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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.

Complementary to conventional and phase x-ray radiography, dark-field imaging has become central in visualizing diffusive scattering signals that come from spatially unresolved texture within an object. To date, most diffusive dark-field retrieval methods require either the acquisition of multiple images at the cost of higher radiation dose or significant amounts of computational memory and time. In this work, a simple method of x-ray diffusive dark-field retrieval is presented, applicable to any single-mask imaging setup, with only one exposure of the sample. The approach, which is based on a model of geometric and diffusive reverse-flow conservation, is implicit and noniterative. This numerically fast methodology is applied to experimental x-ray images acquired using both a random mask and a grid mask, giving high-quality reconstructions that are very stable in the presence of noise. The method should be useful for high-speed imaging and/or imaging with low-flux sources.

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.

Discrete delta functions define the limits of attainable spatial resolution for all imaging systems. Here we construct broad, multi-dimensional discrete functions that replicate closely the action of a Dirac delta function for convolution under aperiodic boundary conditions. These arrays spread the energy of a sharp probe beam to simultaneously sample multiple points across the volume of a large object, without losing image sharpness. Applying these point-spread functions in any computational imaging system can reveal the underlying structure of objects less intrusively and with equal or better signal-to-noise ratio. These multi-dimensional arrays are related to previously known, but relatively rarely employed, one-dimensional integer Huffman sequences. Practical probes can now be made that are larger than the object under measure. Such arrays can be applied to ghost imaging, which has demonstrated potential to greatly improve signal-to-noise ratios and reduce the total dose required for tomographic imaging. The discrete arrays built here parallel the self-adjoint or Hermitian functions of the continuum that underpin classical wave theory and quantum mechanics.

Spatial resolution in standard phase-contrast X-ray imaging is limited by the finite number and size of detector pixels. As a result, this limits the size of features that can be seen directly in projection images or tomographic reconstructions. Dark-field imaging allows information regarding such features to be obtained, as the reconstructed image is a measure of the position-dependent small-angle X-ray scattering of incident rays from the unresolved microstructure. In this paper we utilize an intrinsic speckle-tracking-based X-ray imaging technique to obtain the effective dark-field signal from a wood sample. This effective dark-field signal is 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. We here assume 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 dark-field signal, is numerically stabilised using a “weighted determinants” approach. Effective dark-field projection images are presented, as well as the dark-field tomographic reconstructions obtained using Fokker-Planck implicit speckle-tracking.

Propagation-based imaging is an X-ray phase-contrast method that does not require any additional optical elements in the imaging system. Instead, phase-contrast effects are observed as measurable intensity variations after an incident X-ray wave field has propagated some distance between the object and plane of detection. Subsequent phase-retrieval processing of the recorded images is a necessary step in order to obtain quantitative information and can greatly increase the image signal-to-noise ratio, in particular in computed tomography. Most commonly this is done by approximating the object as a single material, which cannot always be justified. An example of this is the presence of bones and soft-tissues in biomedical imaging, which motivated the work presented here. We developed the combination of spectral X-ray imaging and propagation-based X-ray phasecontrast imaging to overcome the limitations of a single-material approximation. The addition of spectral data enables phase-retrieval on multi-material objects, as well as the use of processing methods well-known from conventional dual-energy imaging, such as material decomposition, virtual monochromatic images and effective atomic-number mapping. Here, we present first computed tomography results of spectral propagation-based phase-contrast. We discuss how the phase retrieval process can be interpreted as low- and high-pass filters. Lastly, we take a look at current and future technological challenges, and the potential application areas of spectral propagation-based phase-contrast imaging.

A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We employ different spatial wave forms to understand the role played by both wave oscillations and spatial wave decay in the walking dynamics. We observe steady walking motion as well as unsteady motions such as oscillating walking, self-trapped oscillations and irregular walking. We explore the dynamical and statistical aspects of irregular walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion., Comment: 18 pages, 12 figures

Ghost imaging has recently been successfully achieved in the X-ray regime. Due to the penetrating power of X-rays this immediately opens up the possibility of ghost-tomography. No research into this topic currently exists in the literature. Here, we present adaptations of conventional X-ray tomography techniques to this new ghost-imaging scheme. Several numerical implementations for tomography through X-ray ghost-imaging are considered. Specific attention is paid to schemes for reducing the noise-like artefacts of the resulting tomographic reconstruction, issues related to dose fractionation, and considerations regarding the ensemble of illuminating masks used for ghost-imaging. Each theme is explored through a series of numerical simulations, and several suggestions offered for practical realisations of ghost-tomography.

The beam diffusing properties of stacked layers of diffuser material were evaluated experimentally and compared to a Gaussian random phase screen model. The model was found to give promising accuracy in combination with a Lorentzian auto-correlation model. The tail behaviour of the angular scattering distribution as a function of number of diffusing layers was particularly well described by the model, and in the case of an amorphous carbon diffuser, the model could describe the whole of the scattering distribution convincingly. (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Classical ghost imaging is a computational imaging technique that employs patterned illumination. It is very similar in concept to the single-pixel camera in that an image may be reconstructed from a set of measurements even though all imaging photons or particles that pass through that sample are never recorded with a position resolving detector. The method was first conceived and applied for visible-wavelength photons and was subsequently translated to other probes such as x rays, atomic beams, electrons, and neutrons. In the context of classical ghost imaging using penetrating probes that enable transmission measurement, we here consider several questions relating to the achievable signal-to-noise ratio (SNR). This is compared with the SNR for conventional imaging under scenarios of constant radiation dose and constant experiment time, considering both photon shot noise and per-measurement electronic readout noise. We show that inherent improved SNR capabilities of classical ghost imaging are limited to a subset of these scenarios and are actually due to increased dose (Fellgett advantage). An explanation is also presented for recent results published in the literature that are not consistent with these findings.

We outline some basics of imaging using both fully-coherent and partially-coherent X-ray beams, with an emphasis on phase-contrast imaging. We open with some of the basic notions of X-ray imaging, including the vacuum wave equations and the physical meaning of the intensity and phase of complex scalar fields. The projection approximation is introduced, together with the concepts of attenuation contrast and phase contrast. We also outline the multi-slice approach to X-ray propagation through thick samples or optical elements, together with the Fresnel scaling theorem. Having introduced the fundamentals, we then consider several aspects of the forward problem, of modeling the formation of phase-contrast X-ray images. Several topics related to this forward problem are considered, including the transport-of-intensity equation, arbitrary linear imaging systems, shift-invariant linear imaging systems, the transfer-function formalism, blurring induced by finite source size, the space–frequency model for partially-coherent fields, and the Fokker–Planck equation for paraxial X-ray imaging. Having considered these means for modeling the formation of X-ray phase-contrast images, we then consider aspects of the associated inverse problem of phase retrieval. This concerns how one may decode phase-contrast images to gain information regarding the sample-induced attenuation and phase shift.

It has been argued that in atomic-resolution transmission electron microscopy (TEM) of sparse weakly scattering structures, such as small biological molecules, multiple electron scattering usually has only a small effect, while the in-molecule Fresnel diffraction can be significant due to the intrinsically shallow depth of focus. These facts suggest that the three-dimensional reconstruction of such structures from defocus image series collected at multiple rotational orientations of a molecule can be effectively performed for each atom separately, using the incoherent first Born approximation. The corresponding reconstruction method, termed here Differential Holographic Tomography, is developed theoretically and demonstrated computationally on several numerical models of biological molecules. It is shown that the method is capable of accurate reconstruction of the locations of atoms in a molecule from TEM data collected at a small number of random orientations of the molecule, with one or more defocus images per orientation. Possible applications to cryogenic electron microscopy and other areas are briefly discussed., 32 pages, 8 figures, version 5c (a few typos have been found in the previous version and fixed in the current version)

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.

A simple coherent-imaging method due to Paganin et al is widely employed for phase–amplitude reconstruction of samples using a single paraxial x-ray propagation-based phase-contrast image. The method assumes that the sample-to-detector distance is sufficiently small for the associated Fresnel number to be large compared to unity. The algorithm is particularly effective when employed in a tomographic setting, using a single propagation-based phase-contrast image for each projection. Here we develop a simple extension of the method, which improves the reconstructed contrast of very fine sample features. This provides first-principles motivation for boosting fine spatial detail associated with high Fourier frequencies, relative to the original method, and was inspired by several recent works employing empirically-obtained Fourier filters to a similar end.

IUCrJ 7(6), 1114 - 1123 (2020). doi:10.1107/S2052252520013019, 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., Published by Chester

Signal-to-noise ratio, spatial resolution and information capacity of tomographic coherent diffractive imaging are investigated; the results are expected to be useful for the design and analysis of synchrotron and XFEL-based diffractive imaging experiments., It is shown that the average signal-to-noise ratio (SNR) in the three-dimensional electron-density distribution of a sample reconstructed by coherent diffractive imaging cannot exceed twice the square root of the ratio of the mean total number of scattered photons detected during the scan and the number of spatially resolved voxels in the reconstructed volume. This result leads to an upper bound on Shannon’s information capacity of this imaging method by specifying the maximum number of distinguishable density distributions within the reconstructed volume when the radiation dose delivered to the sample and the spatial resolution are both fixed. If the spatially averaged SNR in the reconstructed electron density is fixed instead, the radiation dose is shown to be proportional to the third or fourth power of the spatial resolution, depending on the sampling of the three-dimensional diffraction space and the scattering power of the sample.

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.

Arbitrary distributions of radiant exposure may be written by transversely scanning a single known spatially random screen that is normally illuminated by spatially but not necessarily temporally uniform radiation or matter wave fields. The arbitrariness of the written pattern of radiant exposure holds up to both a spatial resolution that is dictated by the characteristic transverse length scale of the illuminated spatially random screen and a background term that grows linearly with the number of random-illumination patterns. Two classes of the method are developed. One assumes the distance between the illuminated random mask and the target plane to be sufficiently small that the effects of diffraction may be neglected. The other accounts for the effects of Fresnel diffraction in the regime of large Fresnel number. Numerical simulations are provided for both variants of the method. Contrast and the signal-to-noise ratio are also considered. The method may be parallelized and is suited to both magnifying and demagnifying geometries. Possible applications include spatial light modulators and intensity projectors for those matter and radiation wave fields for which such devices do not exist, printing or microfabrication in both two and three spatial dimensions, and lithography.

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 rapid development of mechanistic, trait-based models has resulted in increasingly reliable predictions of the functional diversity of individuals in populations and communities. However, a focus on individuals’ traits differs from the prevailing focus on species in much of community ecology. We sought to identify correlative links between species richness and size diversity, focusing on size diversity as one component of functional diversity. These links could be used to extend individual, size-based models to predict patterns of species richness. We used the distribution of the sizes of individuals in a community – the individual–size distribution (ISD) – as a measure of size diversity, and constructed Bayesian regression models with species richness as the response variable and ISDs as the predictor variables. We used two methods to include ISDs in our analyses. First, we summarized the ISD with five common diversity indices and used these indices as predictor variables in our analyses. Second, we used functional data analysis to include the entire ISD (a continuous function) as a predictor variable in our analyses. Analyses of diversity indices identified consistent, positive associations between species richness and size diversity. Analyses of entire ISDs revealed that these associations were driven by numbers of small- and medium-sized individuals. In general, a combination of diversity indices predicted species richness as well as or better than continuous ISDs. However, models with ISDs as predictor variables were less sensitive to technical details of model fitting (e.g. discretization method) than those based on diversity indices, and the use of ISDs avoids the arbitrary selection of one or several diversity indices. Our use of functional data analysis allows any trait distribution to be included as a variable in statistical analyses, and has the potential to reveal new diversity patterns in ecology.

A practical experimental procedure for transmission X-ray ghost imaging (XGI) using synchrotron light is presented. The authors demonstrate the method, discuss data acquisition and analysis, and measure the point-spread function of an XGI system. The generalization of the methods for future experiments is also discussed., An experimental procedure for transmission X-ray ghost imaging using synchrotron light is presented. Hard X-rays from an undulator were divided by a beamsplitter to produce two copies of a speckled incident beam. Both beams were simultaneously measured on an indirect pixellated detector and the intensity correlation between the two copies was used to retrieve the ghost image of samples placed in one of the two beams, without measuring the samples directly. Aiming at future practical uses of X-ray ghost imaging, the authors discuss details regarding data acquisition, image reconstruction strategies and measure the point-spread function of the ghost-imaging system. This approach may become relevant for applications of ghost imaging with X-ray sources such as undulators in storage rings, free-electron lasers and lower-coherence laboratory facilities.

Electric field mapping using segmented detectors in the scanning transmission electron microscope has recently been achieved at the nanometre scale. However, converting these results to quantitative field measurements involves assumptions whose validity is unclear for thick specimens. We consider three approaches to quantitative reconstruction of the projected electric potential using segmented detectors: a segmented detector approximation to differential phase contrast and two variants on ptychographical reconstruction. Limitations to these approaches are also studied, particularly errors arising from detector segment size, inelastic scattering, and non-periodic boundary conditions. A simple calibration experiment is described which corrects the differential phase contrast reconstruction to give reliable quantitative results despite the finite detector segment size and the effects of plasmon scattering in thick specimens. A plasmon scattering correction to the segmented detector ptychography approaches is also given. Avoiding the imposition of periodic boundary conditions on the reconstructed projected electric potential leads to more realistic reconstructions.

Aim Data on ecological communities are often condensed into single-valued diversity indices, which support comparisons among ecosystems but may discard important information. At the other extreme, some studies retain full data on the identities of all species present, which retains maximum information on community structure but occludes comparisons among ecosystems. We sought to determine whether the analysis of species abundance distributions and individual size distributions could support more detailed inferences than diversity indices while remaining sufficiently general to identify fundamental ecological responses in multiple ecosystems. Location The United States and south-eastern Australia. Methods We analysed changes in species abundance distributions and individual size distributions along environmental gradients. We compared analyses of distributions with analyses of five diversity indices and with a distance-based multivariate analysis of species composition. We used data from 1438 fish communities in the United States and from 840 bird communities in the western United States and south-eastern Australia. These communities were sampled over 10–20 years and included 298 species of birds and 488 species of fishes. Results Analyses of species abundance distributions and individual size distributions identified environmental associations that were not found in analyses of diversity indices, and revealed common associations with shrubs in bird communities on different continents and with temperature in communities of birds and fishes. Individual size distributions were associated more closely with environmental variables than were species abundance distributions. Main conclusions Our novel identification of ecological associations suggests that analyses of community distributions might reveal fundamental ecological processes common to multiple ecosystems. Close associations between individual sizes and environmental variables suggest that a greater focus on size in ecological analyses might assist in the development of general models of ecological communities.

We develop a means for speckle-based phase imaging of the projected thickness of a single-material object, under the assumption of illumination by spatially random time-independent x-ray speckles. These speckles are generated by passing x rays through a suitable spatially random mask. The method makes use of a single image obtained in the presence of the object, which serves to deform the illuminating speckle field relative to a reference speckle field (which only needs to be measured once) obtained in the presence of the mask and the absence of the object. The method implicitly rather than explicitly tracks speckles, and utilizes the transport-of-intensity equation to give a closed-form solution to the inverse problem of determining the complex transmission function of the object. Implementation using x-ray synchrotron data shows the method to be robust and efficient with respect to noise. Applications include x-ray phase--amplitude radiography and tomography, as well as time-dependent imaging of dynamic and radiation-sensitive samples using low-flux sources.

We present a pixel-specific, measurement-driven correction that effectively reduces errors in detector response that give rise to the ring artifacts commonly seen in X-ray computed tomography (CT) scans. This correction is easy to implement, suppresses CT artifacts significantly, and is effective enough for use with both absorption and phase contrast imaging. It can be used as a standalone correction or in conjunction with existing ring artifact removal algorithms to further improve image quality. We validate this method using two X-ray CT data sets acquired using monochromatic sources, showing post-correction signal-to-noise increases of up to 55%, and we define an image quality metric to use specifically for the assessment of ring artifact suppression.

We consider a realistic model for calculating the cross-spectral density of partially coherent beams from an x-ray undulator in a modern storage ring. This two-point coherence function is seen to have a speckled structure associated with the presence of x-ray coherence vortices and domain walls. Such cross-spectral density speckle is associated with a network of spatial pairs of points for which there is zero correlation. X-ray coherence vortices and domain walls are seen to emerge naturally as the number of coherent modes required increases. An understanding of the existence and nature of such correlation singularities enhances our ability to exploit partially coherent x-ray radiation from new or upgraded synchrotron sources, for both imaging and diffraction applications.

The notions of spatial resolution and signal-to-noise are central to most forms of imaging. However, it appears that rigorous definitions of these quantities, which would be general enough to be useful in a broad range of imaging problems, while being also sufficiently specific to enable precise quantitative evaluation of the relevant properties of imaging systems, have been somewhat lacking. This is particularly true in respect to many modern forms of imaging that include digital processing of the acquired imaging data as an integral step leading to final images presented to the enduser. In the present paper, both the well-known historical definitions of spatial resolution and some more recent approaches suitable for many modern computational imaging techniques are discussed. An intrinsic duality of the spatial resolution and signal-to-noise exists in almost all types of imaging, with the related uncertainty relationship determining the inevitable trade-offs between the two quantities. Examples are presented with applications to some forms of X-ray imaging. Relations to Shannon and Fisher information capacity of imaging systems and super-resolution are also briefly discussed.

Standard histopathological examination is the gold standard for many disease diagnoses although the technique is limited with no full 3D volume analysis possible. Three dimensional X-ray Phase-Contrast Imaging(PCI) methods have been under constant and fast developments in the recent decades due to their superior performance for imaging low density objects and their ability to provide complementary information compared to attenuation based imaging. Despite the progresses, X-ray Phase Contrast Tomography still encounters remaining challenges to overcome on its way to become a routine non-invasive technique allowing the 3D assessment of tissue architecture in laboratory set-ups. Speckle Based Imaging (SBI) forms a new class of X-ray PCI techniques, sensitive to the first derivative of the phase. The set-up involved and the simplicity of implementation provide many advantages to SBI such as having no field of view and no resolution limitation in addition to have low requirements on the beam coherences. These advantages make SBI a good candidate for the transfer on conventional sources. In this work, we present preliminary results obtained on a conventional μCT and their comparison with data acquired at the European Synchrotron. We used a new phase retrieval algorithm based on optical energy conservation. We applied the method on both phantoms and biological samples in order to evaluate its quantitativeness for a transfer. A comparison to previously available speckle tracking algorithms is also performed. We demonstrate that the combination of the phase retrieval method with a standard μCT can achieve high resolution and high contrast within a few minutes, with a comparable image quality to the results obtained using synchrotron light.

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.

Ghost imaging is demonstrated using a poly-energetic reactor source of thermal neutrons. The method presented enables position resolution to be incorporated, into a variety of neutron instruments that are not position resolving. In an imaging context, ghost imaging can be beneficial for dose reduction and resolution enhancement. We also demonstrate a super-resolution variant of the method, namely a parallel form of neutron ghost imaging, with the ability to significantly increase the spatial resolution of a pixelated detector such as a CCD or CMOS camera. Extensions of our neutron ghost-imaging protocol are discussed in detail and include neutron ghost tomography, neutron ghost microscopy, dark-field neutron ghost imaging, and isotope-resolved color neutron ghost imaging via prompt gamma-ray bucket detection., Comment: 10 page, journal article format