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As a general rule of thumb the resolution of a light microscope (i.e. the ability to discern objects) is predominantly described by the full width at half maximum (FWHM) of its point spread function (psf)-the diameter of the blurring density at half of its maximum. Classical wave optics suggests a linear relationship between FWHM and resolution also manifested in the well known Abbe and Rayleigh criteria, dating back to the end of 19th century. However, during the last two decades conventional light microscopy has undergone a shift from microscopic scales to nanoscales. This increase in resolution comes with the need to incorporate the random nature of observations (light photons) and challenges the classical view of discernability, as we argue in this paper. Instead, we suggest a statistical description of resolution obtained from such random data. Our notion of discernability is based on statistical testing whether one or two objects with the same total intensity are present. For Poisson measurements we get linear dependence of the (minimax) detection boundary on the FWHM, whereas for a homogeneous Gaussian model the dependence of resolution is nonlinear. Hence, at small physical scales modeling by homogeneous gaussians is inadequate, although often implicitly assumed in many reconstruction algorithms. In contrast, the Poisson model and its variance stabilized Gaussian approximation seem to provide a statistically sound description of resolution at the nanoscale. Our theory is also applicable to other imaging setups, such as telescopes.

I am grateful to the members of the committee for agreeing to evaluate my thesis, in particular, to Prof. Tatyana Krivobokova for taking up the duty of Korreferentin.I find myself lucky to have colleagues at the IMS. I think that the institute is a friendly, supportive and an inspiring place to work.A big thanks goes to my office mates: Florian Pein, Marco Seiler and Miguel del Álamo Ruiz for helping me with various (non)mathematical issues. Especially when I started and when I was sick. I would like to thank Natalia Khizanishvili and Heiner Keilholz for the same.My sincere gratitude goes to the RTG 2088 for organizing nice workshops, lecture series and conference funding. I would also like to thank the DFG, which partially supported my PhD through CRC 755.I would also like to acknowledge many great teachers outside of Göttingen: Virginija

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