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X-ray tomography is widely used for three-dimensional structure determination in many areas of science, from the millimeter to the nanometer scale. The resolution and quality of the 3D reconstruction is limited by the availability of alignment parameters that correct for the mechanical shifts of the sample or sample stage for the images that constitute a scan. In this paper we describe an algorithm for marker-free, fully automated and accurately aligned and reconstructed X-ray tomography data. Our approach solves the tomographic reconstruction jointly with projection data alignment based on a rigid-body deformation model. We demonstrate the robustness of our method on both synthetic phantom and experimental data and show that our method is highly efficient in recovering relatively large alignment errors without prior knowledge of a low resolution approximation of the 3D structure or a reasonable estimate of alignment parameters.

X-ray-based computed tomography is a well established technique for determining the three-dimensional structure of an object from its two-dimensional projections. In the past few decades, there have been significant advancements in the brightness and detector technology of tomography instruments at synchrotron sources. These advancements have led to the emergence of new observations and discoveries, with improved capabilities such as faster frame rates, larger fields of view, higher resolution and higher dimensionality. These advancements have enabled the material science community to expand the scope of tomographic measurements towards increasingly in situ and in operando measurements. In these new experiments, samples can be rapidly evolving, have complex geometries and restrictions on the field of view, limiting the number of projections that can be collected. In such cases, standard filtered back-projection often results in poor quality reconstructions. Iterative reconstruction algorithms, such as model-based iterative reconstructions (MBIR), have demonstrated considerable success in producing high-quality reconstructions under such restrictions, but typically require high-performance computing resources with hundreds of compute nodes to solve the problem in a reasonable time. Here, tomoCAM, is introduced, a new GPU-accelerated implementation of model-based iterative reconstruction that leverages non-uniform fast Fourier transforms to efficiently compute Radon and back-projection operators and asynchronous memory transfers to maximize the throughput to the GPU memory. The resulting code is significantly faster than traditional MBIR codes and delivers the reconstructive improvement offered by MBIR with affordable computing time and resources. tomoCAM has a Python front-end, allowing access from Jupyter-based frameworks, providing straightforward integration into existing workflows at synchrotron facilities., (open access.)

One of the brilliant ideas of John Spence when he saw the first diffraction patterns from the Linac Coherent Light Source was that one could solve the crystallographic phase problem by utilising the intensities between Bragg peaks. Because these intensities are due to the Fourier transform of the shape of the crystal, the approach came to be known as "shape-transform phasing." Shape-transform phasing was developed over the next ten years and formed the basis for many other interesting ideas and pursuits. Here we describe the current best implementation of the original idea using a lattice occupancy formalism and show that certain types of crystal defects can also be modelled via this approach, allowing the molecular structure to be recovered from the additional information offered by the inter-Bragg intensities from these crystal defects., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 Elsevier B.V. All rights reserved.)

A method for recovering complex structure factors from many simultaneously excited Bragg beam in- tensities is described. The method is applied to simulated transmission electron diffraction data over a wide range of crystal thickness and beam energies. The method is based on iterated projections between structure and scattering matrices, which are related by a matrix unit ary transformation, exponential, which we invert. The algorithm removes multiple-scattering perturbations from diffraction data and might be extended to other fields, including X-ray and neutron diffraction and cryo-electron microscopy. Because coherent multiple scattering involves interference between Bragg beams, the method also solves the phase problem. Unlike dynamical inversion from electron microscope images or ptychography data, the method, which starts with Bragg beam intensities, provides complex structure factors unaffected by focusing errors or resolution limitations imposed by lenses. We provide inversions from simulated data with 441 simultaneously excited Bragg beams over a range of thickness and beam energy. We discuss the retrieval of chirality information from enantiomorphs, the efficient incorporation of symmetry information using the irreducible representation of the group of structure matrices, and the effect of HOLZ lines to provide three-dimensional information., (Copyright © 2021 Elsevier B.V. All rights reserved.)

Coefficients for translational and rotational diffusion characterize the Brownian motion of particles. Emerging X-ray photon correlation spectroscopy (XPCS) experiments probe a broad range of length scales and time scales and are well-suited for investigation of Brownian motion. While methods for estimating the translational diffusion coefficients from XPCS are well-developed, there are no algorithms for measuring the rotational diffusion coefficients based on XPCS, even though the required raw data are accessible from such experiments. In this paper, we propose angular-temporal cross-correlation analysis of XPCS data and show that this information can be used to design a numerical algorithm (Multi-Tiered Estimation for Correlation Spectroscopy [MTECS]) for predicting the rotational diffusion coefficient utilizing the cross-correlation: This approach is applicable to other wavelengths beyond this regime. We verify the accuracy of this algorithmic approach across a range of simulated data., Competing Interests: The authors declare no competing interest., (Copyright © 2021 the Author(s). Published by PNAS.)

A method for recovering complex structure factors from many simultaneously excited Bragg beam in- tensities is described. The method is applied to simulated transmission electron diffraction data over a wide range of crystal thickness and beam energies. The method is based on iterated projections between structure and scattering matrices, which are related by a matrix unit ary transformation, exponential, which we invert. The algorithm removes multiple-scattering perturbations from diffraction data and might be extended to other fields, including X-ray and neutron diffraction and cryo-electron microscopy. Because coherent multiple scattering involves interference between Bragg beams, the method also solves the phase problem. Unlike dynamical inversion from electron microscope images or ptychography data, the method, which starts with Bragg beam intensities, provides complex structure factors unaffected by focusing errors or resolution limitations imposed by lenses. We provide inversions from simulated data with 441 simultaneously excited Bragg beams over a range of thickness and beam energy. We discuss the retrieval of chirality information from enantiomorphs, the efficient incorporation of symmetry information using the irreducible representation of the group of structure matrices, and the effect of HOLZ lines to provide three-dimensional information., (Copyright © 2021 Elsevier B.V. All rights reserved.)

An iterated projection algorithm (N-Phaser) is developed that reconstructs a scattering potential from N-beam multiple Bragg scattered intensities. The method may be used to eliminate multiple scattering artifacts from electron diffraction data, solving the phase problem and increasing the thicknesses of samples used in materials science, solid-state chemistry, and small molecule crystallography. For high-energy transmission electron diffraction, we show that the algorithm recovers accurate complex structure factors from a wide range of thicknesses, orientations, and relativistic beam energies, and does not require known thickness or atomic-resolution data if sufficient multiple scattering occurs. Extensions to Cryo-electron microscopy and Micro-electron diffraction are suggested.

Diffraction patterns from small protein crystals illuminated by highly coherent X-rays often contain measurable interference signals between Bragg peaks. This coherent `shape transform' signal introduces enough additional information to allow the molecular densities to be determined from the diffracted intensities directly, without prior information or resolution restrictions. However, the various correlations amongst molecular occupancies/vacancies at the crystal surface result in a subtle yet critical problem in shape transform phasing whereby the sublattices of symmetry-related molecules exhibit a form of partial coherence amongst lattice sites when an average is taken over many crystal patterns. Here an iterative phase retrieval algorithm is developed which is capable of treating this problem; it is demonstrated on simulated data.

Fluctuation X-ray scattering (FXS) is an emerging experimental technique in which X-ray solution scattering data are collected from particles in solution using ultrashort X-ray exposures generated by a free-electron laser (FEL). FXS experiments overcome the low data-to-parameter ratios associated with traditional solution scattering measurements by providing several orders of magnitude more information in the final processed data. Here we demonstrate the practical feasibility of FEL-based FXS on a biological multiple-particle system and describe data-processing techniques required to extract robust FXS data and significantly reduce the required number of snapshots needed by introducing an iterative noise-filtering technique. We showcase a successful ab initio electron density reconstruction from such an experiment, studying the Paramecium bursaria Chlorella virus (PBCV-1)., Competing Interests: The authors declare no conflict of interest., (Copyright © 2018 the Author(s). Published by PNAS.)

Fluctuation X-ray scattering (FXS) is an emerging experimental technique in which solution scattering data are collected using X-ray exposures below rotational diffusion times, resulting in angularly anisotropic X-ray snapshots that provide several orders of magnitude more information than traditional solution scattering data. Such experiments can be performed using the ultrashort X-ray pulses provided by a free-electron laser source, allowing one to collect a large number of diffraction patterns in a relatively short time. Here, we describe a test data set for FXS, obtained at the Linac Coherent Light Source, consisting of close to 100 000 multi-particle diffraction patterns originating from approximately 50 to 200 Paramecium Bursaria Chlorella virus particles per snapshot. In addition to the raw data, a selection of high-quality pre-processed diffraction patterns and a reference SAXS profile are provided.

We use extremely bright and ultrashort pulses from an x-ray free-electron laser (XFEL) to measure correlations in x rays scattered from individual bioparticles. This allows us to go beyond the traditional crystallography and single-particle imaging approaches for structure investigations. We employ angular correlations to recover the three-dimensional (3D) structure of nanoscale viruses from x-ray diffraction data measured at the Linac Coherent Light Source. Correlations provide us with a comprehensive structural fingerprint of a 3D virus, which we use both for model-based and ab initio structure recovery. The analyses reveal a clear indication that the structure of the viruses deviates from the expected perfect icosahedral symmetry. Our results anticipate exciting opportunities for XFEL studies of the structure and dynamics of nanoscale objects by means of angular correlations.

Free-electron lasers now have the ability to collect X-ray diffraction patterns from individual molecules; however, each sample is delivered at unknown orientation and may be in one of several conformational states, each with a different molecular structure. Hit rates are often low, typically around 0.1%, limiting the number of useful images that can be collected. Determining accurate structural information requires classifying and orienting each image, accurately assembling them into a 3D diffraction intensity function, and determining missing phase information. Additionally, single particles typically scatter very few photons, leading to high image noise levels. We develop a multitiered iterative phasing algorithm to reconstruct structural information from single-particle diffraction data by simultaneously determining the states, orientations, intensities, phases, and underlying structure in a single iterative procedure. We leverage real-space constraints on the structure to help guide optimization and reconstruct underlying structure from very few images with excellent global convergence properties. We show that this approach can determine structural resolution beyond what is suggested by standard Shannon sampling arguments for ideal images and is also robust to noise., Competing Interests: The authors declare no conflict of interest.

Fluctuation X-ray scattering (FXS) is an extension of small- and wide-angle X-ray scattering in which the X-ray snapshots are taken below rotational diffusion times. This technique, performed using a free electron laser or ultrabright synchrotron source, provides significantly more experimental information compared with traditional solution scattering methods. We develop a multitiered iterative phasing algorithm to determine the underlying structure of the scattering object from FXS data.

X-ray nanocrystallography allows the structure of a macromolecule to be determined from a large ensemble of nanocrystals. However, several parameters, including crystal sizes, orientations, and incident photon flux densities, are initially unknown and images are highly corrupted with noise. Autoindexing techniques, commonly used in conventional crystallography, can determine orientations using Bragg peak patterns, but only up to crystal lattice symmetry. This limitation results in an ambiguity in the orientations, known as the indexing ambiguity, when the diffraction pattern displays less symmetry than the lattice and leads to data that appear twinned if left unresolved. Furthermore, missing phase information must be recovered to determine the imaged object's structure. We present an algorithmic framework to determine crystal size, incident photon flux density, and orientation in the presence of the indexing ambiguity. We show that phase information can be computed from nanocrystallographic diffraction using an iterative phasing algorithm, without extra experimental requirements, atomicity assumptions, or knowledge of similar structures required by current phasing methods. The feasibility of this approach is tested on simulated data with parameters and noise levels common in current experiments.