Forensic microtrace investigation relies on time- and labour-intensive microscopic analyses. To aid forensic experts in their investigations, an image recognition model for microtrace localisation and classification is needed. In this work, we use deep learning to automate trace recognition in images captured with automated microscopy. We localise and classify fibres, hairs, skin, glass and sand in microscopy scans through pixel-wise classification of tape-lift samples. As deep learning requires extensive amounts of annotated training data, we additionally investigate various pretraining strategies to minimise the required annotation workload. We compare ImageNet pretraining, pretraining with self-supervised learning and a sequential application of these approaches. We find that pretrained models are able to reduce the required annotated data twofold compared to models trained from scratch while retaining the prediction accuracy. While our ImageNet-pretrained models outperform our self-supervised-pretrained models, we achieve the highest accuracy by combining the two approaches, resulting in a factor 4 reduction of manual annotated microtraces or a 65 % improvement in recognition and localisation accuracy (mean intersection over union increases from 0.34 to 0.56 due to pretraining) when training on only 2.2 dm² of annotated tape lift scans. The developed models offer a solid fundament for automated analysis of forensic microtrace scans.

To assist in medicine development and microbiological research, microscopy has been an important tool ever since the seventeenth century. Fluorescence microscopy is able to provide the specificity and contrast needed for biological imaging, and the physical resolution limit caused by diffraction can be circumvented through super-resolution microscopy. By combining modulated excitation with sparse activation of fluorescent emitters and subsequent localization of emitter positions, modulation enhanced single molecule localization microscopy achieves a localization precision in the order of magnitude of nanometres to Angstroms, thereby making the invisible visible.

While these super-resolution methods allow us to access the nanoscale, their findings are accompanied by the statistical uncertainty about whether the molecule positions that we retrieve correspond to the true underlying positions of emitters that are truly present in the sample. A fundamental objective of super-resolution microscopy is thus to give certainty about the localization uncertainty with which the position of a single molecule can be determined. To make the uncertain certain in single-molecule localization, the Cramer-Rao lower bound is commonly used. The Cramer-Rao lower bound represents the theoretical minimum uncertainty with which unbiased estimators can localize emitters. However, the Cramer-Rao lower bound leads to narrowly applicable, improperly represented or mathematically incorrect characterizations of the localization precision of modulation enhanced single-molecule localization microscopy.

To address this, new and generalizable image formation models are needed. In addition, we need to develop statistical tools that represent the full estimator distribution, as well as the uncertainty of localization methods that use biased estimators.

In this dissertation, we address these issues through three major contributions. As our first contribution, we derive a new and generalizable image formation model that integrates modulation enhanced localization in existing setups that use a spinning disk in the illumination- and emission paths, leading to the theoretical design of a new method called SpinFlux. In the SpinFlux analysis, emitters are localized in the recordings from a sequence of individual pattern acquisitions, taking knowledge about the pattern into account. SpinFlux shows its merit when the excitation intensity is modulated to incorporate the maximum amount of information, reaching a 3.5-fold local precision improvement over single-molecule localization microscopy when using donut-shaped illumination patterns. Combined with the versatility of the image formation model to incorporate arbitrary illumination patterns, this makes SpinFlux the method of choice for local refinements of the localization precision.

Secondly, we analyse the occurrence of multimodality in three-dimensional multiple emitter imaging by reconstructing the full posterior distribution of localization. We develop a Bayesian three-dimensional localization method called three-dimensional reversible jump Markov chain Monte Carlo, which approximates the posterior density of emitter positions rather than giving point estimates. We show that astigmatic multiple emitter imaging results in a multimodal posterior distribution when two emitters are separated by less than the standard deviation of the in-focus point spread function, which causes ambiguous solutions to the estimation problem. This motivates the importance of including appropriately chosen uncertainty measures in localization algorithms. In particular, estimation of the full posterior distribution makes it possible to detect cases where the localization uncertainty for individual emitters is not accurately represented by Gaussian uncertainty ellipses, which would be misrepresented by the Cramer-Rao lower bound.

Lastly, we quantify and analyse the localization precision of iterative localization microscopy methods, such as MINFLUX. These methods are able to locally improve the localization precision around an emitter position by using prior information derived from measurements in earlier iterations. As the Cramer-Rao lower bound requires estimators to be unbiased, it cannot incorporate prior information, making it inapplicable to iterative localization microscopy. However, the Bayesian Van Trees inequality circumvents this mathematical limitation, and is therefore an appropriate bound to analyse iterative localization microscopy. By taking modulation- and background imperfections into account, we show that the improvement of iterative methods over single-molecule localization is at most fivefold. The Van Trees inequality allows us to nuance existing precision limits for methods resembling MINFLUX when affected by modulation- and background imperfections, by showing that the precision of these methods is not maximized by minimizing the pattern distance, nor exponentially improved by increasing the iteration count.

Based on these findings we argue that, in order to reflect the statistical uncertainty of the localization process, emitter position estimates in single-molecule localization microscopy should be presented in the context of the estimation uncertainty. Image formation models and uncertainty quantification should be tailored to the application, letting the particularities of the application determine the choice of appropriate mathematical tools. As shown in this dissertation, this attitude towards uncertainty leads to new experimental methods to improve the localization precision, and it advances our fundamental understanding of localization uncertainty in super-resolution microscopy.

Modulation enhanced single-molecule localization microscopy (meSMLM), where emitters are sparsely activated with sequentially applied patterned illumination, increases the localization precision over single-molecule localization microscopy (SMLM). The precision improvement of modulation enhanced SMLM is derived from retrieving the position of an emitter relative to individual illumination patterns, which adds to existing point spread function information from SMLM. Here, we introduce SpinFlux: modulation enhanced localization for spinning disk confocal microscopy. SpinFlux uses a spinning disk with pinholes in its illumination and emission paths, to sequentially illuminate regions in the sample during each measurement. The resulting intensity-modulated emission signal is analyzed for each individual pattern to localize emitters with improved precision. We derive a statistical image formation model for SpinFlux and we quantify the theoretical minimum localization uncertainty in terms of the Cramér-Rao lower bound. Using the theoretical minimum uncertainty, we compare SpinFlux to localization on Fourier reweighted image scanning microscopy reconstructions. We find that localization on image scanning microscopy reconstructions with Fourier reweighting ideally results in a global precision improvement of 2.1 over SMLM. When SpinFlux is used for sequential illumination with three patterns around the emitter position, the localization precision improvement over SMLM is twofold when patterns are focused around the emitter position. If four donut-shaped illumination patterns are used for SpinFlux, the maximum local precision improvement over SMLM is increased to 3.5. Localization of image scanning microscopy reconstructions thus has the largest potential for global improvements of the localization precision, where SpinFlux is the method of choice for local refinements.

In single-molecule microscopy, a big question is how precisely we can estimate the location of a single molecule. Our research shows that by using iterative localisation microscopy and factoring in the prior information, we can boost precision and reduce the number of photons needed. Leveraging the Van Trees inequality aids in determining the optimal precision achievable. Our approach holds promise for wider application in discerning the optimal precision across diverse imaging scenarios, encompassing various illumination strategies, point spread functions and overarching control methodologies.

Image scanning microscopy (ISM) achieves resolution beyond the diffraction limit by a factor of √2. However, prior ISM research predominantly employs scalar diffraction theory, neglecting critical physical effects such as polarization, aberrations, and Stokes shift. This paper presents a comprehensive vectorial ISM point spread function (PSF) model that accounts for these phenomena. By considering the effect of polarization in emission and excitation paths, as well as aberrations and Stokes shift, our model provides a more accurate representation of ISM. We analyze the differences between scalar and vectorial theories in ISM and investigate the impact of pinhole size and aberration strength on resolution. At a numerical aperture of 1.2, the full width half maximum (FWHM) discrepancy between scalar and vectorial ISM PSFs can reach 45 nm, representing a 30% deviation from the vectorial model. Additionally, we explore multiphoton excitation in ISM and observe increased FWHM for 2-photon and 3-photon excitation compared to 1-photon excitation. The FWHM of the 2-photon excitation ISM PSF increases by 20% and the FWHM of the 3-photon excitation ISM PSF increases by 28% compared to the 1-photon excitation ISM. In addition, we found that the optimal sweep factor for 2-photon ISM is 1.22, and the optimal sweep factor of 3-photon ISM is 1.12 instead of the 2 predicted by the one-photon scalar ISM theory. Our work improves the understanding of ISM and contributes to its advancement as a high-resolution imaging technique.

Single-molecule localization microscopy requires sparse activation of emitters to circumvent the diffraction limit. In densely labeled or thick samples, overlap of emitter images is inevitable. Single-molecule localization of these samples results in a biased parameter estimate with a wrong model of the number of emitters. On the other hand, multiple emitter fitting suffers from point spread function degeneracy, which increases model and parameter uncertainty. To better estimate the model, parameters and uncertainties, a three-dimensional Bayesian multiple emitter fitting algorithm was constructed using Reversible Jump Markov Chain Monte Carlo. It reconstructs the posterior density of both the model and the parameters, namely the three-dimensional position and photon intensity, of overlapping emitters. The ability of the algorithm to separate two emitters at varying distance was evaluated using an astigmatic point spread function. We found that for astigmatic imaging, the posterior distribution of the emitter positions is multimodal when emitters are within two times the in-focus standard deviation of the point spread function. This multimodality describes the ambiguity in position that astigmatism introduces in localization microscopy. Biplane imaging was also tested, proving capable of separating emitters up to 0.75 times the in-focus standard deviation of the point spread function while staying free of multimodality. The posteriors seen in astigmatic and biplane imaging demonstrate how the algorithm can identify point spread function degeneracy and evaluate imaging techniques for three-dimensional multiple-emitter fitting performance.

We study predictive control for blood glucose regulation in patients with type 1 diabetes mellitus. We determine optimal control actions for insulin and glucagon infusion via linear time-varying model predictive control (LTV MPC) and dynamic linerization around the state trajectory predicted. Through in silico implementation of a comprehensive nonlinear model, we show that our proposed controller is able to reject meal disturbances, retain normoglycemia afterwards and significantly outperform standard linearized MPC.

Modulation enhanced single-molecule localization microscopy (meSMLM) methods improve the localization precision by using patterned illumination to encode additional position information. Iterative meSMLM (imeSMLM) methods iteratively generate prior information on emitter positions, used to locally improve the localization precision during subsequent iterations. The Cramér-Rao lower bound cannot incorporate prior information to bound the best achievable localization precision because it requires estimators to be unbiased. By treating estimands as random variables with a known prior distribution, the Van Trees inequality (VTI) can be used to bound the best possible localization precision of imeSMLM methods. An imeSMLM method is considered, where the positions of in-plane standing-wave illumination patterns are controlled over the course of multiple iterations. Using the VTI, we analytically approximate a lower bound on the maximum localization precision of imeSMLM methods that make use of standing-wave illumination patterns. In addition, we evaluate the maximally achievable localization precision for different illumination pattern placement strategies using Monte Carlo simulations. We show that in the absence of background and under perfect modulation, the information content of signal photons increases exponentially as a function of the iteration count. However, the information increase is no longer exponential as a function of the iteration count under non-zero background, imperfect modulation, or limited mechanical resolution of the illumination positioning system. As a result, imeSMLM with two iterations reaches at most a fivefold improvement over SMLM at 8 expected background photons per pixel and 95% modulation contrast. Moreover, the information increase from imeSMLM is balanced by a reduced signal photon rate. Therefore, SMLM outperforms imeSMLM when considering an equal measurement time and illumination power per iteration. Finally, the VTI is an excellent tool for the assessment of the performance of illumination control and is therefore the method of choice for optimal design and control of imeSMLM methods.

Ultrasound localization microscopy (ULM) is a vascular imaging method that provides a 10-fold improvement in resolution compared to ultrasound Doppler imaging. Because typical ULM acquisitions accumulate large numbers of synthetic microbubble (MB) trajectories over hundreds of cardiac cycles, transient hemodynamic variations such as pulsatility get averaged out. Here we introduce two independent processing methods to retrieve pulsatile flow characteristics from MB trajectories sampled at kilohertz frame rates and demonstrate their potential on a simulated dataset. The first approach follows a Lagrangian description of the flow. We filter the MB trajectories to eliminate ULM localization grid artifacts and successfully recover the pulsatility fraction Pf with a root mean square error (RMSE) of 3.3%. Our second approach follows a Eulerian description of the flow. It relies on the accumulation of MB velocity estimates as observed from a stationary observer. We show that pulsatile flow gives rise to a bimodal velocity distribution with peaks indicating the maximum and minimum velocities of the cardiac cycle. In this second method, we recovered the pulsatility fraction Pf by measuring the location of these distribution peaks with a RMSE of 5.2%. We evaluated the impact of the MB localization precision σ on our ability to retrieve the bimodal signature of a pulsatile flow. Together, our results demonstrate that pulsatility can be retrieved from ULM acquisitions at kilohertz frame rate and that the estimation of the pulsatility fraction improves with localization precision.

Single-photon avalanche diode (SPAD) arrays can be used for single-molecule localization microscopy (SMLM) because of their high frame rate and lack of readout noise. SPAD arrays have a binary frame output, which means photon arrivals should be described as a binomial process rather than a Poissonian process. Consequentially, the theoretical minimum uncertainty of the localizations is not accurately predicted by the Poissonian Cramér-Rao lower bound (CRLB). Here, we derive a binomial CRLB and benchmark it using simulated and experimental data. We show that if the expected photon count is larger than one for all pixels within one standard deviation of a Gaussian point spread function, the binomial CRLB gives a 46% higher theoretical uncertainty than the Poissonian CRLB. For typical SMLM photon fluxes, where no saturation occurs, the binomial CRLB predicts the same uncertainty as the Poissonian CRLB. Therefore, the binomial CRLB can be used to predict and benchmark localization uncertainty for SMLM with SPAD arrays for all practical emitter intensities.