Statistical Multiscale Parameter Selection Strategies

参数选择是任何（统计）正则化过程中的最后但非常重要的步骤，以便确定统计逆问题中给定正则化重建的分辨率水平。在这个项目中，我们的目标是充分的数据驱动的参数选择方法是基于统计多尺度分析的残差。而在项目的第一部分，各种正则化方法的正则化参数的选择策略已经在第二部分中进行了研究，我们将构建和研究局部自适应统计多尺度正则化作为形状约束的方法。理论分析的方法，包括收敛速度估计和几乎肯定的最大值的基本部分和过程的分布限制。这将应用于细胞的纳米级荧光显微镜成像（与S。Hell，MPI Biophysical Chemistry）和磁共振成像中的全自动图像重建（与J.Frahm，MPI Biophysical Chemistry，BiomedNMR合作）。

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).