Geometric reconstruction methods for electron tomography

Electron tomography is becoming an increasingly important tool in materials science for studying the three-dimensional morphologies and chemical compositions of nanostructures. The image quality obtained by many current algorithms is seriously affected by the problems of missing wedge artefacts and nonlinear projection intensities due to diffraction effects. The former refers to the fact that data cannot be acquired over the full $180^\circ$ tilt range; the latter implies that for some orientations, crystalline structures can show strong contrast changes. To overcome these problems we introduce and discuss several algorithms from the mathematical fields of geometric and discrete tomography. The algorithms incorporate geometric prior knowledge (mainly convexity and homogeneity), which also in principle considerably reduces the number of tilt angles required. Results are discussed for the reconstruction of an InAs nanowire

Robust phase retrieval with the swept approximate message passing (prSAMP) algorithm

In phase retrieval, the goal is to recover a complex signal from the magnitude of its linear measurements. While many well-known algorithms guarantee deterministic recovery of the unknown signal using i.i.d. random measurement matrices, they suffer serious convergence issues some ill-conditioned matrices. As an example, this happens in optical imagers using binary intensity-only spatial light modulators to shape the input wavefront. The problem of ill-conditioned measurement matrices has also been a topic of interest for compressed sensing researchers during the past decade. In this paper, using recent advances in generic compressed sensing, we propose a new phase retrieval algorithm that well-adopts for both Gaussian i.i.d. and binary matrices using both sparse and dense input signals. This algorithm is also robust to the strong noise levels found in some imaging applications

Projected Multi-Agent Consensus Equilibrium (PMACE) for Distributed Reconstruction with Application to Ptychography

Multi-Agent Consensus Equilibrium (MACE) formulates an inverse imaging problem as a balance among multiple update agents such as data-fitting terms and denoisers. However, each such agent operates on a separate copy of the full image, leading to redundant memory use and slow convergence when each agent affects only a small subset of the full image. In this paper, we extend MACE to Projected Multi-Agent Consensus Equilibrium (PMACE), in which each agent updates only a projected component of the full image, thus greatly reducing memory use for some applications.We describe PMACE in terms of an equilibrium problem and an equivalent fixed point problem and show that in most cases the PMACE equilibrium is not the solution of an optimization problem. To demonstrate the value of PMACE, we apply it to the problem of ptychography, in which a sample is reconstructed from the diffraction patterns resulting from coherent X-ray illumination at multiple overlapping spots. In our PMACE formulation, each spot corresponds to a separate data-fitting agent, with the final solution found as an equilibrium among all the agents. Our results demonstrate that the PMACE reconstruction algorithm generates more accurate reconstructions at a lower computational cost than existing ptychography algorithms when the spots are sparsely sampled

Refractive shape from light field distortion

Acquiring transparent, refractive objects is challenging as these kinds of objects can only be observed by analyzing the distortion of reference background patterns. We present a new, single image approach to reconstructing thin transparent surfaces, such as thin solids or surfaces of fluids. Our method is based on observing the distortion of light field background illumination. Light field probes have the potential to encode up to four dimensions in varying colors and intensities: spatial and angular variation on the probe surface; commonly employed reference patterns are only two-dimensional by coding either position or angle on the probe. We show that the additional information can be used to reconstruct refractive surface normals and a sparse set of control points from a single photograph

Phase Retrieval for Partially Coherent Observations

Phase retrieval is in general a non-convex and non-linear task and the corresponding algorithms struggle with the issue of local minima. We consider the case where the measurement samples within typically very small and disconnected subsets are coherently linked to each other - which is a reasonable assumption for our objective of antenna measurements. Two classes of measurement setups are discussed which can provide this kind of extra information: multi-probe systems and holographic measurements with multiple reference signals. We propose several formulations of the corresponding phase retrieval problem. The simplest of these formulations poses a linear system of equations similar to an eigenvalue problem where a unique non-trivial null-space vector needs to be found. Accurate phase reconstruction for partially coherent observations is, thus, possible by a reliable solution process and with judgment of the solution quality. Under ideal, noise-free conditions, the required sampling density is less than two times the number of unknowns. Noise and other observation errors increase this value slightly. Simulations for Gaussian random matrices and for antenna measurement scenarios demonstrate that reliable phase reconstruction is possible with the presented approach.Comment: 12 pages, 14 figure