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.

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.

Directional dark-field retrieval with single-grid x-ray imaging.

Directional dark-field imaging is an emerging x-ray modality that is sensitive to unresolved anisotropic scattering from sub-pixel sample microstructures. A single-grid imaging setup can be used to capture dark-field images by looking at changes in a grid pattern projected upon the sample. By creating analytical models for the experiment, we have developed a single-grid directional dark-field retrieval algorithm that can extract dark-field parameters such as the dominant scattering direction, and the semi-major and -minor scattering angles. We show that this method is effective even in the presence of high image noise, allowing for low-dose and time-sequence imaging.

X-Ray Dark-Field and Phase Retrieval Without Optics, via the Fokker-Planck Equation.

Emerging methods of x-ray imaging that capture phase and dark-field effects are equipping medicine with complementary sensitivity to conventional radiography. These methods are being applied over a wide range of scales, from virtual histology to clinical chest imaging, and typically require the introduction of optics such as gratings. Here, we consider extracting x-ray phase and dark-field signals from bright-field images collected using nothing more than a coherent x-ray source and a detector. Our approach is based on the Fokker-Planck equation for paraxial imaging, which is the diffusive generalization of the transport-of-intensity equation. Specifically, we utilize the Fokker-Planck equation in the context of propagation-based phase-contrast imaging, where we show that two intensity images are sufficient for successful retrieval of both the projected thickness and the dark-field signal associated with the sample. We show the results of our algorithm using both a simulated dataset and an experimental dataset. These demonstrate that the x-ray dark-field signal can be extracted from propagation-based images, and that sample thickness can be retrieved with better spatial resolution when dark-field effects are taken into account. We anticipate the proposed algorithm will be of benefit in biomedical imaging, industrial settings, and other non-invasive imaging applications.

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.

Precise phase retrieval for propagation-based images using discrete mathematics.

The ill-posed problem of phase retrieval in optics, using one or more intensity measurements, has a multitude of applications using electromagnetic or matter waves. Many phase retrieval algorithms are computed on pixel arrays using discrete Fourier transforms due to their high computational efficiency. However, the mathematics underpinning these algorithms is typically formulated using continuous mathematics, which can result in a loss of spatial resolution in the reconstructed images. Herein we investigate how phase retrieval algorithms for propagation-based phase-contrast X-ray imaging can be rederived using discrete mathematics and result in more precise retrieval for single- and multi-material objects and for spectral image decomposition. We validate this theory through experimental measurements of spatial resolution using computed tomography (CT) reconstructions of plastic phantoms and biological tissues, using detectors with a range of imaging system point spread functions (PSFs). We demonstrate that if the PSF substantially suppresses high spatial frequencies, the potential improvement from utilising the discrete derivation is limited. However, with detectors characterised by a single pixel PSF (e.g. direct, photon-counting X-ray detectors), a significant improvement in spatial resolution can be obtained, demonstrated here at up to 17%.

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.

Spectral propagation-based x-ray phase-contrast computed tomography.

Purpose: Propagation-based x-ray imaging (PBI) is a phase-contrast technique that is employed in high-resolution imaging by introducing some distance between sample and detector. PBI causes characteristic intensity fringes that have to be processed with appropriate phase-retrieval algorithms, which has historically been a difficult task for objects composed of several different materials. Spectral x-ray imaging has been introduced to PBI to overcome this issue and to potentially utilize the spectral nature of the data for material-specific imaging. We aim to explore the potential of spectral PBI in three-dimensional computed tomography (CT) imaging in this work. Approach: We demonstrate phase-retrieval for experimental high-resolution spectral propagation-based CT data of a simple two-component sample, as well as a multimaterial capacitor test sample. Phase-retrieval was performed using an algorithm based on the Alvarez-Macovski model. Virtual monochromatic (VMI) and effective atomic number images were calculated after phase-retrieval. Results: Phase-retrieval results from the spectral data set show a distinct gray-level for each material with no residual phase-contrast fringes. Several representations of the phase-retrieved data are provided. The VMI is used to display an attenuation-equivalent image at a chosen display energy of 80 keV, to provide good separation of materials with minimal noise. The effective atomic number image shows the material composition of the sample. Conclusions: Spectral photon-counting detector technology has already been shown to be compatible with spectral PBI, and there is a foreseeable need for robust phase-retrieval in high-resolution, spectral x-ray CT in the future. Our results demonstrate the feasibility of phase-retrieval for spectral PBI CT.

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.

Quantitative material decomposition using linear iterative near-field phase retrieval dual-energy x-ray imaging.

This paper expands the linear iterative near-field phase retrieval (LIPR) formalism to achieve quantitative material thickness decomposition. Propagation-based phase contrast x-ray imaging with subsequent phase retrieval has been shown to improve the signal-to-noise ratio (SNR) by factors of up to hundreds compared to conventional x-ray imaging. This is a key step in biomedical imaging, where radiation exposure must be kept low without compromising the SNR. However, for a satisfactory phase retrieval from a single measurement, assumptions must be made about the object investigated. To avoid such assumptions, we use two measurements collected at the same propagation distance but at different x-ray energies. Phase retrieval is then performed by incorporating the Alvarez-Macovski (AM) model, which models the x-ray interactions as being comprised of distinct photoelectric and Compton scattering components. We present the first application of dual-energy phase retrieval with the AM model to monochromatic experimental x-ray projections at two different energies for obtaining split x-ray interactions. Our phase retrieval method allows us to separate the object investigated into the projected thicknesses of two known materials. Our phase retrieval output leads to no visible loss in spatial resolution while the SNR improves by factors of 2 to 10. This corresponds to a possible x-ray dose reduction by a factor of 4 to 100, under the Poisson noise assumption.

Material Decomposition Using Spectral Propagation-Based Phase-Contrast X-Ray Imaging.

Material decomposition in X-ray imaging uses the energy-dependence of attenuation to digitally decompose an object into specific constituent materials, generally at the cost of enhanced image noise. Propagation-based X-ray phase-contrast imaging is a developing technique that can be used to reduce image noise, in particular from weakly attenuating objects. In this paper, we combine spectral phase-contrast imaging with material decomposition to both better visualize weakly attenuating features and separate them from overlying objects in radiography. We derive an algorithm that performs both tasks simultaneously and verify it against numerical simulations and experimental measurements of ideal two-component samples composed of pure aluminum and poly(methyl methacrylate). Additionally, we showcase first imaging results of a rabbit kitten's lung. The attenuation signal of a thorax, in particular, is dominated by the strongly attenuating bones of the ribcage. Combined with the weak soft tissue signal, this makes it difficult to visualize the fine anatomical structures across the whole lung. In all cases, clean material decomposition was achieved, without residual phase-contrast effects, from which we generate an un-obstructed image of the lung, free of bones. Spectral propagation-based phase-contrast imaging has the potential to be a valuable tool, not only in future lung research, but also in other systems for which phase-contrast imaging in combination with material decomposition proves to be advantageous.

Spectral x-ray imaging: Conditions under which propagation-based phase-contrast is beneficial.

Energy-resolved attenuation data in spectral x-ray imaging enables material decomposition, in which the different materials inside an object can be identified and separated virtually. Material decomposition has the drawback of increased noise in the resulting material images relative to the measured images. Recently, spectral x-ray imaging was combined with propagation-based x-ray phase-contrast imaging, an x-ray technique that has the potential to greatly reduce image noise by utilizing wave-optical effects. The net combined effects on image noise of performing spectral material decomposition with phase-contrast are not yet well understood, and we provide a detailed theoretical investigation of this topic here. In particular, we investigate how the addition of phase-contrast in spectral imaging affects material decomposition compared to using conventional spectral attenuation data. We show how the underlying equations can be rearranged into parts that resemble low- and high-pass filters on the input images, from which we are able to identify different energy-dependent cases where phase-contrast is or is not advantageous. Our results suggest that the benefits of phase-contrast in the context of material decomposition are primarily restricted to x-ray energies under a certain threshold, where that threshold depends on the given material combination, and sits in a region where photoelectric absorption dominates x-ray attenuation. Additionally, we show that decomposition of the electron density using an image basis spanned by functions of the Alvarez-Macovski model benefits from phase-contrast, regardless of the x-ray energies. All our findings are based purely on theoretical considerations, and can, therefore, be used to determine the feasibility and utility of propagation-based phase-contrast in spectral x-ray imaging ahead of any data collection.

Material decomposition from a single x-ray projection via single-grid phase contrast imaging.

This study describes a new approach for material decomposition in x-ray imaging, utilizing phase contrast both to increase sensitivity to weakly attenuating samples and to act as a complementary measurement to attenuation, therefore allowing two overlaid materials to be separated. The measurements are captured using the single-exposure, single-grid x-ray phase contrast imaging technique, with a novel correction that aims to remove propagation-based phase effects seen at sharp edges in the attenuation image. The use of a single-exposure technique means that images can be collected in a high-speed sequence. Results are shown for both a known two-material sample and for a biological specimen.

Quantification of muco-obstructive lung disease variability in mice via laboratory X-ray velocimetry.

To effectively diagnose, monitor and treat respiratory disease clinicians should be able to accurately assess the spatial distribution of airflow across the fine structure of lung. This capability would enable any decline or improvement in health to be located and measured, allowing improved treatment options to be designed. Current lung function assessment methods have many limitations, including the inability to accurately localise the origin of global changes within the lung. However, X-ray velocimetry (XV) has recently been demonstrated to be a sophisticated and non-invasive lung function measurement tool that is able to display the full dynamics of airflow throughout the lung over the natural breathing cycle. In this study we present two developments in XV analysis. Firstly, we show the ability of laboratory-based XV to detect the patchy nature of cystic fibrosis (CF)-like disease in ß-ENaC mice. Secondly, we present a technique for numerical quantification of CF-like disease in mice that can delineate between two major modes of disease symptoms. We propose this analytical model as a simple, easy-to-interpret approach, and one capable of being readily applied to large quantities of data generated in XV imaging. Together these advances show the power of XV for assessing local airflow changes. We propose that XV should be considered as a novel lung function measurement tool for lung therapeutics development in small animal models, for CF and for other muco-obstructive diseases.

Photon-counting, energy-resolving and super-resolution phase contrast X-ray imaging using an integrating detector.

This work demonstrates the use of a scientific-CMOS (sCMOS) energy-integrating detector as a photon-counting detector, thereby eliminating dark current and read-out noise issues, that simultaneously provides both energy resolution and sub-pixel spatial resolution for X-ray imaging. These capabilities are obtained by analyzing visible light photon clouds that result when X-ray photons produce fluorescence from a scintillator in front of the visible light sensor. Using low-fluence monochromatic X-ray projections to avoid overlapping photon clouds, the centroid of individual X-ray photon interactions was identified. This enabled a tripling of the spatial resolution of the detector to 6.71 ± 0.04 µm. By calculating the total charge deposited by this interaction, an energy resolution of 61.2 ± 0.1% at 17 keV was obtained. When combined with propagation-based phase contrast imaging and phase retrieval, a signal-to-noise ratio of up to 15 ± 3 was achieved for an X-ray fluence of less than 3 photons/mm2.

Real-time in vivo imaging of regional lung function in a mouse model of cystic fibrosis on a laboratory X-ray source.

Most measures of lung health independently characterise either global lung function or regional lung structure. The ability to measure airflow and lung function regionally would provide a more specific and physiologically focused means by which to assess and track lung disease in both pre-clinical and clinical settings. One approach for achieving regional lung function measurement is via phase contrast X-ray imaging (PCXI), which has been shown to provide highly sensitive, high-resolution images of the lungs and airways in small animals. The detailed images provided by PCXI allow the application of four-dimensional X-ray velocimetry (4DxV) to track lung tissue motion and provide quantitative information on regional lung function. However, until recently synchrotron facilities were required to produce the highly coherent, high-flux X-rays that are required to achieve lung PCXI at a high enough frame rate to capture lung motion. This paper presents the first translation of 4DxV technology from a synchrotron facility into a laboratory setting by using a liquid-metal jet microfocus X-ray source. This source can provide the coherence required for PCXI and enough X-ray flux to image the dynamics of lung tissue motion during the respiratory cycle, which enables production of images compatible with 4DxV analysis. We demonstrate the measurements that can be captured in vivo in live mice using this technique, including regional airflow and tissue expansion. These measurements can inform physiological and biomedical research studies in small animals and assist in the development of new respiratory treatments.

Model-Based Iterative Reconstruction for Propagation-Based Phase-Contrast X-Ray CT including Models for the Source and the Detector.

Propagation-based phase-contrast X-ray computed tomography is a valuable tool for high-resolution visualization of biological samples, giving distinct improvements in terms of contrast and dose requirements compared to conventional attenuation-based computed tomography. Due to its ease of implementation and advances in laboratory X-ray sources, this imaging technique is increasingly being transferred from synchrotron facilities to laboratory environments. This however poses additional challenges, such as the limited spatial coherence and flux of laboratory sources, resulting in worse resolution and higher noise levels. Here we extend a previously developed iterative reconstruction algorithm for this imaging technique to include models for the reduced spatial coherence and the signal spreading of efficient scintillator-based detectors directly into the physical forward model. Furthermore, we employ a noise model which accounts for the full covariance statistics of the image formation process. In addition, we extend the model describing the interference effects such that it now matches the formalism of the widely-used single-material phase-retrieval algorithm, which is based on the sample-homogeneity assumption. We perform a simulation study as well as an experimental study at a laboratory inverse Compton source and compare our approach to the conventional analytical approaches. We find that the modeling of the source and the detector inside the physical forward model can tremendously improve the resolution at matched noise levels and that the modeling of the covariance statistics reduces overshoots (i.e. incorrect increase / decrease in sample properties) at the sample edges significantly.

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.

X-ray Fokker-Planck equation for paraxial imaging.

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.