Solving inverse problems is an essential task in a variety of imaging applications, ranging from the nanometer scale in molecular imaging, to macroscopic scales in Astrophysics. This thesis represents project research on problems on both ends of this scale.

Single molecule localization microscopy (SMLM) is an imaging technique that allows the resolution of biological samples to be improved far beyond the diffraction limit of light. The main idea is the temporal separation of the emission and subsequent data fitting to determine the positions of individual emitters with nanometer precision. Cryo-SMLM, where the sample of interest is imaged at cryogenic temperatures, is currently a very active area of research. It has been demonstrated to produce higher-quality data, allowing for an even further improved resolution. However, the drawback is a non-Gaussian intensity pattern produced by frozen dipoles as well as sample drift due to greatly prolonged imaging times compared to normal SMLM. In this thesis, we introduce a vectorial imaging model that takes into account the intensity pattern produced by fixed dipole emitters in cryo- SMLM. We demonstrate that by introducing deliberate astigmatic distortion, optimal localization precision is achievable even in the presence of substantial defocus. Further, we present a localization method for cryo-SMLM that accounts for distortions due to sample drift. We demonstrate that nanometer precision is feasible even in case of very large sample drifts.

On the other end of the imaging scale, galactic archaeology aims to study the evolution of galaxies by retrieving chemical and dynamical properties from spectral data. Currently used methods such as pPXF make model simplifications and analyze the data in a very restrictive way, ignoring global information such as smoothness. The main challenge in this regard has been the prohibitive computational complexity of modelling the full datase

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