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Recently, the regularized functional matching pursuit (RFMP) was introduced as a greedy algorithm for linear ill-posed inverse problems. This algorithm incorporates the Tikhonov-Phillips regularization which implies the necessity of a parameter choice. In this paper, some known parameter choice methods are evaluated with respect to their performance in the RFMP and its enhancement, the regularized orthogonal functional matching pursuit (ROFMP). As an example of a linear inverse problem, the downward continuation of gravitational field data from the satellite orbit to the Earth's surface is chosen, because it is exponentially ill-posed. For the test scenarios, different satellite heights with several noise-to-signal ratios and kinds of noise are combined. The performances of the parameter choice strategies in these scenarios are analyzed. For example, it is shown that a strongly scattered set of data points is an essentially harder challenge for the regularization than a regular grid. The obtained results yield, as a first orientation, that the generalized cross validation, the L-curve method and the residual method could be most appropriate for the RFMP and the ROFMP.

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Single molecule localization microscopy (SMLM) has enormous potential for resolving subcellular structures below the diffraction limit of light microscopy: Localization precision in the low digit nanometer regime has been shown to be achievable. In order to record localization microscopy data, however, sample fixation is inevitable to prevent molecular motion during the rather long recording times of minutes up to hours. Eventually, it turns out that preservation of the sample’s ultrastructure during fixation becomes the limiting factor. We propose here a workflow for data analysis, which is based on SMLM performed at cryogenic temperatures. Since molecular dipoles of the fluorophores are fixed at low temperatures, such an approach offers the possibility to use the orientation of the dipole as an additional information for image analysis. In particular, assignment of localizations to individual dye molecules becomes possible with high reliability. We quantitatively characterized the new approach based on the analysis of simulated oligomeric structures. Side lengths can be determined with a relative error of less than 1% for tetramers with a nominal side length of 5 nm, even if the assumed localization precision for single molecules is more than 2 nm.