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The Fokker–Planck equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This forms a natural generalisation of the transport-of-intensity equation. The x-ray Fokker–Planck equation can simultaneously account for both propagation-based phase contrast, and the diffusive effects of sample-induced small-angle x-ray scattering, when forming an x-ray image of a thin sample. Two derivations are given for the Fokker–Planck equation associated with x-ray imaging, together with a Kramers–Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample micro-structure. These equations may be applied to the forward problem of modelling image formation in the presence of both coherent and diffusive energy transport. They may also be used to formulate associated inverse problems of retrieving the phase shifts due to a sample placed in an x-ray beam, together with the diffusive properties of the sample. The domain of applicability for the Fokker–Planck and Kramers–Moyal equations for paraxial imaging is at least as broad as that of the transport-of-intensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons.

X-ray imaging has conventionally relied upon attenuation to provide contrast. In recent years, two complementary modalities have been added; (a) phase contrast, which can capture low-density samples that are difficult to see using attenuation, and (b) dark-field x-ray imaging, which reveals the presence of sub-pixel sample structures. These three modalities can be accessed using a crystal analyser, a grating interferometer or by looking at a directly-resolved grid, grating or speckle pattern. Grating and grid-based methods extract a differential phase signal by measuring how far a feature in the illumination has been shifted transversely due to the presence of a sample. The dark-field signal is extracted by measuring how the visibility of the structured illumination is decreased, typically due to the presence of sub-pixel structures in a sample. The strength of the dark-field signal may depend on the grating period, the pixel size and the set-up distances, and additional dark-field signal contributions may be seen as a result of strong phase effects or other factors. In this paper we show that the finite-difference form of the Fokker–Planck equation can be applied to describe the drift (phase signal) and diffusion (dark-field signal) of the periodic or structured illumination used in phase contrast x-ray imaging with gratings, in order to better understand any cross-talk between attenuation, phase and dark-field x-ray signals. In future work, this mathematical description could be used as a basis for new approaches to the inverse problem of recovering both phase and dark-field information.

The size of the smallest detectable sample feature in an x-ray imaging system is usually restricted by the spatial resolution of the system. This limitation can now be overcome using the diffusive dark-field signal, which is generated by unresolved phase effects or the ultra-small-angle x-ray scattering from unresolved sample microstructures. A quantitative measure of this dark-field signal can be useful in revealing the microstructure size or material for medical diagnosis, security screening and materials science. Recently, we derived a new method to quantify the diffusive dark-field signal in terms of a scattering angle using a single-exposure grid-based approach. In this manuscript, we look at the problem of quantifying the sample microstructure size from this single-exposure dark-field signal. We do this by quantifying the diffusive dark-field signal produced by 5 different sizes of polystyrene microspheres, ranging from 1.0 to 10.8 µm, to investigate how the strength of the extracted dark-field signal changes with the sample microstructure size, S . We also explore the feasibility of performing single-exposure dark-field imaging with a simple equation for the optimal propagation distance, given microstructure with a specific size and thickness, and show consistency between this model and experimental data. Our theoretical model predicts that the dark-field scattering angle is inversely proportional to S , which is also consistent with our experimental data.

In recent years, a novel x-ray imaging modality has emerged that reveals unresolved sample microstructure via a “dark-field image”, which provides complementary information to conventional “bright-field” images, such as attenuation and phase-contrast modalities. This x-ray dark-field signal is produced by unresolved microstructures scattering the x-ray beam resulting in localised image blur. Dark-field retrieval techniques extract this blur to reconstruct a dark-field image. Unfortunately, the presence of non-dark-field blur such as source-size blur or the detector point-spread-function can affect the dark-field retrieval as they also blur the experimental image. In addition, dark-field images can be degraded by the artefacts induced by large intensity gradients from attenuation and propagation-based phase contrast, particularly around sample edges. By measuring any non-dark-field blurring across the image plane and removing it from experimental images, as well as removing attenuation and propagation-based phase contrast, we show that a directional dark-field image can be retrieved with fewer artefacts and more consistent quantitative measures. We present the details of these corrections and provide “before and after” directional dark-field images of samples imaged at a synchrotron source. This paper utilises single-grid directional dark-field imaging, but these corrections have the potential to be broadly applied to other x-ray imaging techniques.

Braided rivers, while uncommon internationally, are significant in terms of their unique ecosystems and as vital freshwater resources at locations where they occur. With an increasing awareness of the connected nature of surface water and groundwater, there have been many studies examining groundwater–surface water exchange in various types of waterbodies, but significantly less research has been conducted in braided rivers. Thus, there is currently limited understanding of how characteristics unique to braided rivers, such as channel shifting, expanding and narrowing margins, and a high degree of heterogeneity affect groundwater–surface water flow paths. This article provides an overview of characteristics specific to braided rivers, including a map showing the regions where braided rivers are mainly found at the global scale: Alaska, Canada, the Japanese and European Alps, the Himalayas, Russia, and New Zealand. To the authors' knowledge, this is the first map of its kind. This is followed by a review of prior studies that have investigated groundwater–surface water interactions in braided rivers and their associated aquifers. The various methods used to characterise these processes are discussed with emphasis on their effectiveness in achieving the studies' objectives and their applicability in braided rivers. We also discuss additional methods that appear promising to apply in braided river settings. The aim is to provide guidance on methodologies most suitable for future work in braided rivers. In many cases, previous studies found a multi-method approach useful to produce more robust results and compare data collected at various scales. Given the challenges of working directly in braided rivers, there is considerable scope for the increased use of remote sensing techniques. There is also opportunity for new approaches to modelling braided rivers using integrated techniques that incorporate the complex river bed terrain and geomorphology of braided rivers explicitly. We also identify a critical need to improve the conceptual understanding of hyporheic exchange in braided rivers, rates of recharge to and from braided rivers, and historical patterns of dry and low-flow periods in these rivers.

Speckle-based phase-contrast X-ray imaging (SB-PCXI) can reconstruct high-resolution images of weakly-attenuating materials that would otherwise be indistinguishable in conventional attenuation-based X-ray imaging. The experimental setup of SB-PCXI requires only a sufficiently coherent X-ray source and spatially random mask, positioned between the source and detector. The technique can extract sample information at length scales smaller than the imaging system’s spatial resolution; this enables multimodal signal reconstruction. “Multimodal Intrinsic Speckle-Tracking” (MIST) is a rapid and deterministic formalism derived from the paraxial-optics form of the Fokker–Planck equation. MIST simultaneously extracts attenuation, refraction, and small-angle scattering (diffusive dark-field) signals from a sample and is more computationally efficient compared to alternative speckle-tracking approaches. Hitherto, variants of MIST have assumed the diffusive dark-field signal to be spatially slowly varying. Although successful, these approaches have been unable to well-describe unresolved sample microstructure whose statistical form is not spatially slowly varying. Here, we extend the MIST formalism such that this restriction is removed, in terms of a sample’s rotationally-isotropic diffusive dark-field signal. We reconstruct multimodal signals of two samples, each with distinct X-ray attenuation and scattering properties. The reconstructed diffusive dark-field signals have superior image quality—as measured by the naturalness image quality evaluator, signal-to-noise ratio, and azimuthally averaged power-spectrum—compared to our previous approaches which assume the diffusive dark-field to be a slowly varying function of transverse position. Our generalisation may assist increased adoption of SB-PCXI in applications such as engineering and biomedical disciplines, forestry, and palaeontology, and is anticipated to aid the development of speckle-based diffusive dark-field tensor tomography.

Phase contrast x-ray imaging (PCXI) provides high-contrast images of weakly-attenuating structures like the lungs. PCXI, when paired with 4D X-ray Velocimetry (XV), can measure regional lung function and non-invasively assess the efficacy of emerging therapeutics. Bacteriophage therapy is an emerging antimicrobial treatment option for lung diseases such as cystic fibrosis (CF), particularly with increasing rates of multi-drug-resistant infections. Current efficacy assessment in animal models is highly invasive, typically requiring histological assessment. We aim to use XV techniques as non-invasive alternatives to demonstrate efficacy of bacteriophage therapy for treating Pseudomonas aeruginosa CF lung infections, measuring functional changes post-treatment. Time-resolved in vivo PCXI-CT scans of control, Pseudomonas-infected, and phage-treated mouse lungs were taken at the Australian Synchrotron Imaging and Medical Beamline. Using XV we measured local lung expansion and ventilation throughout the breath cycle, analysing the skew of the lung expansion distribution. CT images allowed visualisation of the projected air volume in the lungs, assessing structural lung damage. XV analysis demonstrated changes in lung expansion between infection and control groups, however there were no statistically significant differences between treated and placebo groups. In some cases where structural changes were not evident in the CT scans, XV successfully detected changes in lung function.

There are various imaging setups capable of capturing dark-field images, each with its own capabilities and limitations. Across all these setups, the underlying physical mechanism responsible for dark-field contrast is generally defined similarly: small-angle X-ray scattering from sub-resolution structures, multiple refraction from large resolvable features and/or edge effects. Despite this conceptual consistency, there remains a lack of studies that directly compare different dark-field imaging setups using identical samples. We have recently developed a `family' of dark-field imaging techniques that, although they have different experimental setups, each extract dark-field contrast by directly resolving dark-field-associated local image blurring with a high-resolution camera. In this paper, we provide a qualitative comparison of the capabilities of this family of methods, covering both different experimental acquisition methods and image-retrieval algorithms for this subset of dark-field imaging techniques. We acquired X-ray imaging data from two test samples using synchrotron propagation-based, single-grid and speckle-based setups, and retrieved dark-field images from each dataset using an appropriate algorithm. The dark-field retrieval methods differed in their computational framework: single-shot algorithms were applied for the single-grid and speckle-based approaches, an additional multi-exposure method was used for speckle data, and a dual-energy method was applied to the propagation-based data. We find that all approaches successfully retrieve dark-field contrast in microstructure-dense regions, as expected. Differences arise at high-spatial-frequency sample features, such as edges, and some approaches exhibit artefacts. We attribute these differences to variations in experimental parameters ( e.g. pixel size, sample-to-detector distance and X-ray energy) and to the algorithmic assumptions underlying each technique ( e.g. the single-material assumption and treatment of phase effects). This study aims to guide users of dark-field imaging in selecting the most suitable technique for their imaging goals. To this end, we provide a summary table and highlight opportunities for future research into the sources of dark-field contrast across emerging methods.

Propagation-based phase-contrast X-ray imaging is a promising technique for in vivo medical imaging, offering lower radiation doses than traditional attenuation-based imaging. Previous studies have focused on X-ray energies below 50keV for small-animal imaging and mammography. Here, we investigate the feasibility of high-energy propagation-based computed tomography for human adult-scale lung imaging at the Australian Synchrotron’s Imaging and Medical Beamline. This facility is uniquely positioned for human lung imaging, offering a large field of view, high X-ray energies, and supporting clinical infrastructure. We imaged an anthropomorphic chest phantom (LungMan) between 50keV and 80keV across the range of possible sample-to-detector distances, with a photon-counting and an integrating detector. Strong phase-contrast fringes were observed with the photon-counting detector, even at high X-ray energies and a large pixel size relative to previous work, whereas the integrating detector with lower spatial resolution showed no clear phase effects. Measured X-ray phase-shifting properties of LungMan aligned well with reference soft tissue values, validating the phantom for phase-contrast studies. Imaging quality assessments suggest an optimal configuration at approximately 70keV and the longest available propagation distance of 7.5m, indicating potential benefit in positioning the patient in an upstream hutch. This study represents the first step towards clinical adult lung imaging at the Australian Synchrotron.