Data-Driven PSF Modeling using B-Splines

Abstract

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.