On recovery guarantees for angular synchronization

The angular synchronization problem of estimating a set of unknown angles from their known noisy pairwise differences arises in various applications. It can be reformulated as a optimization problem on graphs involving the graph Laplacian matrix. We consider a general, weighted version of this problem, where the impact of the noise differs between different pairs of entries and some of the differences are erased completely; this version arises for example in ptychography. We study two common approaches for solving this problem, namely eigenvector relaxation and semidefinite convex relaxation. Although some recovery guarantees are available for both methods, their performance is either unsatisfying or restricted to the unweighted graphs. We close this gap, deriving recovery guarantees for the weighted problem that are completely analogous to the unweighted version.Comment: 20 pages, 5 figure

Regularized Newton Methods for X-ray Phase Contrast and General Imaging Problems

Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.Comment: (C)2016 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibite