Single Molecule Localization Microscopy (SMLM) has enabled researchers to breakthrough the diffraction limit and obtain nanometer resolution images of macromolecular structures. But due to the time involved in obtaining ample data for proper image, the technique is venerable to many problem including fluctuations due to thermal gradients from surrounding which cause the frames to drift. SMLM relies on the stochastic blinking of fluorophore probes. Thus drift in SMLM could be explicitly modelled as a stochastic state space process. These models could be used to perform drift correction. Two state space models are proposed relying on different properties of SMLM.

The first model utilizes shifting of underlying image structure. The state space model for this property is constructed using shift matrices. A system identification method along with image reconstruction method is also derived to form the drift compensation algorithm for this model. This algorithm is further developed to provide adequate performance within low computational time. The second model relies on the position of emitter molecules and utilizes linking or pairing of fluorophore probes in succeeding frame to obtain the output data. Drift compensation algorithm for this model is constructed using Prediction Error Methods (PEM) and Kalman (RTS) smoother. The drift correction algorithm for these two models are also bench-marked with existing algorithms to obtain insight into performance. Furthermore, other properties of these algorithms are explored using simulation dataset and recommendation are provided for improvement and further research.

For a long time, scientists believed that it is not possible to achieve a better microscopic resolution than half the wavelength of light (~ 200 nm). During the last decades, various super-resolution fluorescence microscopy techniques have been developed, offering high-resolution imaging beyond the diffraction limit. It has been established as one of the major tools currently available for the study and analysis of biological tissues. The observation of, previously limited and unclear, cellular structures has demonstrated the great promise of super-resolution fluorescence microscopy in exemplifying biological processes at the cellular level. Inherently, single-molecule localization microscopy (SMLM) is based upon acquiring a sequence of diffraction-limited images, consisting of localizations of sparsely distributed emitting fluorophores. Next, precise and accurate estimates of the spatial coordinates of the fluorophores, placed on the structures of interest, are made. Subsequently, all recovered emitter positions from each frame are combined, and a super-resolution image is generated. One of the most crucial questions in SMLM concerns the precision with which the location of a single molecule can be determined. The Cramér–Rao lower bound (CRLB) is used to quantify the estimation performance in terms of localization precision. Recently, several modulation enhanced methods have shown resolution improvement by combining structured patterned illumination with SMLM techniques, such as MINFLUX and SIMFLUX. However, the analysis of images by SMLM and these modulation enhanced methods requires sparse activation regions to prevent overlap between active emitters. Recently, several methods have been developed with the ability to analyze higher density data, allowing partial overlap between emitters. In this work, we propose a maximum likelihood-based, modulation enhanced SMLM method able to localize multiple active emitters within a single region of interest (ROI), which we refer to as Multi- SIMFLUX. The mathematical framework of the algorithm is comprehensively built up throughout this work. The performance of the algorithm has been evaluated, and validated under a wide range of realistic imaging conditions. This involves various active-emitter densities, different emitter- and background intensities, numerous emitter configurations, a multitude of ROI-sizes and point spread function widths. As theoretically expected, our approach significantly outperforms conventional 2DGaussian SMLM with a twofold improvement factor in terms of localization precision for various emitter densities and intensities. At higher active emitter imaging densities, the extension from a single to a multi-emitter approach showed a clear improvement in recall fraction.

Super Resolution Microscopy is a technique to image objects with resolution higher than the diffraction limit, the fundamental limit to the resolution of a microscope. One of the techniques used for Super Resolution Microscopy is Single Molecule Localization Microscopy (SMLM). For this method, a sample is labeled using fluorophores (fluorescent proteins or dyes), and each individual fluorophore is imaged individually. The fluorophores can be localized with a precision much higher than the diffraction limit, since the localization precision is only limited by the amount of photons collected. By combining the localizations of all the fluorophores, the complete object can be reconstructed with resolution higher than the diffraction limit. To achieve a high localization precision, a model for the Point Spread Function (PSF) of the microscope is required. The PSF describes how the microscope distorts the image of the fluorophore. This model is used to estimate the fluorophore location in the sample. This means the model needs to be very accurate and unbiased. Furthermore, because an experiment contains hundred of thousands of images and the model needs to be evaluated multiple times per localization, the model also needs to be fast to evaluate. Traditional fitters model the PSF with a Gaussian approximation. However, more recent studies showed the errors introduced by this approximation. More accurate models have been introduced, such as C-Spline based approximations. These C-Splines require over a million spline coefficients, and therefore lots of calibration data. In this research, it is investigated if Simplex B-Splines can be used to better approximate the PSF. Simplex B-Splines consist of multiple B-Splines defined on a triangulation of simplices. This allows for a very flexible model structure and high accuracy using a relatively low number of parameters. Most applications assume the PSF is constant over the field of view. However, due to varying aberrations or refractive index mismatches the PSF actually changes with the fluorophore position in the field of view. If these changes in the PSF are not corrected for, the precision is reduced or a significant bias is introduced. B-splines allow the model to be extended to higher dimensions, and can therefore also model these changes to the PSF. Different metrics are used to compare models. Using a statistical (chi squared) test the model quality can be assessed. Localization precision is verified by implementing the model in a MLE algorithm, and by showing that the standard deviation reaches the theoretical lower bound for precision, the CRLB.
In this research it is shown that the performance of B-Spline models is comparable to state-of-the-art methods, but uses 144 times fewer spline coefficients. Furthermore, the B-Splines were also successfully extended to higher dimensions, to reduce the effect of a varying PSF.