X-ray Fokker-Planck equation for paraxial imaging

X-ray phase and dark-field computed tomography without optical elements

X-ray diffusive dark-field imaging, which allows spatially unresolved microstructure to be mapped across a sample, is an increasingly popular tool in an array of settings. Here, we present a new algorithm for phase and dark-field computed tomography based on the x-ray Fokker-Planck equation. Needing only a coherent x-ray source, sample, and detector, our propagation-based algorithm can map the sample density and dark-field/diffusion properties of the sample in 3D. Importantly, incorporating dark-field information in the density reconstruction process enables a higher spatial resolution reconstruction than possible with previous propagation-based approaches. Two sample exposures at each projection angle are sufficient for the successful reconstruction of both the sample density and dark-field Fokker-Planck diffusion coefficients. We anticipate that the proposed algorithm may be of benefit in biomedical imaging and industrial settings.

Dark-field and directional dark-field on low-coherence x ray sources with random mask modulations: validation with SAXS anisotropy measurements

Phase-contrast imaging, dark-field, and directional dark-field imaging are recent x ray imaging modalities that have been demonstrated to reveal different information and contrast from those provided by conventional x ray imaging. Access to these new types of images is currently limited because the acquisitions require coherent sources such as synchrotron radiation or complicated optical setups. This Letter demonstrates the possibility of efficiently performing phase-contrast, dark-field, and directional dark-field imaging on a low-coherence laboratory system equipped with a conventional x ray tube, using a simple, fast, and robust single-mask technique.

On the quantification of sample microstructure using single-exposure x-ray dark-field imaging via a single-grid setup

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, [Formula: see text]. 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 [Formula: see text], which is also consistent with our experimental data.

Multimodal intrinsic speckle-tracking (MIST) to extract images of rapidly-varying diffuse X-ray dark-field

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.

High-speed processing of X-ray wavefront marking data with the Unified Modulated Pattern Analysis (UMPA) model

Wavefront-marking X-ray imaging techniques use e.g., sandpaper or a grating to generate intensity fluctuations, and analyze their distortion by the sample in order to retrieve attenuation, phase-contrast, and dark-field information. Phase contrast yields an improved visibility of soft-tissue specimens, while dark-field reveals small-angle scatter from sub-resolution structures. Both have found many biomedical and engineering applications. The previously developed Unified Modulated Pattern Analysis (UMPA) model extracts these modalities from wavefront-marking data. We here present a new UMPA implementation, capable of rapidly processing large datasets and featuring capabilities to greatly extend the field of view. We also discuss possible artifacts and additional new features.

Dark-field tomography of an attenuating object using intrinsic x-ray speckle tracking

Purpose: We investigate how an intrinsic speckle tracking approach to speckle-based x-ray imaging is used to extract an object's effective dark-field (DF) signal, which is capable of providing object information in three dimensions. Approach: The effective DF signal was extracted using a Fokker-Planck type formalism, which models the deformations of illuminating reference beam speckles due to both coherent and diffusive scatter from the sample. Here, we assumed that (a) small-angle scattering fans at the exit surface of the sample are rotationally symmetric and (b) the object has both attenuating and refractive properties. The associated inverse problem of extracting the effective DF signal was numerically stabilized using a "weighted determinants" approach. Results: Effective DF projection images, as well as the DF tomographic reconstructions of the wood sample, are presented. DF tomography was performed using a filtered back projection reconstruction algorithm. The DF tomographic reconstructions of the wood sample provided complementary, and otherwise inaccessible, information to augment the phase contrast reconstructions, which were also computed. Conclusions: An intrinsic speckle tracking approach to speckle-based imaging can tomographically reconstruct an object's DF signal at a low sample exposure and with a simple experimental setup. The obtained DF reconstructions have an image quality comparable to alternative x-ray DF techniques.

Quantifying the x-ray dark-field signal in single-grid imaging

X-ray dark-field imaging reveals the sample microstructure that is unresolved when using conventional methods of x-ray imaging. In this paper, we derive a new method to extract and quantify the x-ray dark-field signal collected using a single-grid imaging set-up, and relate the signal strength to the number of sample microstructures, N. This was achieved by modelling sample-induced changes to the shadow of the upstream grid, and fitting experimental data to this model. Our results suggested that the dark-field scattering angle from our spherical microstructures deviates slightly from the theoretical model of N, which was consistent with results from other experimental methods. We believe the approach outlined here can equip quantitative dark-field imaging of small samples, particularly in cases where only one sample exposure is possible, either due to sample movement or radiation dose limitations. Future directions include an extension into directional dark-field imaging.

Semi-classical Monte Carlo algorithm for the simulation of X-ray grating interferometry

Traditional simulation techniques such as wave optics methods and Monte Carlo (MC) particle transport cannot model both interference and inelastic scattering phenomena within one framework. Based on the rules of quantum mechanics to calculate probabilities, we propose a new semi-classical MC algorithm for efficient and simultaneous modeling of scattering and interference processes. The similarities to MC particle transport allow the implementation as a flexible c++ object oriented extension of EGSnrc-a well-established MC toolkit. In addition to previously proposed Huygens principle based transport through optics components, new variance reduction techniques for the transport through gratings are presented as transport options to achieve the required improvement in speed and memory costs necessary for an efficient exploration (system design-dose estimations) of the medical implementation of X-ray grating interferometry (GI), an emerging imaging technique currently subject of tremendous efforts towards clinical translation. The feasibility of simulation of interference effects is confirmed in four academic cases and an experimental table-top GI setup. Comparison with conventional MC transport show that deposited energy features of EGSnrc are conserved.

Arbitrary-Order Finite-Time Corrections for the Kramers-Moyal Operator

With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers-Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the different terms in a stochastic differential equation. With the full representation of this operator, we are able to separate finite-time corrections of the power-series expansion of arbitrary order into terms with and without derivatives of the Kramers-Moyal coefficients. We arrive at a closed-form solution expressed through conditional moments, which can be extracted directly from time-series data with a finite sampling intervals. We provide all finite-time correction terms for parametric and non-parametric estimation of the Kramers-Moyal coefficients for discontinuous processes which can be easily implemented-employing Bell polynomials-in time-series analyses of stochastic processes. With exemplary cases of insufficiently sampled diffusion and jump-diffusion processes, we demonstrate the advantages of our arbitrary-order finite-time corrections and their impact in distinguishing diffusion and jump-diffusion processes strictly from time-series data.

Dark-field signal extraction in propagation-based phase-contrast imaging

A method for extracting the dark-field signal in propagation-based phase-contrast imaging is proposed. In the case of objects consisting predominantly of a single material, or several different materials with similar ratios of the real decrement to the imaginary part of the complex refractive index, the proposed method requires a single image for extraction of the dark-field signal in two-dimensional projection imaging. In the case of three-dimensional tomographic imaging, the method needs only one image to be collected at each projection angle. Initial examples using simulated and experimental data indicate that this method can improve visualization of small sharp features inside a larger object, e.g. the visualization of microcalcifications in propagation-based x-ray breast cancer imaging. It is suggested that the proposed approach may be useful in other forms of biomedical imaging, where it can help one to obtain additional small-angle scattering information without increasing the radiation dose to the sample.

Emphysema quantified: mapping regional airway dimensions using 2D phase contrast X-ray imaging

We have developed an analyser-based phase contrast X-ray imaging technique to measure the mean length scale of pores or particles that cannot be resolved directly by the system. By combining attenuation, phase and ultra-small angle X-ray scattering information, the technique was capable of measuring differences in airway dimension between lungs of healthy mice and those with mild and severe emphysema. Our measurements of airway dimensions from 2D images showed a 1:1 relationship to the actual airway dimensions measured using micro-CT. Using 80 images, the sensitivity and specificity were measured to be 0.80 and 0.89, respectively, with the area under the ROC curve close to ideal at 0.96. Reducing the number of images to 11 slightly decreased the sensitivity to 0.75 and the ROC curve area to 0.90, whilst the specificity remained high at 0.89.

Modeling classical wavefront sensors

We present an image formation model for deterministic phase retrieval in propagation-based wavefront sensing, unifying analysis for classical wavefront sensors such as Shack-Hartmann (slopes tracking) and curvature sensors (based on Transport-of-Intensity Equation). We show how this model generalizes commonly seen formulas, including Transport-of-Intensity Equation, from small distances and beyond. Using this model, we analyze theoretically achievable lateral wavefront resolution in propagation-based deterministic wavefront sensing. Finally, via a prototype masked wavefront sensor, we show simultaneous bright field and phase imaging numerically recovered in real-time from a single-shot measurement.

Applying the Fokker-Planck equation to grating-based x-ray phase and dark-field imaging

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.