novel fast particle fusion method tailored to single molecule localization microscopy (SMLM). This method first registers particles based on Joint Registration of Multiple Point Clouds (JRMPC) and then classifies and reconnects misaligned locally optimally clustered sets of particles. This approach significantly reduces computational cost compared to earlier template free methods in particular for a large number of particles.. This advancement enables more detailed and accurate reconstructions of super-particles, enhancing the capabilities of SMLM.

The second part of the dissertation deals with a data analysis of nuclear pore complexes (NPCs) reconstructed by the earlier developed particle fusion technique. By fusing thousands of NPCs labeled at nucleoporin Nup96 and analyzing the high-resolution reconstructions, we reveal intricate details of the NPC structure, in particular the unit structure of Nup96. This analysis showcases the potential of SRM in combination with advanced data analysis to contribute to structural biology on the length scale below 10 nm.

The third part focuses on the influence of the dipole orientation on the localization accuracy and precision of MINFLUX. We simulate the imaging process with a physically realistic vector diffraction Point Spread Function (PSF) model and then localize the emitters based on the simplified Gaussian doughnut PSF model used in MINFLUX so far. Our study, including dipoles with free and fixed orientations and key simulation parameters, reveals the need for more refined modeling to overcome the bias, especially for fixed dipole orientations and background fluorescence. This investigation helps to understand the limitations of MINFLUX in its current form and paves the way for future improvements of the technique.

Finally, we discuss potential future directions for improving SRM techniques. These include refining the fast particle fusion method by incorporating localization uncertainties and prior knowledge, optimizing experimental parameters in MINFLUX, and developing advanced localization strategies to improve accuracy and efficiency. By addressing these future challenges, SRM technologies can move closer to the goal of 1 nm resolution in super-resolution imaging.

Localization microscopy has circumvented the diffraction limit by sequentially imaging individual light emitting molecules at a time. The position of these individual molecules can be determined and a super-resolution reconstruction is made with improved resolution. Normally freely rotating emitters are used such that the point spread function (PSF) is rotationally symmetric and only minor errors in the localization process are made by approximating the PSF with a Gaussian. The precision with which the individual emitters can be localized scales with the 1/√N, N the number of detected photons so that more detected photons leads to a better localization precision. However, the emission of fluorescent molecules is limited by photobleaching, a light induced chemical reaction to a permanent non-fluorescent state. In this thesis we investigate the effect of cooling the sample to cryogenic temperatures with liquid nitrogen. This reduces the chemical reaction rates and improves photostability more than 100 fold. To use localization microscopy it is necessary to switch the fluorescent molecules between an on-state and off-state, this turns out to be difficult at cryogenic temperatures. Standard methods used at room temperature in aqueous media do not work. As the molecules are frozen in place at cryogenic temperatures we use polarized light to selectively image molecules with certain orientations at a time. To realize this it is necessary to generate pure linear polarization with an arbitrary orientation in the sample plane. By calibrating the phase difference induced by the dichroic mirrors this can be achieved, effectively modulating the fluorescence of fixed dipole emitters at cryogenic temperatures. The addition of an orthogonal linearly polarized stimulated emission depletion (STED) beam narrows the orientational distribution of fluorescing molecules. This method does induce some degree of sparsity, however, it is not enough for localization microscopy of dense biological samples. Furthermore, the STED process reduces the photon yield of single molecules. This is presumably caused by the long dark-state recovery measured on fluorescent molecules in vacuum and at cryogenic temperatures. Localization microscopy of fixed or orientationally constrained emitters has long been avoided as the orientation of individual molecules leads to bias in the localizations. There are various ways to eliminate this bias but they reduce the amount of information that can be extracted from the sample. By fixing the orientation of fluorescent emitters to biomolecules of interest they become reporters for the orientation of the biomolecules. We have devised the so-called Vortex PSF with which the orientation, 3D position and degree of rotational constraint can be extracted from a single image. Alternatively the orientation of single-molecules can be probed with varying polarization states over multiple frames achieving a better precision with less photons.

The time taken to generate a super-resolution image and the quality of the final synthetic image depends on the performance of the localization algorithm which is used in the localization microscopy pipeline. The most precise and accurate algorithms are mostly iterative and they take a long time to generate the localization list while the faster ‘one-shot algorithms’ are not very accurate and precise. A deep learning method smNet (single-molecule Net) was developed by Zhang et al which was claimed to perform one-shot localization with precision close to the theoretical limit and very accurately, along with performing aberration estimation and dipole-emitter orientation angle estimation. The deep
learning model smNet was trained either by augmenting experimental data or using simulated data generated with an erroneously simplified simulation model and a phase retrieval method. The purpose of this work was to characterize the performance of smNet when it was trained with simulated images generated using an accurate vector model for a range of physical conditions. Along with the characterization of smNet’s performance in doing 3D localization and aberration estimation with the accurate vector model, a pipeline was also designed which made the training process of smNet more efficient and computationally cheaper while performing accurate and precise 3D localization and aberration estimation.
The pipeline was designed to implement the concept of simulator learning where a smNet model could be trained on simulated data and used to perform 3D localization and aberration estimation directly on experimental data without any retraining or domain adaptation techniques.

Single molecule localization microscopy (SMLM) shows promise for quantitative structural analysis of subcellular complexes and organelles with a resolution well below the diffraction limit. This superresolution microscopy technique relies on the blinking events of fluorescent molecules that labeled the structure of interest and are spatiotemporally spread over the entire field of view and time. Once hundred thousands frames of these sparse events are recorded, single molecule positions are localized with nanometer precision to form a 2D/3D point set of coordinates. Therefore, SMLM images are not conventional pixelated images but rather spatial point patterns. Photon scarcity and incomplete labeling of the imaged structure, however, limit the resolution that can possibly be achieved by means of SMLM. Moreover, due to experimental limitations the axial resolution is typically ~2-3 times worse than the lateral resolution in conventional setups. Inspired by single particle analysis (SPA) in cryo-electron microscopy (cryo-EM), proper alignment of repeated structures ("particle fusion") in a 2D/3D SMLM measurement can overcome these limiting factors and so push for isotropic resolution. The existing approaches for particle fusion in SMLM can be classified into customized routines that are borrowed from SPA in EM or methods that use strong prior knowledge about the structure to be reconstructed. While the first approaches are completely ignoring the differences in image formation model between EM and SMLM, the second ones are highly prone to the template-bias problem. In this thesis, a dedicated particle fusion pipeline for 2D/3D SMLM data is proposed. The approach properly considers the pointillistic nature of the SMLM modality and takes into account the localization uncertainties. Furthermore, while it does not require any prior knowledge about the underlying structure of the particles, it can incorporate certain features such as symmetry into the fusion process. Owing to the novel all-to-all registration scheme, the application of the devised pipeline on experimental data with very poor labeling density has been successfully demonstrated. The requirements for successful particle fusion for different SMLM modalities, namely PAINT and STORM, have been characterized through extensive study on 2D and 3D experimental and simulation data. In 2D, an FRC resolution of 3.3 nm on DNA-origami nanostructures has been achieved, and, in 3D, it was demonstrated how the combination of SMLM as a light microscopy technique and a computational approach enables structural analysis of the Nuclear Pore Complex. Future advances of SMLM rely highly on computational routines after data acquisition. Advanced data analysis techniques such as particle fusion can help pushing the boundaries of structural biology using light microscopy.

This thesis explores a novel optical architecture for Whole Slide Imaging (WSI). This new architecture allows for multi-focal (3D) image acquisitions in a single scan pass. The multi-focal imaging capability is used to demonstrate 3D phase imaging and 3D imaging of thick tissue sections on a prototype scanner. Further, instrumentation for the extension of WSI to fluorescence imaging is developed: a technologically robust and cost effective method based on LEDs and a highly efficient method based on a multi-line laser illumination source. A WSI system is an optical instrument aimed at creating digital images of bio- logical samples mounted on a microscopy slide at a high throughput. WSI systems image tissues over large fields of view (∼ few cm), in 2D or in 3D (up to hundred layers of μm thickness), and at cellular resolution (∼1 μm). They are applied in high throughput screening in biology, and for novel computer aided medical diagnoses in the field of digital pathology. The optical architecture explored in this thesis is based on a tilted multi-line image sensor concept, originating from Philips. The goal of multi-line image ac- quisition is to enable closed-loop autofocus scanning, but it also allows for multi- focal image acquisitions. The core of this scanner concept lies in a novel design for a multi-line image sensor. The sensor is experimentally characterized for gain and noise, and a system model is developed to find the optimum signal-to-noise ratio (SNR) given the available photo-electron flux. This showed that images with a very high SNR of 292 can be acquired, provided that a sufficiently high photon flux can be realized. Two major issues with the sensor were found in our exper- imental characterization. At high line rates, the sensor showed missing symbols, leading to non-linearities in the read out. Further, the sensor showed high frequent fluctuations in the gain. 3D phase imaging and 3D imaging of thick specimens are two novel contrast modalities based on computational imaging techniques and are enabled by the availability of multi-focal images. For both techniques simplified algorithms were developed compatible with parallel processing at very high speeds. For 3D imaging of thick specimens a deconvolution technique is developed for improving the in- herently low axial contrast. 3D phase imaging is realized by a simplified algorithm for Quantitative Phase Tomography (QPT). QPT imaging is found to be able to im- age the sites labeled for Fluorescence in situ Hybridization (FISH) imaging, and provide additional structural information on unlabeled tissues or tissues stained for immunofluorescence. A system design study is presented showing that the in- plane transfer function has the character of a band-pass spatial frequency filter. The major opportunity for WSI systems to become compatible with fluores- cence imaging is addressed by the development of two imaging modalities. First, a widefield fluorescence WSI system with an LED illumination source is developed and built. A color sequential illumination strategy in combination with multi-band dichroics is used for multi-color imaging using a single monochromatic sensor. The main speed limitation is formed by the exposure time required to capture enough photo-electrons for a decent SNR. Based on the experimental results, a system with 96 Time Delayed Integration (TDI) lines is estimated to achieve a rea- sonable throughput of about 130 kPixel/s. This makes scanning possible of an area of 15 × 15 mm2 in three colors in about 23 min. Second, a novel optical architecture for multi-focal fluorescence image acqui- sitions based on a laser illumination source is proposed and realized in a proto- type. Illumination PSF engineering using diffractive optics is applied to generate a set of parallel scan lines in object space, that span a plane conjugate to a tilted im- age sensor. An important new element in the design is the use of higher order astig- matism to improve the uniformity of peak intensity and line width along the scan lines. Focusing the illumination on the sample provides a very high illumination efficiency and a confocal suppression of background. This optical architecture is projected to ultimately achieve a throughput of several hundreds MPixel/s, which would enable scanning an area of 15 × 15 mm2 in 8 layers in less than a minute. This thesis is concluded with an outlook to opportunities for future research in WSI systems. The challenges and some potential solutions for using a general purpose scientific CMOS (sCMOS) camera for multi-line scanning of a tilted ob- ject plane, and some opportunities for extension of WSI techniques to Light Sheet Microscopy (LSM) and Structured Illumination Microscopy (SIM) are discussed. In summary, this thesis investigates the imaging qualities and extension to computational imaging modalities of a brightfield WSI system and describes two approaches for fluorescence WSI.

A limiting factor with regard to resolution in OPT is the limited depth of field (DoF) due to light detection with a Gaussian beam profile. The further a source in the sample is removed from the centre of rotation in the focal plane, the more distorted the image is in tangential direction due to the limited DoF. The goal of this research is to extend the depth of field in Optical Projection Tomography. The DoF limitations due to diffraction are mainly caused by the Gaussian beam shape and its inherent limitations such as a small high intensity spot and a small region of focus. An alternative to Gaussian beams is found sporadically in the literature in the form of non-diffracting beams. Non-diffracting beams are beams that propagate without diffraction and show regenerative properties after obstruction. The Bessel beam is a non-diffracting beam that is rotationally symmetric and displays a transversal high-intensity core. It can be generated without energy loss with a lens shaped like a rotationally symmetric prism, called an axicon.
A propagation simulation of the axicon generated Bessel beam, using the Hankel transform as a rotationally symmetric alternative to the 2D Fourier transform, is performed and used to verify the analytic description of the axicon generated Bessel beam. A numerical OPT simulation shows that OPT reconstructions of point sources show virtually no blurring, but do show concentric rings due to the intensity distribution of the Bessel beam. These rings can be removed by deconvolution of the projection or deconvolution of the reconstruction of simulated OPT results with the imaging point spread function (PSF) of the axicon-generated Bessel beam. The PSF describes the response of the imaging system to a point source.
Practical work is presented with the imaging set-up of an axicon with an objective lens as described earlier. The resolution of the PSF is analysed over paraxial distance from the objective lens for both coherent and incoherent illumination. The same is done for a resolution target in transmission. Comparison of the Bessel system with Gaussian models show that the DoF increase shown by the Bessel system is significant in all cases. It is found that for sources with spatially narrow intensity distributions (near-point source) the PSF resolution matches theoretical predictions. However, as the spatial light source distribution increases slightly, Bessel distributions overlap spatially. This creates artefacts and deteriorates the resolution
It is concluded that extended depth of field in OPT can be achieved with non-diffracting axicon generated Bessel beams. However, for objects larger than point sources the resolution deteriorates. Furthermore, the means of illumination are a major influence on the resulting images when using an axicon.
Further research on the optimization of illumination is recommended. Additionally, recommendations for further research on the significance of self-regeneration are made. Recommended applications for use of Bessel beams in optical imaging are those where a large DoF is desired and high resolution is less of a priority, or imaging and OPT of very sparse but large samples.