Effects of unresolvable edges in grating-based X-ray differential phase imaging

Correcting directional dark field x-ray imaging artefacts using position dependent image deblurring and attenuation removal

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.

[X-Ray Dark-field radiography: Does this carry potential for diagnosing gout in exotic pets?]

OBJECTIVE

The aim of this study is to evaluate whether X-ray dark-field (DF) radiography is useful for the diagnosis of gout in birds and reptiles and whether this preclinical model could be helpful to establish this non-invasive imaging method in human medicine.

MATERIAL AND METHODS

A total of 18 limbs originating from 11 birds (7 different species) and 7 reptiles (4 different species) with and without suspected joint gout were measured using a grating-based X-ray dark-field setup and conventional X-ray examination, respectively. Each image acquisition generated a dark-field and a conventional absorption x-ray image. The results of the individual scans were compared with the results of a pathological examination and arthrocentesis.

RESULTS

In 5 of the birds and 4 of the reptiles examined, gout was detected by pathologic examination. In each group, uric acid crystals were found in the joints of 3 animals by means of arthrocentesis. The uric acid crystals were detectable in 2 bird and 2 reptile limbs in the dark-field image.

CONCLUSION

The study demonstrated that the urate crystals evoke a clearly visible dark field signal, whereas this was not the case in the conventional radiographs.

CLINICAL RELEVANCE

The results obtained show that uric acid crystal detection using less invasive imaging methods in an animal model with birds and reptiles may expand gout diagnostics not only in veterinary medicine but also in human medicine and possibly replace arthrocentesis if a DF signal is detectable. Preclinical scanners which use X-ray dark-field and phase-contrast radiography already exist for hands and mammography.

X-ray dark-field radiography for in situ gout diagnosis by means of an ex vivo animal study

Gout is the most common form of inflammatory arthritis, caused by the deposition of monosodium urate (MSU) crystals in peripheral joints and tissue. Detection of MSU crystals is essential for definitive diagnosis, however the gold standard is an invasive process which is rarely utilized. In fact, most patients are diagnosed or even misdiagnosed based on manifested clinical signs, as indicated by the unchanged premature mortality among gout patients over the past decade, although effective treatment is now available. An alternative, non-invasive approach for the detection of MSU crystals is X-ray dark-field radiography. In our work, we demonstrate that dark-field X-ray radiography can detect naturally developed gout in animals with high diagnostic sensitivity and specificity based on the in situ measurement of MSU crystals. With the results of this study as a potential basis for further research, we believe that X-ray dark-field radiography has the potential to substantially improve gout diagnostics.

Comparison of methods for sensitivity correction in Talbot-Lau computed tomography

Purpose

In Talbot–Lau X-ray phase contrast imaging, the measured phase value depends on the position of the object in the measurement setup. When imaging large objects, this may lead to inhomogeneous phase contributions within the object. These inhomogeneities introduce artifacts in tomographic reconstructions of the object.

Methods

In this work, we compare recently proposed approaches to correct such reconstruction artifacts. We compare an iterative reconstruction algorithm, a known operator network and a U-net. The methods are qualitatively and quantitatively compared on the Shepp–Logan phantom and on the anatomy of a human abdomen. We also perform a dedicated experiment on the noise behavior of the methods.

Results

All methods were able to reduce the specific artifacts in the reconstructions for the simulated and virtual real anatomy data. The results show method-specific residual errors that are indicative for the inherently different correction approaches. While all methods were able to correct the artifacts, we report a different noise behavior.

Conclusion

The iterative reconstruction performs very well, but at the cost of a high runtime. The known operator network shows consistently a very competitive performance. The U-net performs slightly worse, but has the benefit that it is a general-purpose network that does not require special application knowledge.

Generalized reverse projection method for grating-based phase tomography

The reverse projection protocol results in fast phase-contrast imaging thanks to its compatibility with conventional computed-tomography scanning. Many researchers have proposed variants. However, all these reverse projection methods in grating-based phase-contrast imaging are built on the hypothesis of the synchronous phase of reference shifting curves in the whole field of view. The hypothesis imposes uniformity and alignment requirements on the gratings, thus the field of view is generally limited. In this paper, a generalized reverse projection method is presented analytically for the case of non-uniform reference in grating-based phase tomography. The method is demonstrated by theoretical derivation, numerical simulations and synchrotron radiation experiments. The influence of imaging position to sensitivity, and the phase-wrapping phenomenon are also discussed. The proposed method combines the advantages of the high efficiency of the reverse projection method and the universal applicability of the phase-stepping method. The authors believe that the method would be used widely in fast and dose-constrained imaging.

Empirical beam hardening and ring artifact correction for x-ray grating interferometry (EBHC-GI)

PURPOSE

Talbot-Lau grating interferometry enables the use of polychromatic x-ray sources, extending the range of potential applications amenable to phase contrast imaging. However, these sources introduce beam hardening effects not only from the samples but also from the gratings. As a result, grating inhomogeneities due to manufacturing imperfections can cause spectral nonuniformity artifacts when used with polychromatic sources. Consequently, the different energy dependencies of absorption, phase, and visibility contrasts impose challenges that so far have limited the achievable image quality. The purpose of this work was to develop and validate a correction strategy for grating-based x-ray imaging that accounts for beam hardening generated from both the imaged object and the gratings.

METHODS

The proposed two-variable polynomial expansion strategy was inspired by work performed to address beam hardening from a primary modulator. To account for the multicontrast nature of grating interferometry, this approach was extended to each contrast to obtain three sets of correction coefficients, which were determined empirically from a calibration scan. The method's feasibility was demonstrated using a tabletop Talbot-Lau grating interferometer micro-computed tomography (CT) system using CT acquisitions of a water sample and a silicon sample, representing low and high atomic number materials. Spectral artifacts such as cupping and ring artifacts were quantified using mean squared error (MSE) from the beam-hardening-free target image and standard deviation within a reconstructed image of the sample. Finally, the model developed using the water sample was applied to a fixated murine lung sample to demonstrate robustness for similar materials.

RESULTS

The water sample's absorption CT image was most impacted by spectral artifacts, but following correction to decrease ring artifacts, an 80% reduction in MSE and 57% reduction in standard deviation was observed. The silicon sample created severe artifacts in all contrasts, but following correction, MSE was reduced by 94% in absorption, 96% in phase, and 90% in visibility images. These improvements were due to the removal of ring artifacts for all contrasts and reduced cupping in absorption and phase images and reduced capping in visibility images. When the water calibration coefficients were applied to the lung sample, ring artifacts most prominent in the absorption contrast were eliminated.

CONCLUSIONS

The described method, which was developed to remove artifacts in absorption, phase, and normalized visibility micro-CT images due to beam hardening in the system gratings and imaged object, reduced the MSE by up to 96%. The method depends on calibrations that can be performed on any system and does not require detailed knowledge of the x-ray spectrum, detector energy response, grating attenuation properties and imperfections, or the geometry and composition of the imaged object.

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.

Wave optics simulation of grating-based X-ray phase-contrast imaging using 4D Mouse Whole Body (MOBY) phantom

PURPOSE

Demonstrate realistic simulation of grating-based x-ray phase-contrast imaging (GB-XPCI) using wave optics and the four-dimensional Mouse Whole Body (MOBY) phantom defined with non-uniform rational B-splines (NURBS).

METHODS

We use a full-wave approach, which uses wave optics for x-ray wave propagation from the source to the detector. This forward imaging model can be directly applied to NURBS-defined numerical phantoms such as MOBY. We assign the material properties (attenuation coefficient and electron density) of each model part using the data for adult human tissues. The Poisson noise is added to the simulated images based on the calculated photon flux at each pixel.

RESULTS

We simulate the intensity images of the MOBY phantom for eight different grating positions. From the simulated images, we calculate the absorption, differential phase, and normalized visibility contrast images. We also predict how the image quality is affected by different exposure times.

CONCLUSIONS

GB-XPCI can be simulated with the full-wave approach and a realistic numerical phantom defined with NURBS.

Modeling of an X-ray grating-based imaging interferometer using ray tracing

X-ray imaging by means of a grating-based Talbot-Lau interferometer has become an important tool for a wide variety of application areas such as security, medical and materials analysis. Imaging modalities include attenuation, differential phase contrast, and visibility contrast (or so-called dark field). We have developed a novel modeling approach based on ray tracing with commercially available software (Zemax OpticStudio) that yields image projections for all three modalities. The results compare favorably with experimental findings. Our polychromatic ray-based model accommodates realistic 3-D CAD objects with tailored materials properties and also allows for both surface and bulk scattering. As such, the model can simulate imaging of complicated objects as well as assist in a physical understanding of experimental projection details.

Recent Progress in X-ray and Neutron Phase Imaging with Gratings

Under the JST-ERATO project in progress to develop X-ray and neutron phase-imaging methods together, recent achievements have been selected and reviewed after describing the merit and the principle of the phase imaging method. For X-ray phase imaging, recent developments of four-dimensional phase tomography and phase microscopy at SPring-8, Japan are mainly presented. For neutron phase imaging, an approach in combination with the time-of-flight method developed at J-PARC, Japan is described with the description of new Gd grating fabrication.

The trigonometric orthogonality of phase-stepping curves in grating-based x-ray phase-contrast imaging: Integral property and its implications for noise optimization

PURPOSE

Grating-based x-ray phase-contrast imaging (GPCI) is a promising technique for clinical applications as it can provide two newly emerging imaging modalities (differential phase-contrast and dark-field contrast) in addition to the conventional absorption contrast. As far, phase-stepping strategy is the most commonly used approach in GPCI to indirectly acquire differential phase-contrast and dark-field contrast. It is known that the obtained phase-stepping curves (PSCs) have the cosine property and the convolution property, leading to two types of information retrieval approaches in literature: the Fourier component analysis and the multi-order moment analysis. The purpose of this paper is to derive a new property of PSCs and apply the property to noise optimization for information retrieval.

METHODS

Based on the cosine expression of the flat PSC without the sample and the well-established convolution relationship between the flat PSC and the sample PSC, we reveal an important integral property of PSCs: the inner product of PSCs and an arbitrary function contains only zero-order and first-order components in the Fourier series. Furthermore, we apply the property to the direct multi-order moment analysis and propose a set of generalized forms including an optimal one in the presence of noise.

RESULTS

To validate the effectiveness of our analysis, we compared the simulated and real experiment results retrieved by the original direct multi-order moment analysis with the ones retrieved by our proposed noise-optimal form. A significant improvement of noise performance by our method is observed and the improvement ratio in differential phase-contrast is consistent with our theoretical calculation (39.2%).

CONCLUSIONS

In this paper, we reveal a new integral property of the acquired PSCs with and without samples in GPCI, which can be applied to information retrieval approaches like the direct multi-order moment analysis. Then we optimize these approaches to improve the noise performance, offering great potentials of dose reduction in practical applications.

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.

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.

Probing Surface Morphology using X-ray Grating Interferometry

X-ray reflectometry (XRR), a surface-sensitive technique widely used for characterizing surfaces, buried interfaces, thin films, and multilayers, enables determination of the electron density distribution perpendicular to a well-defined surface specularly reflecting X-rays. However, the electron density distribution parallel to the surface cannot be determined from an X-ray reflectivity curve. The electron density correlation in the lateral direction is usually probed by measuring the grazing-incidence small-angle X-ray scattering (GISAXS). GISAXS measurement, however, typically requires using a collimated X-ray point beam to distinguish the GISAXS from the specularly reflected X-rays, and so the sample must be scanned in the lateral direction with the point beam to investigate variations in the surface and interface morphology for a region larger than the size of the beam. In this paper, we report a new approach based on X-ray grating interferometry: an X-ray sheet beam is used instead of an X-ray point beam. A method using this approach can simultaneously provide one-dimensional real-space images of X-ray reflectivity, surface curvature, and ‘dark-field’ contrast with a field-of-view of more than a few millimetres. As a demonstration, a sample having a 400 nm line and space SiO₂ pattern with a depth of 10 nm on its surface was used, and the dark-field contrast due to the unresolved line and space structure, creating GISAXS in the lateral direction, was successfully observed. Quantitative analysis of these contrasts provided the real-space distribution of the structural parameters for a simple model of the grating structure. Our study paves the way to a new approach to structure analysis, providing a quantitative way to investigate real-space variations in surface and interface morphology through wavefront analysis.

Hard X-ray imaging microscopy with self-imaging phenomenon

The self-imaging phenomenon referred to as the Talbot effect in the field of optics was discovered by H.F. Talbot in the 1830s, and is now widely used for imaging using not only visible light but also X-rays, electrons, neutrons, and matter waves. In this review, the author introduces the current progress being made in hard-X-ray imaging microscopy based on the self-imaging phenomenon. Hard-X-ray imaging microscopy is a promising technique for non-destructively visualizing internal structures in specimens with a spatial resolution up to a few tens of nanometers. The use of the self-imaging phenomenon makes it possible to realize highly sensitive phase-contrast X-ray imaging microscopes. These approaches have several advantages over conventional X-ray imaging microscopes, including the widely used Zernike X-ray phase-contrast microscopes, and can provide a powerful way of quantitative visualization with a high spatial resolution and a high sensitivity even for thick specimens.

Implementation of a Talbot-Lau interferometer in a clinical-like c-arm setup: A feasibility study

X-ray grating-based phase-contrast imaging has raised interest regarding a variety of potential clinical applications, whereas the method is feasible using a medical x-ray tube. Yet, the transition towards a clinical setup remains challenging due to the requirement of mechanical robustness of the interferometer and high demands applying to medical equipment in clinical use. We demonstrate the successful implementation of a Talbot-Lau interferometer in an interventional c-arm setup. The consequence of vibrations induced by the rotating anode of the tube is discussed and the prototype is shown to provide a visibility of 21.4% at a tube voltage of 60 kV despite the vibrations. Regarding clinical application, the prototype is mainly set back due to the limited size of the field of view covering an area of 17 mm × 46 mm. A c-arm offers the possibility to change the optical axis according to the requirements of the medical examination. We provide a method to correct for artifacts that result from the angulation of the c-arm. Finally, the images of a series of measurements with the c-arm in different angulated positions are shown. Thereby, it is sufficient to perform a single reference measurement in parking position that is valid for the complete series despite angulation.

Effect of insufficient temporal coherence on visibility contrast in X-ray grating interferometry

X-ray grating interferometry, which has been spotlighted in the last decade as a multi-modal X-ray imaging technique, can provide three independent images, i.e., absorption, differential-phase, and visibility-contrast images. We report on a cause of the visibility contrast, an effect of insufficient temporal coherence, that can be observed when continuous-spectrum X-rays are used. This effect occurs even for a sample without unresolvable random structures, which are known as the main causes of visibility contrast. We performed an experiment using an acrylic cylinder and quantitatively explained the visibility contrast due to this effect.

Design of Acquisition Schemes and Setup Geometry for Anisotropic X-ray Dark-Field Tomography (AXDT)

Anisotropic X-ray Dark-field Tomography (AXDT) is a new imaging technique for reconstructing the three-dimensional scattering profile within a sample using the dark-field signal measured in an X-ray grating interferometry setup. As in any tomographic measurement, the acquisition geometry plays a key role in the accurate reconstruction of the scattering information. More- over, the anisotropic nature of the dark-field signal poses additional challenges for designing the acquisition protocols. In this work, we present an efficient approach to measure scattering orientations spread over the unit sphere and prove its efficacy using the knowledge from conventional tomography. In addition, we conclude (using analytical and experimental results) that placing the gratings such that the grating bars make an angle of 45 degrees with respect to the vertical direction is the optimal setup configuration for AXDT.

A beam hardening and dispersion correction for x-ray dark-field radiography

PURPOSE

X-ray dark-field imaging promises information on the small angle scattering properties even of large samples. However, the dark-field image is correlated with the object's attenuation and phase-shift if a polychromatic x-ray spectrum is used. A method to remove part of these correlations is proposed.

METHODS

The experimental setup for image acquisition was modeled in a wave-field simulation to quantify the dark-field signals originating solely from a material's attenuation and phase-shift. A calibration matrix was simulated for ICRU46 breast tissue. Using the simulated data, a dark-field image of a human mastectomy sample was corrected for the finger print of attenuation- and phase-image.

RESULTS

Comparing the simulated, attenuation-based dark-field values to a phantom measurement, a good agreement was found. Applying the proposed method to mammographic dark-field data, a reduction of the dark-field background and anatomical noise was achieved. The contrast between microcalcifications and their surrounding background was increased.

CONCLUSIONS

The authors show that the influence of and dispersion can be quantified by simulation and, thus, measured image data can be corrected. The simulation allows to determine the corresponding dark-field artifacts for a wide range of setup parameters, like tube-voltage and filtration. The application of the proposed method to mammographic dark-field data shows an increase in contrast compared to the original image, which might simplify a further image-based diagnosis.

Time resolved X-ray Dark-Field Tomography Revealing Water Transport in a Fresh Cement Sample

Grating-based X-ray dark-field tomography is a promising technique for biomedical and materials research. Even if the resolution of conventional X-ray tomography does not suffice to resolve relevant structures, the dark-field signal provides valuable information about the sub-pixel microstructural properties of the sample. Here, we report on the potential of X-ray dark-field imaging to be used for time-resolved three-dimensional studies. By repeating consecutive tomography scans on a fresh cement sample, we were able to study the hardening dynamics of the cement paste in three dimensions over time. The hardening of the cement was accompanied by a strong decrease in the dark-field signal pointing to microstructural changes within the cement paste. Furthermore our results hint at the transport of water from certain limestone grains, which were embedded in the sample, to the cement paste during the process of hardening. This is indicated by an increasing scattering signal which was observed for two of the six tested limestone grains. Electron microscopy images revealed a distinct porous structure only for those two grains which supports the following interpretation of our results. When the water filled pores of the limestone grains empty during the experiment the scattering signal of the grains increases.

Asymmetric masks for laboratory-based X-ray phase-contrast imaging with edge illumination

We report on an asymmetric mask concept that enables X-ray phase-contrast imaging without requiring any movement in the system during data acquisition. The method is compatible with laboratory equipment, namely a commercial detector and a rotating anode tube. The only motion required is that of the object under investigation which is scanned through the imaging system. Two proof-of-principle optical elements were designed, fabricated and experimentally tested. Quantitative measurements on samples of known shape and composition were compared to theory with good agreement. The method is capable of measuring the attenuation, refraction and (ultra-small-angle) X-ray scattering, does not have coherence requirements and naturally adapts to all those situations in which the X-ray image is obtained by scanning a sample through the imaging system.

A Universal Moiré Effect and Application in X-Ray Phase-Contrast Imaging

A moiré pattern is created by superimposing two black-and-white or gray-scale patterns of regular geometry, such as two sets of evenly spaced lines. We observed an analogous effect between two transparent phase masks in a light beam which occurs at a distance. This phase moiré effect and the classic moiré effect are shown to be the two ends of a continuous spectrum. The phase moiré effect allows the detection of sub-resolution intensity or phase patterns with a transparent screen. When applied to x-ray imaging, it enables a polychromatic far-field interferometer (PFI) without absorption gratings. X-ray interferometry can non-invasively detect refractive index variations inside an object^1-10. Current bench-top interferometers operate in the near field with limitations in sensitivity and x-ray dose efficiency^{2, 5, 7-10}. The universal moiré effect helps overcome these limitations and obviates the need to make hard x-ray absorption gratings of sub-micron periods.

Imaging properties of high aspect ratio absorption gratings for use in preclinical x-ray grating interferometry

X-ray grating interferometry is one among various methods that allow extracting the so-called phase and visibility contrasts in addition to the well-known transmission images. Crucial to achieving a high image quality are the absorption gratings employed. Here, we present an in-depth analysis of how the grating type and lamella heights influence the final images. Benchmarking gratings of two different designs, we show that a frequently used proxy for image quality, a grating's so-called visibility, is insufficient to predict contrast-to-noise ratios (CNRs). Presenting scans from an excised rat lung, we demonstrate that the CNRs obtained for transmission and visibility images anti-correlate. This is explained by the stronger attenuation implied by gratings that are engineered to provide high visibilities by means of an increased lamella height. We show that even the visibility contrast can suffer from this effect when the associated reduced photon flux on the detector is not outweighed by a corresponding gain in visibility. Resulting in an inevitable trade-off between the quality of the two contrasts, the question of how an optimal grating should be designed can hence only be answered in terms of Pareto optimality.

Lens-term- and edge-effect in X-ray grating interferometry

X-ray grating interferometry requires gratings with periods in the micrometer range and allows the acquisition of the dark-field contrast. The analyzer grating is designed to match the period of the interference pattern in order to translate it into a measurable intensity modulation. In this study, we explore the influence of a sample-induced mismatch between the interference pattern and the analyzer grating on the dark-field contrast. We propose a formula for the calculation of the signal due to a period mismatch and present estimations varying periods and detector pixel size. Furthermore, numerical simulations of the X-ray wave-front demonstrate that the wave-front curvature, described by the lens-term, e.g. behind a parabolic lens or edges of a sample can change the period of the interference pattern. Our results give a concrete explanation for the formation of a dark-field contrast from object edges and thus allow a better understanding of the dark-field signal obtained with a grating interferometer.

Interferogram conditioning for improved Fourier analysis and application to X-ray phase imaging by grating interferometry

An interferogram conditioning procedure, for subsequent phase retrieval by Fourier demodulation, is presented here as a fast iterative approach aiming at fulfilling the classical boundary conditions imposed by Fourier transform techniques. Interference fringe patterns with typical edge discontinuities were simulated in order to reveal the edge artifacts that classically appear in traditional Fourier analysis, and were consecutively used to demonstrate the correction efficiency of the proposed conditioning technique. Optimization of the algorithm parameters is also presented and discussed. Finally, the procedure was applied to grating-based interferometric measurements performed in the hard X-ray regime. The proposed algorithm enables nearly edge-artifact-free retrieval of the phase derivatives. A similar enhancement of the retrieved absorption and fringe visibility images is also achieved.

Effect of beam hardening on a visibility-contrast image obtained by X-ray grating interferometry

X-ray grating interferometry has been highlighted in the last decade as a multi-modal X-ray phase-imaging technique for providing absorption, differential phase, and visibility-contrast images. It has been mainly reported that the visibility contrast in the visibility-contrast image originates from unresolvable random microstructures. In this paper, we show that the visibility contrast is even reduced by a uniform sample with flat surfaces due to the so-called "beam-hardening effect", which has to be taken into account when X-rays with a continuous spectrum is used. We drive a criterion for determining whether the beam-hardening effect occurs or not, and propose a method for correcting the effect of beam hardening on a visibility-contrast image.