Fixed Point Analysis of Douglas-Rachford Splitting for Ptychography and Phase Retrieval
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Douglas-Rachford Splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are linearly stable if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas-Rachford (CDR) algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsistent. Moreover, the fixed point analysis decisively leads to the selection of an optimal step size which in turn renders the Gaussian DRS method in a particularly simple form with no tuning parameter (Averaged Projection-Reflection). When applied to the challenging problem of blind ptychography, which seeks to recover both the object and the probe simultaneously, the DRS methods converge geometrically and globally when properly initialized.
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