Signal-to-noise and spatial resolution in in-line imaging. 1. Basic theory, numerical simulations and planar experimental images.

Signal-to-noise ratio and spatial resolution are quantitatively analysed in the context of in-line (propagation based) X-ray phase-contrast imaging. It is known that free-space propagation of a coherent X-ray beam from the imaged object to the detector plane, followed by phase retrieval in accordance with Paganin's method, can increase the signal-to-noise in the resultant images without deteriorating the spatial resolution. This results in violation of the noise-resolution uncertainty principle and demonstrates `unreasonable' effectiveness of the method. On the other hand, when the process of free-space propagation is performed in software, using the detected intensity distribution in the object plane, it cannot reproduce the same effectiveness, due to the amplification of photon shot noise. Here, it is shown that the performance of Paganin's method is determined by just two dimensionless parameters: the Fresnel number and the ratio of the real decrement to the imaginary part of the refractive index of the imaged object. The relevant theoretical analysis is performed first, followed by computer simulations and then by a brief test using experimental images collected at a synchrotron beamline. More extensive experimental tests will be presented in the second part of this paper.

Young's double-slit interference with single hard X-ray photons.

Double-slit interference experiments using monochromatic hard X-rays with the energy of 25 keV are presented. The experiments were performed at a synchrotron source with a distance of 110 m between the interferometer and the detector to produce an interference pattern with a sufficiently broad period that could be adequately sampled by a photon-counting detector with 75 micrometre pixels. In the single-particle version of the experiment, over one million image frames with a single registered photon in each one were collected. The sum of these frames showed a clear presence of the interference pattern with the expected period. Subsequent analysis provided an objective estimation of the minimal number of detected photons required to determine, in accordance with the Rose criterion, the presence of the photon interference. Apart from a general theoretical interest, these investigations were aimed at exploring the possibility of medical X-ray phase-contrast imaging in photon-counting regime at minimal radiation doses.

Unified fast reconstruction algorithm for conventional, phase-contrast, and diffraction tomography.

A unified method for three-dimensional reconstruction of objects from transmission images collected at multiple illumination directions is described. The method may be applicable to experimental conditions relevant to absorption-based, phase-contrast, or diffraction imaging using x rays, electrons, and other forms of penetrating radiation or matter waves. Both the phase retrieval (also known as contrast transfer function correction) and the effect of Ewald sphere curvature (in the cases with a shallow depth of field and significant in-object diffraction) are incorporated in the proposed algorithm and can be taken into account. Multiple scattering is not treated explicitly but can be mitigated as a result of angular averaging that constitutes an essential feature of the method. The corresponding numerical algorithm is based on three-dimensional gridding which allows for fast computational implementation, including a straightforward parallelization. The algorithm can be used with any scanning geometry involving plane-wave illumination. A software code implementing the proposed algorithm has been developed, tested on simulated and experimental image data, and made publicly available.

Method for virtual optical sectioning and tomography utilizing shallow depth of field.

A method is proposed for high-resolution, three-dimensional reconstruction of internal structures of objects from planar transmission images. The described approach can be used with any form of radiation or matter waves, in principle, provided that the depth of field is smaller than the thickness of the sample. The physical optics basis for the method is elucidated, and the reconstruction algorithm is presented in detail. A simulated example demonstrates an application of the method to three-dimensional electron transmission imaging of a nanoparticle under realistic radiation dose and spatial resolution constraints. It is envisaged that the method can be applicable in high-resolution transmission electron microscopy, soft x-ray microscopy, ultrasound imaging, and other areas.

A Method for High-Resolution Three-Dimensional Reconstruction with Ewald Sphere Curvature Correction from Transmission Electron Images.

A method for three-dimensional reconstruction of objects from defocused images collected at multiple illumination directions in high-resolution transmission electron microscopy is presented. The method effectively corrects for the Ewald sphere curvature by taking into account the in-particle propagation of the electron beam. Numerical simulations demonstrate that the proposed method is capable of accurately reconstructing biological molecules or nanoparticles from high-resolution defocused images under conditions achievable in single-particle electron cryo-microscopy or electron tomography with realistic radiation doses, non-trivial aberrations, multiple scattering, and other experimentally relevant factors. The physics of the method is based on the well-known Diffraction Tomography formalism, but with the phase-retrieval step modified to include a conjugation of the phase (i.e., multiplication of the phase by a negative constant). At each illumination direction, numerically backpropagating the beam with the conjugated phase produces maximum contrast at the location of individual atoms in the molecule or nanoparticle. The resultant algorithm, Conjugated Holographic Reconstruction, can potentially be incorporated into established software tools for single-particle analysis, such as, for example, RELION or FREALIGN, in place of the conventional contrast transfer function correction procedure, in order to account for the Ewald sphere curvature and improve the spatial resolution of the three-dimensional reconstruction.

Comparison of propagation-based CT using synchrotron radiation and conventional cone-beam CT for breast imaging.

OBJECTIVES: To evaluate and compare the image quality of propagation-based phase-contrast computed tomography (PB-CT) using synchrotron radiation and conventional cone-beam breast computed tomography (CBBCT) based on various radiological image quality criteria. METHODS: Eight excised breast tissue samples of various sizes and containing different lesion types were scanned using PB-CT at a synchrotron facility and using CBBCT at a university-affiliated breast imaging centre. PB-CT scans were performed at two different mean glandular dose (MGD) levels: standard (5.8 mGy) and low (1.5 mGy), for comparison with CBBCT scans at the standard MGD (5.8 mGy). Image quality assessment was carried out using six quality criteria and six independent medical imaging experts in a reading room with mammography workstations. The interobserver agreement between readers was evaluated using intraclass correlation coefficient (ICC), and image quality was compared between the two breast imaging modalities using the area under the visual grading characteristic curve (AUCVGC). RESULTS: Interobserver agreement between the readers showed moderate reliability for five image criteria (ICC: ranging from 0.488 to 0.633) and low reliability for one criterion (image noise) (ICC 0.307). For five image quality criteria (overall quality, perceptible contrast, lesion sharpness, normal tissue interfaces, and calcification visibility), both standard-dose PB-CT images (AUCVGC 0.958 to 1, p ≤ .05) and low dose PB-CT images (AUCVGC 0.785 to 0.834, p ≤ .05) were of significantly higher image quality than standard-dose CBBCT images. CONCLUSIONS: Synchrotron-based PB-CT can achieve a significantly higher radiological image quality at a substantially lower radiation dose compared with conventional CBBCT. KEY POINTS: • PB-CT using synchrotron radiation results in higher image quality than conventional CBBCT for breast imaging. • PB-CT using synchrotron radiation requires a lower radiation dose than conventional CBBCT for breast imaging. • PB-CT can help clinicians diagnose patients with breast cancer.

BERTHA: Implementation of a four-component Dirac-Kohn-Sham relativistic framework.

In this paper, we present and review the most recent computational advances in the BERTHA code. BERTHA can be regarded as the state of the art in fully relativistic four-component Dirac-Kohn-Sham (DKS) software. Thanks to the implementation of various parallelization and memory open-ended distribution schemes in combination with efficient "density fitting" algorithms, it greatly reduces the computational burden of four-component DKS calculations. We also report the newly developed OpenMP version of the code, that, together with the berthmod Python module, provides a significant leap forward in terms of usability and applicability of the BERTHA software. Some applications of the recently developed natural orbitals for chemical valence/charge displacement bonding analysis and the real-time time dependent DKS implementation are also reported.

On the "unreasonable" effectiveness of transport of intensity imaging and optical deconvolution.

The effectiveness of reconstructive imaging using the homogeneous transport of intensity equation may be regarded as "unreasonable," because it has been shown to significantly increase signal-to-noise ratio while preserving spatial resolution, compared to equivalent conventional absorption-based imaging techniques at the same photon fluence. We reconcile this surprising behavior by analyzing the propagation of noise in typical in-line holography experiments. This analysis indicates that novel imaging techniques may be designed that produce high signal-to-noise images at low radiation doses without sacrificing spatial resolution.

On the van Cittert-Zernike theorem for intensity correlations and its applications.

A reciprocal relationship between the autocovariance of the light intensity in the source plane and in the far-field detector plane is presented in a form analogous to the classical van Cittert-Zernike theorem, but involving intensity correlation functions. A "classical" version of the reciprocity relationship is considered first, based on the assumption of circular Gaussian statistics of the complex amplitudes in the source plane. The result is consistent with the theory of Hanbury Brown-Twiss interferometry, but it is shown to be also applicable to estimation of the source size or the spatial resolution of the detector from the noise power spectrum of flat-field images. An alternative version of the van Cittert-Zernike theorem for intensity correlations is then derived for a quantized electromagnetic beam in a coherent state, which leads to Poisson statistics for the intrinsic intensity of the beam.

Recent advances and perspectives in four-component Dirac-Kohn-Sham calculations.

We review recent theoretical and computational advances in the full relativistic four-component Dirac-Kohn-Sham (DKS) approach and its application to the calculation of the electronic structure of chemical systems containing many heavy atoms. We describe our implementation of an all-electron DKS approach based on the use of G-spinor basis sets, Hermite Gaussian functions, state-of-the-art density-fitting techniques and memory distributed parallelism. This approach has enormously extended the applicability of the DKS method, including for example large clusters of heavy atoms, and opens the way for future key developments. We examine the current limitations and future possible applications of the DKS approach, including the implementation of four-current density functionals and real-time propagation schemes. This would make possible to describe molecules in strong fields, accurately accounting for relativistic kinematic effects and spin-orbit coupling.

Propagation-Based Phase-Contrast CT of the Breast Demonstrates Higher Quality Than Conventional Absorption-Based CT Even at Lower Radiation Dose.

RATIONALE AND OBJECTIVES: Propagation-based phase-contrast CT (PB-CT) is an advanced X-ray imaging technology that exploits both refraction and absorption of the transmitted X-ray beam. This study was aimed at optimizing the experimental conditions of PB-CT for breast cancer imaging and examined its performance relative to conventional absorption-based CT (AB-CT) in terms of image quality and radiation dose. MATERIALS AND METHODS: Surgically excised breast mastectomy specimens (n = 12) were scanned using both PB-CT and AB-CT techniques under varying imaging conditions. To evaluate the radiological image quality, visual grading characteristics (VGC) analysis was used in which 11 breast specialist radiologists compared the overall image quality of PB-CT images with respect to the corresponding AB-CT images. The area under the VGC curve was calculated to measure the differences between PB-CT and AB-CT images. RESULTS: The highest radiological quality was obtained for PB-CT images using a 32 keV energy X-ray beam and by applying the Homogeneous Transport of Intensity Equation phase retrieval with the value of its parameter γ set to one-half of the theoretically optimal value for the given materials. Using these optimized conditions, the image quality of PB-CT images obtained at 4 mGy and 2 mGy mean glandular dose was significantly higher than AB-CT images at 4 mGy (AUCVGC = 0.901, p = 0.001 and AUCVGC = 0.819, p = 0.011, respectively). CONCLUSION: PB-CT achieves a higher radiological image quality compared to AB-CT even at a considerably lower mean glandular dose. Successful translation of the PB-CT technique for breast cancer imaging can potentially result in improved breast cancer diagnosis.

Effect of x-ray energy on the radiological image quality in propagation-based phase-contrast computed tomography of the breast.

Purpose: Breast cancer is the most common cancer in women in developing and developed countries and is responsible for 15% of women's cancer deaths worldwide. Conventional absorption-based breast imaging techniques lack sufficient contrast for comprehensive diagnosis. Propagation-based phase-contrast computed tomography (PB-CT) is a developing technique that exploits a more contrast-sensitive property of x-rays: x-ray refraction. X-ray absorption, refraction, and contrast-to-noise in the corresponding images depend on the x-ray energy used, for the same/fixed radiation dose. The aim of this paper is to explore the relationship between x-ray energy and radiological image quality in PB-CT imaging. Approach: Thirty-nine mastectomy samples were scanned at the imaging and medical beamline at the Australian Synchrotron. Samples were scanned at various x-ray energies of 26, 28, 30, 32, 34, and 60 keV using a Hamamatsu Flat Panel detector at the same object-to-detector distance of 6 m and mean glandular dose of 4 mGy. A total of 132 image sets were produced for analysis. Seven observers rated PB-CT images against absorption-based CT (AB-CT) images of the same samples on a five-point scale. A visual grading characteristics (VGC) study was used to determine the difference in image quality. Results: PB-CT images produced at 28, 30, 32, and 34 keV x-ray energies demonstrated statistically significant higher image quality than reference AB-CT images. The optimum x-ray energy, 30 keV, displayed the largest area under the curve ( AUC VGC ) of 0.754 ( p = 0.009 ). This was followed by 32 keV ( AUC VGC = 0.731 , p ≤ 0.001 ), 34 keV ( AUC VGC = 0.723 , p ≤ 0.001 ), and 28 keV ( AUC VGC = 0.654 , p = 0.015 ). Conclusions: An optimum energy range (around 30 keV) in the PB-CT technique allows for higher image quality at a dose comparable to conventional mammographic techniques. This results in improved radiological image quality compared with conventional techniques, which may ultimately lead to higher diagnostic efficacy and a reduction in breast cancer mortalities.

Ptychographic X-ray speckle tracking.

A method is presented for the measurement of the phase gradient of a wavefront by tracking the relative motion of speckles in projection holograms as a sample is scanned across the wavefront. By removing the need to obtain an undistorted reference image of the sample, this method is suitable for the metrology of highly divergent wavefields. Such wavefields allow for large magnification factors that, according to current imaging capabilities, will allow for nanoradian angular sensitivity and nanoscale sample projection imaging. Both the reconstruction algorithm and the imaging geometry are nearly identical to that of ptychography, except that the sample is placed downstream of the beam focus and that no coherent propagation is explicitly accounted for. Like other X-ray speckle tracking methods, it is robust to low-coherence X-ray sources, making it suitable for laboratory-based X-ray sources. Likewise, it is robust to errors in the registered sample positions, making it suitable for X-ray free-electron laser facilities, where beam-pointing fluctuations can be problematic for wavefront metrology. A modified form of the speckle tracking approximation is also presented, based on a second-order local expansion of the Fresnel integral. This result extends the validity of the speckle tracking approximation and may be useful for similar approaches in the field.

speckle-tracking: a software suite for ptychographic X-ray speckle tracking.

In recent years, X-ray speckle-tracking techniques have emerged as viable tools for wavefront metrology and sample imaging applications. These methods are based on the measurement of near-field images. Thanks to their simple experimental setup, high angular sensitivity and compatibility with low-coherence sources, these methods have been actively developed for use with synchrotron and laboratory light sources. Not only do speckle-tracking techniques give the potential for high-resolution imaging, but they also provide rapid and robust characterization of aberrations of X-ray optical elements, focal spot profiles, and sample position and transmission properties. In order to realize these capabilities, software implementations are required that are equally rapid and robust. To address this need, a software suite has been developed for the ptychographic X-ray speckle-tracking technique, an X-ray speckle-based method suitable for highly divergent wavefields. The software suite is written in Python 3, with an OpenCL back end for GPU and multi-CPU core processing. It is accessible as a Python module, through the command line or through a graphical user interface, and is available as source code under Version 3 or later of the GNU General Public License.

Effect of radiation damage and illumination variability on signal-to-noise ratio in X-ray free-electron laser single-particle imaging.

The deterioration of both the signal-to-noise ratio and the spatial resolution in the electron-density distribution reconstructed from diffraction intensities collected at different orientations of a sample is analysed theoretically with respect to the radiation damage to the sample and the variations in the X-ray intensities illuminating different copies of the sample. The simple analytical expressions and numerical estimates obtained for models of radiation damage and incident X-ray pulses may be helpful in planning X-ray free-electron laser (XFEL) imaging experiments and in analysis of experimental data. This approach to the analysis of partially coherent X-ray imaging configurations can potentially be used for analysis of other forms of imaging where the temporal behaviour of the sample and the incident intensity during exposure may affect the inverse problem of sample reconstruction.

Noise-resolution uncertainty principle in classical and quantum systems.

We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is then demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such as for example photon number squeezed states in quantum optics, can overcome the "classical" noise-resolution uncertainty limit. This uncertainty relationship, which is fundamentally different from the Heisenberg and related uncertainty principles, can define an upper limit for the information capacity of communication and imaging systems. It is expected to be useful in a variety of problems in classical and quantum optics and imaging.

PyBERTHART: A Relativistic Real-Time Four-Component TDDFT Implementation Using Prototyping Techniques Based on Python.

We present a real-time time-dependent four-component Dirac-Kohn-Sham (RT-TDDKS) implementation based on the BERTHA code. This new implementation takes advantage of modern software engineering, including the prototyping techniques. The software design follows a three step approach: (i) the prototype implementation of a time-propagation algorithm in nonrelativistic real-time TDDFT within the Psi4NumPy framework, which provides a suitable environment for the creation of a clear, readable, and easy to test reference code in Python, (ii) the design of an original Python application programming interface for the relativistic four-component code BERTHA (PyBERTHA), which has an efficient computational kernel for relativistic integrals written in FORTRAN, and (iii) the porting of the time-propagation scheme enveloped within the Psi4NumPy framework to PyBERTHA. The propagation scheme consequently resides in a single readable Python computer code that is easy to maintain and in which the key quantities, such as the Dirac-Kohn-Sham and dipole matrices, can be accessed directly from the PyBERTHA module. For linear algebra operations (matrix-matrix multiplications and diagonalization) we use the highly optimized procedures implemented in the popular NumPy library. The overhead introduced by the Python interface to BERTHA is almost negligible (less than 1% evaluated on the SCF procedure), and the interoperability between different programming languages (FORTRAN, C, and Python) does not affect the numerical stability of the time-propagation scheme. Our new RT-TDDKS implementation has been employed to investigate the stability of the time-propagation procedure in combination with a density-fitting algorithm (both for the Coulomb and for the exchange-correlation matrix construction), which are employed in BERTHA to speed up the Dirac-Kohn-Sham matrix evaluation. On the basis of systematic calculations, employing several density-fitting basis sets of increasing accuracy, we showed that quantitative agreement can be achieved in combination with extended-fitting basis sets, with an error in the Coulomb energy below 1 µ-hartree. Convergence of the transition energies increasing of quality of the fitting basis sets has been also observed. Our data suggest that the error in the Coulomb energy may also represent a good estimate of the fitting basis set quality for real-time electron dynamics simulations. Further, we study the applicability of the RT-TDDKS method in combination with both weak- and extreme strong-field regime. Numerical results of excited-state transitions for the Group 12 atoms are reported and compared with a previous real-time Dirac-Kohn-Sham implementation (Repisky et al. J. Chem. Theory Comput. 2015, 11, 980-991). Finally, calculations of high harmonic generation in the hydrogen molecule and Au dimer have been also carried out. We were able to generate high harmonics with relatively well-defined peaks up to the 21st and 13th order in the case of H2 and Au2, respectively. Our findings show that the four-component structure of the Dirac-Kohn-Sham Hamiltonian provides a suitable theoretical framework, with no intrinsic unfavorable features, to study molecules in the strong-field regime.

Recovery of undamaged electron-density maps in the presence of damage-induced partial coherence in single-particle imaging.

Resolving the electronic structure of single biological molecules in their native state was among the primary motivations behind X-ray free-electron lasers. The ultra-short pulses they produce can outrun the atomic motion induced by radiation damage, but the electronic structure of the sample is still significantly modified from its original state. This paper explores the decoherence of the scattered signal induced by temporal evolution of the electronic structure in the sample molecule. It is shown that the undamaged electron density of a single-molecule sample can often be retrieved using only the two most occupied modes from the coherent mode decomposition of the partially coherent diffraction fluence.

Structural dynamics in proteins induced by and probed with X-ray free-electron laser pulses.

X-ray free-electron lasers (XFELs) enable crystallographic structure determination beyond the limitations imposed upon synchrotron measurements by radiation damage. The need for very short XFEL pulses is relieved through gating of Bragg diffraction by loss of crystalline order as damage progresses, but not if ionization events are spatially non-uniform due to underlying elemental distributions, as in biological samples. Indeed, correlated movements of iron and sulfur ions were observed in XFEL-irradiated ferredoxin microcrystals using unusually long pulses of 80 fs. Here, we report a femtosecond time-resolved X-ray pump/X-ray probe experiment on protein nanocrystals. We observe changes in the protein backbone and aromatic residues as well as disulfide bridges. Simulations show that the latter's correlated structural dynamics are much slower than expected for the predicted high atomic charge states due to significant impact of ion caging and plasma electron screening. This indicates that dense-environment effects can strongly affect local radiation damage-induced structural dynamics.

Astigmatic phase retrieval: an experimental demonstration.

We present the first experimental demonstration of the astigmatic phase retrieval technique, in which the diffracted wavefield is distorted by cylindrical curvature in two orthogonal directions. A charge-one vortex, a charge-two vortex, and a simple test image are all correctly reconstructed.