Reconstruction algorithms for x-ray nanocrystallography via solution of the twinning problem

X-ray nanocrystallography is an emerging technique for imaging nanoscale objects that alleviates the large crystallization requirement of conventional crystallography by collecting diffraction patterns from a large ensemble of smaller and easier to build nanocrystals, which are typically delivered to the x-ray beam via a liquid jet. In order to determine the structure of an imaged object, several parameters must first be determined, including the crystal sizes, incident photon flux densities, and crystal orientations. Autoindexing techniques, which have been used extensively to orient conventional crystals, only determine the orientation of the nanocrystals up to symmetry of the crystal lattice, which is often greater than the symmetry of the diffraction information, resulting in what is known as the twinning problem. In addition, the image data is corrupted by large degrees of shot noise due to low collected signal, background signal due to the liquid jet and detector electronics, as well as other sources of noise. Furthermore, diffraction only measures the magnitudes of the Fourier transform of the object and, thus, one must recover phase information in order to invert the data and recover a three-dimensional reconstruction of the constituent molecular structure. Previous approaches for handling the twinning problem have mainly relied on having a known similar structure available, which may not be present for fundamentally new structures. We present a series of techniques to determine the crystal sizes, incident photon flux densities, and crystal orientations in the presence of large amounts of noise common in experiments. Additionally, by using a new sampling strategy, we demonstrate that phase information can be computed from nanocrystallographic diffraction images using only Fourier magnitude information, via a compressive phase retrieval algorithm. We demonstrate the feasibility of this new approach by testing it on simulated data with parameters and noise levels common in current experiments

Reconstruction algorithms for x-ray nanocrystallography via solution of the twinning problem

X-ray nanocrystallography is an emerging technique for imaging nanoscale objects that alleviates the large crystallization requirement of conventional crystallography by collecting diffraction patterns from a large ensemble of smaller and easier to build nanocrystals, which are typically delivered to the x-ray beam via a liquid jet. In order to determine the structure of an imaged object, several parameters must first be determined, including the crystal sizes, incident photon flux densities, and crystal orientations. Autoindexing techniques, which have been used extensively to orient conventional crystals, only determine the orientation of the nanocrystals up to symmetry of the crystal lattice, which is often greater than the symmetry of the diffraction information, resulting in what is known as the twinning problem. In addition, the image data is corrupted by large degrees of shot noise due to low collected signal, background signal due to the liquid jet and detector electronics, as well as other sources of noise. Furthermore, diffraction only measures the magnitudes of the Fourier transform of the object and, thus, one must recover phase information in order to invert the data and recover a three-dimensional reconstruction of the constituent molecular structure. Previous approaches for handling the twinning problem have mainly relied on having a known similar structure available, which may not be present for fundamentally new structures. We present a series of techniques to determine the crystal sizes, incident photon flux densities, and crystal orientations in the presence of large amounts of noise common in experiments. Additionally, by using a new sampling strategy, we demonstrate that phase information can be computed from nanocrystallographic diffraction images using only Fourier magnitude information, via a compressive phase retrieval algorithm. We demonstrate the feasibility of this new approach by testing it on simulated data with parameters and noise levels common in current experiments

Inversion of dynamical Bragg intensities to complex structure factors by iterated projections. For Ultramic. 2020. ("Pico" Festschrift, May 2021).

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

Inversion of Many-Beam Bragg Intensities for Phasing by Iterated Projections: Removal of Multiple Scattering Artifacts from Diffraction Data.

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

Inversion of Many-Beam Bragg Intensities for Phasing by Iterated Projections: Removal of Multiple Scattering Artifacts from Diffraction Data.

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

AMBIGUITY FUNCTION AND FRAME THEORETIC PROPERTIES OF PERIODIC ZERO AUTOCORRELATION WAVEFORMS

Constant Amplitude Zero Autocorrelation (CAZAC) waveforms u are analyzed in terms of the ambiguity function Au. Elementary number theoretic considerations illustrate that peaks in Au are not stable under small pertubations in its domain. Further, it is proved that the analysis of vector-valued CAZAC waveforms depends on methods from the theory of frames. Fi-nally, techniques are introduced to characterize the structure of Au, to compute u in terms of Au, and to evaluate MSE for CAZAC waveforms