Footnote:
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Description:
The two-dimensional phase retrieval problem arises in many areas of experimental physics, e.g. in x-ray microscopy. The central theme of this thesis is the application of sparsity constraints in the two-dimensional discrete phase retrieval problem. It provides a framework for the utilization of sparsifying transforms, such as the discrete shearlet transform, which is an extension of the wavelet transform that is especially suited for the efficient representation of so-called cartoon like images. Based on the relaxed averaged alternating reflections (RAAR) algorithm, a reconstruction algorithm is proposed which incorporates shrinkage mappings of frame coefficients. For tight frames we show that the resulting operator is the proximity operator of a proper, lower-semicontinuous, convex function. Furthermore, bounds on the iterates of the newly developed algorithm as well as Césaro convergence are proven for arbitrary frames. The thesis concludes with a numerical evaluation of simulated measurement data for x-ray microscopy experiments in the near-field regime contaminated by Poisson noise.