Statistical Multiscale Parameter Selection Strategies

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Parameter selection is a final but very important step in any (statistical) regularisation process in order to determine the level of resolution of a given regularized reconstruction in a statistical inverse problem. In this project we aim for fully data driven parameter selection methods which are based on a statistical multiscale analysis of the residuals. Whereas in the first part of the project selection strategies of regularisation parameters for various regularisation methods have been investigated in the second part we will construct and investigate methods for locally adaptive statistical multiscale regularisation as a shape constraint. Theoretical analysis of the methods involves convergence rates for the resulting estimators and almost sure and distributional limits for maxima of the underlying partial sum processes. This will be applied to nanoscale fluorescence microscopy imaging of cells (in collaboration with S. Hell, MPI Biophysical Chemistry) and to the fully automatic image reconstruction in Magnetic Resonance Imaging (in collaboration with J. Frahm, MPI Biophysical Chemistry, BiomedNMR).