Single shot x-ray dark-field and differential phase contrast imaging with a single-mask set-up

Abstract

Single-mask X-ray imaging simultaneously captures attenuation, differential phase contrast (DPC), and dark-field images in a single exposure by capturing multiple contrast features like dark field and differential phase contrast using mask alignment with relatively low-resolution detectors and smaller magnification. New light transport models are used for various system configurations to effectively extract contrast features from various single mask X-ray configurations, thereby eliminating the need for highly coherent X-ray sources, ultra-high-resolution detectors, and intricate gratings, and as such simplifying the imaging process and significantly reducing costs and improving the prospects of clinical translation. Three variations of the single-mask setup are each optimized for different contrast types, offering flexibility and efficiency in a variety of applications.

Description

[0001] UNITED STATES PATENT APPLICATION FOR:

[0002] SINGLE SHOT X-RAY DARK-FIELD AND DIFFERENTIAL PHASE CONTRAST IMAGING WITH A SINGLE-MASK SET-UP

[0003] RELATED APPLICATIONS

[0004] This application claims the benefit of United States (US) Patent Application Number 63 / 712,760, filed October 28, 2024.

[0005] GOVERNMENT LICENSE RIGHTS

[0006] This invention was made with government support from the National Institutes of Health (NIH) National Institute of Biomedical Imaging and Bioengineering (NIB IB) grant R01 EB EB029761, the US Department of Defense (DOD) Congressionally Directed Medical Research Program (CDMRP) Breakthrough Award BC 151607, and the National Science Foundation CAREER Award 1652892. The government has certain rights in the invention.

[0007] FIELD OF THE INVENTION

[0008] The embodiments herein relate to imaging systems, devices, and methods and, more particularly, to X-ray imaging.

[0009] BACKGROUND

[0010] X-ray imaging has long been a fundamental tool in both medical diagnostics and materials science due to its ability to reveal internal structures with high spatial resolution. Traditional X-ray imaging techniques predominantly rely on attenuation contrast, where differences in the absorption of X-rays by various tissue types or materials generate an image. However, this method often falls short when distinguishing between materials with similar attenuation coefficients, especially for light-element materials like soft tissues.

[0011] Additionally, its ability to detect microstructures is constrained by the resolution limits of conventional imaging systems.

[0012] X-ray phase-contrast imaging (XPCI) and dark-field (DF) imaging have emerged as powerful complementary techniques that offer enhanced contrast, particularly for soft tissues and materials with microstructures (See, e.g., References 1, 2, 3, 4, 5). Phase-contrast imaging is sensitive to the phase shift of X-ray wave that passes through the medium, which exploits variations in the refractive index of different materials at the length scale of tens of micrometers. These variations induce beam refraction at the micro-radian scale See, e.g., References 4, 5). This type of techniques offers enhanced visualization through signatures like differential phase contrast (DPC) and Laplacian phase signatures, etc.

[0013] Dark-field imaging, on the other hand, is sensitive to ultra-small-angle X-ray scattering (USAXS) induced by sub-pixel-scale (microns or sub-micron level) microstructures within the medium. As X-rays pass through, they undergo multiple refractions caused by these microstructures, leading to the scattering of X-ray photons within an extremely narrow angular range, which appears as local blurring on images (See, e.g., References 5, 6). This phenomenon highlights the presence of microstructures that are invisible in traditional absorption-based imaging, thereby introducing a new dimension of contrast in X-ray imaging.

[0014] Most XPCI techniques follow a common principle: a reference intensity pattern is generated and its distortions, caused by refraction and USAXS, are analyzed to retrieve the phase and dark-field signals. Specifically, x-ray attenuation locally reduces the average brightness of the pattern, refraction causes slight local displacements, and scattering causes local blurring, thereby locally decreasing the contrast of the pattern. Therefore, XPCI methods can therefore be classified according to two key aspects: (1) how the reference pattern is generated; and (2) how changes to that pattern are detected.

[0015] The detection of reference pattern change remains particularly difficult, since the X-ray refraction angles are typically at micro- to sub-micro-radian ranges for soft materials. Capturing these effects often requires high resolution detectors or multiple steps of data acquisitions with precise movements of the x-ray optics in the beam path.

[0016] Propagation-based (PB) imaging (See, e.g., References 7, 8) is the simplest technique for phase contrast imaging, as it does not require additional optical elements in the beam path. The method works by increasing the sample-to-detector distance, allowing phase contrast patterns to form as the propagation of X-rays. Despite its simplicity, PB imaging requires a coherent or partially coherent X-ray source and longer exposure times to produce high-quality phase-contrast images. Also, retrieving the phase signal can be challenging, as it usually has additional requirements, like single-material assumption (5Fe, e g., Reference 9), multiple exposures (See, e.g., Reference 10), or spectral imaging (See, e.g., Reference 11), etc.

[0017] Grating interferometry (See, e.g., References 12, 13) typically employs three delicate gratings. The diffraction of the second grating (Talbot effect) forms the reference pattern and analyzed by the stepping of the third grating with multiple exposures. The method is compatible with conventional X-ray tubes by incorporating a source grating to create an array of spatially coherent sources. However, its clinical translation faces challenges due to the complexity of the setup, stringent fabrication requirements for gratings with micron-level pitch, and the need for multiple exposures with precise grating movements. Furthermore, the technique is typically optimized for a single wavelength, which limits its performance with broad-spectrum X-ray sources (See, e.g., References 14-16).

[0018] Modulation-based imaging techniques, including speckle-based (See, e.g., References 17, 18), beam-tracking (See, e.g., References 19-21), and mesh-based methods (See, e.g., Reference 22), employ a single optical element to generate a reference intensity pattern. This pattern can arise from either a random texture such as sandpaper (speckle-based) or a periodic pattern (beam tracking and mesh-based) (See, e.g., Reference 23). Despite the simplicity, these techniques rely heavily on ultra-high-resolution detectors (See, e.g., References 17, 20, 21) or high geometric magnification (See, e.g., Reference 19) to resolve the subtle deformations of the reference pattern caused by refraction and scattering, which remains a significant limitation for translation to clinical and industrial imaging systems.

[0019] The edge-illumination (or coded-aperture) method (See, e.g., Reference 24) uses two lower-pitch masks instead of delicate gratings: the first generates a reference pattern, which is analyzed by the stepping of the second mask with multiple exposures. Unlike gratingbased methods, it does not rely on the Talbot effect, making it more tolerant to polychromatic source. Despite its advantages, it still requires precise alignment, multiple exposures with precise mask movements (See, e.g., Reference 25) or spectral imaging (See, e.g., Reference 26) to extract dark-field and differential phase images.

[0020] In summary, most of these conventional techniques demand complex and costly setups, often requiring one or more of the following: highly coherent X-ray sources, ultra-high-resolution detectors, sophisticated X-ray gratings, ultra-precise alignment of components, and multiple exposures to retrieve phase contrast images. These stringent requirements have hindered the broader adoption of these methods in clinical and industrial applications. Thus, there is a need for an X-ray configuration that effectively captures multiple contrast features with a single exposure using relatively low-resolution detectors.

[0021] INCORPORATION BY REFERENCE

[0022] Each patent, patent application, and / or publication mentioned in this specification is herein incorporated by reference in its entirety to the same extent as if each individual patent, patent application, and / or publication was specifically and individually indicated to be incorporated by reference.

[0023] SUMMARY OF THE INVENTION

[0024] A method for multi-contrast imaging is described herein. The method comprises positioning a mask between a signal source and a sample. The mask comprises a periodic absorption X-ray mask and the signal source comprises an X-ray. The method includes configuring the mask using each of a plurality of mask alignments. Attenuation, differential phase contrast (DPC), and dark-field images of the sample are simultaneously captured at the detector in a single exposure of the mask with the signal. The detector comprises a low-resolution detector including pixel sizes larger than approximately 20 micrometers

[0025] The simultaneously capturing comprises using the plurality of mask alignments to generate a plurality of dimensions of the signal to simultaneously capture at the detector multiple contrast features including the attenuation, the differential phase contrast (DPC), and the dark-field during the single exposure.

[0026] The method comprises optimizing each of the plurality of mask alignments for different types of contrast to provide more dimensions of the signal at the detector for material classification.

[0027] The configuring of the mask comprises configuring the mask using a first mask alignment of the plurality of mask alignments, wherein the first mask alignment comprises positioning the mask such that a period of a projected mask pattern is twice a detector pixel size and a center of each beamlet strip produced by the mask is aligned with a boundary of every other pixel, wherein each beamlet strip is generated by the mask. The method comprises retrieving attenuation and differential phase images from the single exposure of the sample.

[0028] When a sample is not present in a beam path between the signal source and the detector, the first mask alignment creates approximately uniform illumination at each detector pixel and a mask pattern is invisible.

[0029] When a sample is present in the beam path, the beamlet strips are shifted due to refraction by the sample, resulting in intensity variations at corresponding detector pixels manifesting as bright and dark fringes that reveal the mask pattern, wherein the refraction angle or differential phase signal can be detected from the intensity variations.

[0030] The configuring of the mask comprises configuring the mask using a second mask alignment of the plurality of mask alignments, wherein the second mask alignment comprises positioning the mask such that the period of the projected mask pattern is twice the detector pixel size and the center of each beamlet strip is aligned with a center of every other pixel.

[0031] The method comprises retrieving attenuation and dark-field images from the single exposure of the sample.

[0032] When a sample is not present in the beam path, the second mask alignment produces an image showing a high-contrast mask pattern with a period of two pixels, characterized by an effective transparency and contrast.

[0033] When a sample is present in the beam path, photons of the signal scattered by the sample strike dark pixels of the detector and decrease the contrast of the mask pattern locally, wherein a dark-field signal is generated by comparing the intensity contrast between bright and dark pixels with and without the sample.

[0034] The configuring of the mask comprises configuring the mask using a third mask alignment of the plurality of mask alignments, wherein the third mask alignment comprises positioning the mask such that a period of a projected mask pattern is three times the detector pixel size and the center of each beamlet strip is aligned with one in every three-pixel boundaries.

[0035] The method comprises retrieving attenuation, differential phase, and dark-field images from the single exposure of the sample.

[0036] When a sample is not present in the beam path a mask-only image exhibits a pattern with a period of three pixels, where two pixels are bright pixels (equally illuminated) and one pixel remains dark, wherein differential phase signals are extracted from the bright pixels and a dark-field signal is extracted from the dark pixel.

[0037] An imaging system for multi-contrast imaging is described herein. The system comprises a signal source comprising an X-ray and a mask comprising a periodic absorption X-ray mask. The system also includes a low-resolution detector. The mask is positioned between the signal source and a sample and is configured using each of a plurality of mask alignments. The mask is configured to simultaneously capture attenuation, differential phase contrast (DPC), and dark-field images of the sample at a detector in a single exposure of the mask with the signal.

[0038] The mask is configured to simultaneously capture the images using the plurality of mask alignments to generate a plurality of dimensions of the signal to simultaneously capture at the detector multiple contrast features including the attenuation, the differential phase contrast (DPC), and the dark-field.

[0039] Each of the plurality of mask alignments are optimized for different types of contrast to provide more dimensions of the signal at the detector for material classification.

[0040] The mask is configured using a first mask alignment of the plurality of mask alignments, wherein the first mask alignment is configured to position the mask such that a period of a projected mask pattern is twice a detector pixel size and a center of each beamlet strip produced by the mask is aligned with a boundary of every other pixel, wherein each beamlet strip is generated by the mask.

[0041] The mask is configured using a second mask alignment of the plurality of mask alignments, wherein the second mask alignment is configured to position the mask such that the period of the projected mask pattern is twice the detector pixel size and the center of each beamlet strip is aligned with a center of every other pixel.

[0042] The mask is configured using a third mask alignment of the plurality of mask alignments, wherein the third mask alignment is configured to position the mask such that a period of a projected mask pattern is three times the detector pixel size and the center of each beamlet strip is aligned with one in every three-pixel boundaries.

[0043] The detector comprises a low-resolution detector comprising pixel sizes larger than approximately 20 micrometers. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Fig. la shows a diagram for single-mask X-ray differential phase contrast imaging system, under an embodiment.

[0045] Fig. lb shows zoomed in high-resolution (8 pm) x-ray image of the mask where period and opening width of the mask is 53 pm and 20 pm, respectively and the horizontal lines are supporting structures of the mask, under an embodiment.

[0046] Fig. 2a is a diagram for single-mask differential phase contrast (DPC) configuration, under an embodiment.

[0047] Fig. 2b is a zoomed mask image from an experiment under an embodiment in which the mask pattern is invisible since each pixel is equally illuminated by x-ray beamlets, under an embodiment.

[0048] Fig. 3a is an attenuation image, under an embodiment.

[0049] Fig. 3b is a DPC image of the multi-material phantom with single mask DPC configuration, under an embodiment.

[0050] Fig. 4a is a diagram for single-mask dark field (DF) configuration for which the blue strips represent x-ray beamlets created by the mask, under an embodiment.

[0051] Fig. 4b is a zoomed mask image under the embodiment.

[0052] Fig. 5a shows a retrieved attenuation image, under an embodiment.

[0053] Fig. 5b shows a retrieved dark field image of the multi-material phantom with single mask DF configuration, under an embodiment.

[0054] Fig. 6a is a diagram for single-mask dark field and differential phase contrast (DFDPC) configuration in which the blue strips represent x-ray beamlets created by the mask, under an embodiment.

[0055] Fig. 6b is a zoomed mask image from the experiment, in which the white color indicates pixels not fully blocked and receiving photons, under an embodiment.

[0056] Fig. 7a shows a retrieved attenuation image, under an embodiment.

[0057] Fig. 7b shows a retrieved DPC image, under an embodiment.

[0058] Fig. 7c shows a retrieved dark field image of the multi -material phantom with single mask DF-DPC configuration, under an embodiment.

[0059] Fig. 8a shows a retrieved attenuation image, under an embodiment. Fig. 8b shows a dark field image of six capsules filled with diamond powders of different grain sizes, under an embodiment.

[0060] Fig. 9 is a table showing the grit and grain sizes of the diamond powders in each capsule, under an embodiment.

[0061] Fig. 10a shows cross-sectional profiles of retrieved dark field images of capsules phantom with different focal spot size of the x-ray source, S (7 pm), M (20 pm), and L (50 pm), under an embodiment.

[0062] Fig. 10b shows cross-sectional profiles of retrieved dark field image of capsules phantom in different energy windows 15-25 keV, 25-35 keV, 35-45 keV, and 45-60 keV, under an embodiment.

[0063] Fig. Ila shows the attenuation image which provides an overall view of the sample’s structure, under an embodiment.

[0064] Fig. 11b shows the darkfield image which reveals some special features that are not visible in the attenuation image, under an embodiment.

[0065] Fig. 11c shows the combined image which merges information from the attenuation and darkfield signals, offering a comprehensive view of the sample, highlighting both structural and microstructural details, under an embodiment.

[0066] DETAILED DESCRIPTION

[0067] A novel single-mask X-ray imaging setup is described herein that simultaneously captures attenuation, differential phase contrast (DPC), and dark-field images in a single exposure using a low-resolution detector (e.g., pixel sizes larger than approximately 20 micrometers). Most importantly, configurations presented herein enable effectively capturing multiple contrast features like dark field and differential phase contrast using mask alignment with relatively low-resolution detectors and smaller magnification. It is shown that when using new light transport models described herein, contrast features can be effectively extracted from various single mask X-ray configurations. The approach of embodiments herein eliminates the need for highly coherent X-ray sources, ultra-high-resolution detectors, and intricate gratings, and as such simplifies the imaging process and significantly reduces costs and improves the prospects of clinical translation. Three variations of the single-mask setup are presented herein, each optimized for different contrast types, offering flexibility and efficiency in a variety of applications. The versatility of this single-mask approach holds promise for broader use in clinical diagnostics and industrial inspection, making advanced X-ray imaging more accessible and cost-effective in most / all areas using advanced imaging systems and methods.

[0068] The single mask method (See, e.g., References 28, 29) has emerged as a promising approach to x-ray phase contrast imaging. This technique requires only one optical element: a periodic absorption mask positioned between the x-ray source and the sample, typically near the sample itself. Fig. la shows a diagram for single-mask X-ray differential phase contrast imaging system, under an embodiment. The mask creates x-ray beamlets by periodically blocking x-rays with thin strips of heavy element materials, such as gold. Fig. lb shows zoomed in high-resolution (8 pm) x-ray image of the mask where period and opening width of the mask is 53 pm and 20 pm, respectively and the horizontal lines are supporting structures of the mask, under an embodiment. As x-rays pass through the sample, various interactions like attenuation, refraction, and scattering, modify the diffraction pattern formed by the mask on the detector plane. Directly resolving the subtle changes on the reference pattern with the detector requires ultra-high resolution.

[0069] To capture the pattern distortions with a lower resolution detector, the single-mask approach employs a different detection principle based on a specialized alignment between the mask and detector pixels, with each pixel effectively acting as an aperture that samples a distinct portion of the beamlet intensity distribution, which allow sub-pixel distortions caused by refraction and scattering to be encoded into measurable intensity variations.

[0070] Specifically, x-ray attenuation locally reduces the average brightness of the pattern, refraction causes slight local displacements or distortions, and scattering causes local blurring, thereby decreasing the contrast of the pattern stripes. The fringe displacements caused by refraction are typically extremely small, often at the micron or submicron level. However, standard clinical x-ray detectors generally lack the resolution needed to detect such minute displacements accurately. To overcome this limitation, the single-mask method described in detail herein uses a specialized alignment between the mask and the detector pixels to capture these tiny displacements that would be of sub pixel order with a detector resolution close to clinical usage. Building on the light-transport model See, e.g., Reference 27), the single-mask method is configured to simultaneously capturing attenuation and differential phase contrast (DPC) images in a single exposure. This offers high signal-to-noise ratio and dose efficiency in both radiography (See, e.g., Reference 30) and phase contrast micro-Computed Tomography (CT) (See, e.g., References 31, 32).

[0071] Embodiments described herein provide two additional variations of the single-mask setup with the capability to obtain dark-field contrast with or without differential-phase images. These three configurations operate on the same hardware platform and can capture attenuation, DPC, and dark-field signals independently and simultaneously from a single-shot acquisition, without the need for multiple exposures, spectral separation, or ultra-high-resolution detectors.

[0072] Embodiments described herein also provide a unified theoretical framework to describe signal formation across all three configurations, based on Fokker-Planck equation (See, e.g., Reference 33). While transport-of-intensity-based (TIE) models have been applied to X-ray phase-contrast imaging, many of them rely on simplifying assumptions, such as a constant attenuation to phase ratio throughout the object (See, e.g., References 9, 34, 35), or negligible attenuation (See, e.g., Reference 36). Others rely on multiple exposures (See, e.g., Reference 37) or spectral decomposition (See, e.g., References 11, 26) for phase retrieval. Also, some of these existing models exclude dark-field contributions entirely. Moreover, most prior TIE-based models assume that the detector fully resolves the wavefield, which usually needs high-resolution detectors or high magnification. In contrast, the model described in detail herein treats attenuation, phase, and dark-field signals as independent variables without those assumptions. Also, the model presented herein accounts for subpixel beamlet distortions, which enables signal extraction even with limited detector resolution.

[0073] Embodiments described herein include physics models that allow intuitive understanding of signal formation in these different configurations. In addition, the models also allow effective retrieval of attenuation, differential phase, and dark-field images from single exposure without the need for spectral information or complex motion of optical elements. This opens new possibilities for efficient material classification and quantitative imaging across diverse applications, offering practical advantages for both preclinical and potentially clinical translation.

[0074] Light Transport Model

[0075] The formulation for the single-mask DPC configuration is based on the approximated transport-of-intensity equation (TIE) (See, e.g., Reference 38), which described the intensity change after the electromagnetic (EM) wave propagates for a certain distance z

[0076]

[0077] where I(f, 0) is the X-ray intensity before propagation, f = (x,y) is the transverse coordinate, and k is the wave number. The second term describes changes in intensity due to variations of phase shift (p(r, 0) after propagating a distance z, which leads to a redistribution of optical energy, including local converge / di verge and transverse displacement of the beams.

[0078] The X-ray Fokker-Planck equation (See, e.g., Reference 33) is a modification of TIE that adds a term modeling ultra-small-angle scattering. If it is assumed the material properties very slowly within the transverse plane, the equation can be written as:

[0079]

[0080] Here S(r) is defined as the effective scattering coefficient, which denotes the structural property of the sample.

[0081] When applying to a single-mask imaging system, / (f, O) = T (r)M(x), where T and M is the transmission function of the object and the mask, respectively. Similar to previous work (See, e.g., Reference 27), the X-ray intensity measured by a certain detector pixel can be written as the integration within the pixel region:

[0082]

[0083] For simplicity, focus is on a single detector row and 7nis the intensity measured by the nthpixel of this row. Following the process shown in the Supporting Information described herein, the equation can be derived as:

[0084]

[0085] where T(x) and M(x) is the transmission function of the object and the mask, respectively. With this unified formulation in place, various configurations of single-mask phase imaging are described below that offer enhancement and recovery of different signal types.

[0086] Three Configurations of Single-Mask Method

[0087] Single-Mask DPC Configuration

[0088] In the single-mask method (referred to herein as “the DPC configuration”) (See, e.g., References 27, 28), the mask is positioned such that the period of the projected mask pattern is twice the detector pixel size p, and the center of each beamlet strip is aligned with every other pixel boundary. Fig. 2a is a diagram for single-mask differential phase contrast (DPC) configuration, under an embodiment (See also, e.g., Fig. la). The blue strips represent x-ray beamlets created by the mask. Fig. 2b is a zoomed mask image from an experiment under an embodiment in which the mask pattern is invisible since each pixel is equally illuminated by x-ray beamlets.

[0089] Without a sample in the beam path, this specific alignment creates almost uniform illumination of each detector pixel (See, e.g., Fig.2b). When a sample is introduced into the beam path, some beamlets are shifted from their original positions due to refraction by the sample, resulting in intensity variations at the corresponding pixels. These variations manifest as bright and dark fringes in the affected region. Consequently, the refraction angle or differential phase signal can be detected using a low-resolution detector. The detected intensity at each pixel can be derived as shown in the following equation, with the detailed derivation process available in the Supplemental Information described herein: In= weTnl - Ln) - a(—l)nTnDn

[0090] [4] This equation, derived from Fokker-Planck equation, is the same form previously described but derived without the scattering term. In this equation, Tn— T (x„), Dn — d < >(xn), and which represents the average attenuation, differential phase, and Laplacian

[0091]

[0092] phase across the corresponding pixel, respectively.

[0093] Also, the two coefficients in the equation, weand <z, are parameters related to the mask, corresponding to its effective transparency and equivalent contrast, respectively. The formula of the coefficients can be found in the Supplemental Information described herein.

[0094] From Eqn. [4], the following is observed: without the sample (Tn= l, Ln= 0, and Dn— 0), the detector produces a uniform image, and the mask pattern is invisible, with the intensity measured by each pixel determined by the effective transparency weof the mask. Once the sample is introduced into the beam path, the differential phase signal Dncontributes to the second term, resulting in bright and dark fringes on the image, thereby revealing the mask pattern.

[0095] As derived in the Supplemental Information described herein, the signal retrieval equations (See, e.g., Reference 27) are:

[0096]

[0097] where / n=

[0098]

[0099] is the flat-field corrected intensity, in which

[0100]

[0101] is the image with mask and sample, and image with mask only, respectively. The first equation yields the attenuation (with Laplacian enhancement), and the second extracts the DPC signal with system sensitivity proportional to a / we, which can be understood as the DPC sensitivity of the imaging system.

[0102] Figs.3a and 3b collectively show the retrieved attenuation and differential phase images retrieved from a single shot with the single-mask DPC configuration, under an embodiment. More specifically, Fig. 3a is an attenuation image, and Fig. 3b is a DPC image of the multi -material phantom with single mask DPC configuration, under an embodiment. The sample used was a multi-material phantom, which included a graphite rod with a diameter of 3 mm, several plastic beads with a diameter of 2 mm, and diamond powders with grain size ranging from 50 to 80 micrometers. All components were placed inside a centrifuge tube. The DPC image demonstrates superior contrast compared to the attenuation image, effectively highlighting the boundaries between various objects.

[0103] Single Mask Dark-Field Configuration

[0104] The second configuration, referred to as dark-field (DF) configuration, is designed or configured to capture dark-field images. The setup is nearly identical to the DPC configuration previously described herein, in which the mask is positioned so that the period of the projected mask pattern remains twice the pixel size p. The key difference being that the beamlets created by the mask are aligned with the center of every other pixel instead of pixel boundaries. Fig. 4a is a diagram for single-mask dark field (DF) configuration for which the blue strips represent x-ray beamlets created by the mask, under an embodiment. Fig. 4b is a zoomed mask image under the embodiment.

[0105] In the absence of a sample, this alignment produces an image with high-contrast bright and dark patterns (See, e.g., Fig. 4b). When a sample is placed in the beam path, some X-ray photons scattered by the sample will strike the dark pixels, decreasing the contrast of the mask pattern locally. Consequently, the dark-field signal can be extracted by comparing the intensity contrast between bright and dark pixels with and without the sample.

[0106] Following the process in the Supplemental Information described herein, the intensity received by each pixel can be derived as:

[0107] In= [we+ aiC-imCl - L - a3-i TnSn

[0108] [6] where Tnand Lnrepresent the same signals as in the previous section, and Sn= zS(xn) represents the average dark-field signal at the corresponding pixel.

[0109] Also, the coefficients in the equation we> a, and a3) are parameters related to the mask, and the formulas can be found in the Supplemental Information described herein. The coefficient werepresents the equivalent transparency of the mask. Similar to a in Eqn. [4], a1and a3are related to the contrast of the mask. From Eqn. [6], it is seen that without the sample (Tn= l, Ln= 0, and Sn= 0), the image shows a high-contrast mask pattern with a period of two pixels, characterized by the effective transparency weand contrast a1. Once the sample is introduced into the beam path, the dark-field signal Snreduces the local contrast of the mask pattern due to scattering.

[0110] As derived in the Supplemental Information described herein, the retrieval method can be written as:

[0111]

[0112] Similar to the previous section, the right-hand side of the equations is retrieved attenuation (with Laplacian phase enhancement) and dark field image, respectively. The dark field intensity ranges from zero to one, corresponding to no photon, and all photon is scattered to neighboring pixels, respectively. In this configuration, the dark field sensitivity of the imaging system is a3 / c1;related to the transmission function of the mask.

[0113] Figs. 5a and 5b collectively show the retrieved attenuation and dark-field images obtained from a single shot using the single-mask DF configuration. More specifically, Fig.

[0114] 5a shows a retrieved attenuation image, and Fig. 5b shows a retrieved dark field image of the multi-material phantom with single mask DF configuration, under an embodiment. The sample used here is the same as the one used in Fig. 3. From the figure, we can observe that, while the attenuation image (Fig. 5a) is capable of depicting various objects, it falls short in distinguishing between smooth, uniform materials, such as the wall of a centrifuge tube and plastic beads, and materials characterized by microstructures, like graphite and diamond powders. In contrast, the dark field image (Fig. 5b) excels in highlighting materials with microstructures, such as graphite and diamond powders, but show no signal for uniform materials.

[0115] However, for capturing differential phase signals, this configuration is ineffective. Since the beamlets are centered on the pixels, micron-level displacements of the beamlet pattern do not alter the total intensity received by each pixel. Thus, this configuration cannot detect differential phase signals.

[0116] Single Mask DF-DPC Configuration

[0117] The third configuration of the single-mask method described herein, termed the DF-DPC configuration, combines the features of both DPC and DF configurations. In this setup, the mask is positioned such that the period of the projected mask pattern is three times the detector pixel size p, with each beamlet aligned with one in every three-pixel boundaries, as illustrated in Figs. 6a and 6b. Fig. 6a is a diagram for single-mask dark field and differential phase contrast (DFDPC) configuration in which the blue strips represent x-ray beamlets created by the mask, under an embodiment. Fig. 6b is a zoomed mask image from the experiment, in which the white color indicates pixels not fully blocked and receiving photons, under an embodiment. Notably, the same mask used in the firtst two configurations can be used here; only its position needs adjustment to achieve the required magnification for this specific configuration.

[0118] As a result, the mask-only image exhibits a pattern with a period of three pixels, where two pixels are equally illuminated (bright pixels) and one pixel remains dark (shown in Fig. 6b). This arrangement allows for the extraction of differential phase signals from the two bright pixels, while the dark pixels provide the dark-field signal.

[0119] Following the similar process (Equation [3]) (also shown in the Supplemental Information described herein), the light transport model is derived for this configuration. The measured intensity by each pixel can be written as:

[0120]

[0121] where Tn, Dn, Ln, Snrepresents attenuation, differential phase, Laplacian phase, and dark field signal, respectively. Similar to the previous two configurations, the pixel index n corresponds to the position illustrated in Fig. 6a. Similar to the previous two configurations, the coefficients in the equation

[0122] we, alta2, and a3) are parameters related to the mask and can be found in the Supplemental Information described herein. The coefficient werepresents the equivalent transparency of the mask. a, a2, and a3are similar to a, altand a3in Eqn. [4] and [6],

[0123] The retrieval method can be derived following the process shown in the Supplemental Information described herein:

[0124]

[0125] The right-hand side of the equations represent retrieved attenuation, DPC, and dark field image, respectively. The DPC and dark field sensitivity of the imaging system is written as

[0126] respectively.

[0127]

[0128] Figs. 7a-7c collectively show the retrieved attenuation, differential phase, and darkfield images retrieved from a single shot with the single-mask DF-DPC configuration. More particularly, Fig. 7a shows a retrieved attenuation image, Fig. 7b shows a retrieved DPC image, and Fig. 7c shows a retrieved dark field image of the multi-material phantom with single mask DF-DPC configuration, under an embodiment. The sample used is the same as the one used in Figs. 3 and 5. It is seen that the DFDPC produces high-quality attenuation, differential phase, and dark-field images similar to the previous configurations. When comparing configurations, the DF-DPC configuration resulted in higher contrast in DPC images but exhibited lower contrast in dark field images, as opposed to DPC or DF configurations independently.

[0129] Experiment and Results

[0130] Materials and methods used in the embodiments described herein include an X-ray source comprising a polychromatic micro-focus x-ray tube (Hamamatsu L8121-03) operating with three different focal spot sizes (7 pm, 20 pm, and 50 pm). The tube voltage used ranges from 40 to 60 kV. Mask. A mask was used including with gold strips, approximately 53 pm in periodicity, fabricated on a silicon substrate. The width of the opening slit of the mask is around 20 pm. The thickness of the gold layer is 100 pm, which can block approximately 91% of X-ray photons with an energy of 40 keV.

[0131] Embodiments used a detector to collect data, and the detector includes a silicon WidePix photon-counting detector (See, e.g., Reference 39) with a pixel size of 55 pm, which was meticulously calibrated and corrected (See, e g., References 40, 41). The detector sensor measures approximately 70mm * 14mm, comprising five identical chips tiled together, with each chip containing 256*256 pixels. Typical exposures yielded 800-1200 photons per pixel. For all other results shown including phase retrieval, spectral information achievable from the detector was not used. Our efforts at phase retrieval using non-spectral data acquisition are aimed at easy testing and translation of methods to clinical and industrial settings which use non-spectral (energy -integrating) X-ray detectors which are readily available with larger active area or panel which is required for clinical applications.

[0132] The system geometry of embodiments included a source-to-object distance fixed at approximately 60 cm, while the object-to-detector distance varied depending on the configuration: approximately 60 cm for the DPC or DF configurations (resulting in approximately 2x magnification; this ensured the period of the projected mask pattern on the detector plane was twice the detector pitch); and around 120 cm for the DF-DPC configuration (resulting in approximately 3x magnification; this ensured the period of the projected mask pattern on the detector plane was three times the detector pitch). As a result, the period of the projected mask pattern on the detector plane was twice the detector pitch for the DPC or DF configurations, and three times the detector pitch for the DF-DPC configuration.

[0133] In all configurations, the sample was positioned between the mask and the detector, as close to the mask as possible, to maximize the propagation distance from the sample to the detector. The alignment between the mask and the detector is achieved through a multi-step process that can potentially be automated. Initially, the mask is placed in an approximate position such that, with the geometric magnification, the period of the projected mask pattern roughly matches the detector pitch. Next, the mask position is fine-tuned using remotely controlled motors. This fine-tuning is guided by observing the Moire fringes that appear due to slight misalignments. As the alignment improves, the frequency of the Moire pattern gradually decreases. The process continues until the Moire pattern disappears, indicating that the period of the projected mask pattern is well-matched with the detector pitch. Finally, the mask is moved horizontally until it reaches the precise position required for the chosen configuration.

[0134] Using the system configurations described in detail herein to investigate the sensitivity of the dark field signal to different feature sizes, a phantom was used comprising six capsules filled with diamond powders of varying grain sizes (See, e.g., References 42-47).

[0135] Figs. 8a and 8b collectively show the retrieved attenuation and dark-field images using the single-mask DF configuration, under an embodiment. More particularly, Fig. 8a shows a retrieved attenuation image, and Fig. 8b shows a dark field image of six capsules filled with diamond powders of different grain sizes, under an embodiment. Fig. 9 is a table showing the grit and grain sizes of the diamond powders in each capsule, under an embodiment. With reference to Fig. 8a and 8b, the images from left to right correspond to the sample numbers 1-6 in the Table shown in Fig. 9. The measurements were taken with three different focal spot sizes of the x-ray source (S: 7 pm, M: 20 pm, and L: 50 pm) and across different energy windows (15-25 keV, 25-35 keV, 35-45 keV, and 45-60 keV). Figs. 10a and 10b collectively show the cross-sectional plots, under an embodiment. In particular, Fig. 10a shows cross-sectional profiles of retrieved dark field images of capsules phantom with different focal spot size of the x-ray source, S (7 pm), M (20 pm), and L (50 pm), under an embodiment. Fig. 10b shows cross-sectional profiles of retrieved dark field image of capsules phantom in different energy windows 15-25 keV, 25-35 keV, 35-45 keV, and 45-60 keV, under an embodiment.

[0136] From the attenuation image in Fig. 8a, any differences between the six capsules can barely be seen since their infills have almost the same chemical composition. However, the dark field image (Fig. 8b) shows significant differences between each sample. It is observed that Samples 2, 3, and 4 exhibit the strongest dark-field signals, indicating that the X-ray darkfield imaging system has the highest sensitivity to features with sizes ranging from 0.25 to 3 pm. Fig. 9a shows that within the range up to 50 pm, a larger focal spot size does not impact the contrast of the dark field signal. In contrast, according to our previous study (See, e.g., Reference 30), the contrast of DPC images decreases as focal spot size increases.

[0137] Additionally, Fig. 9b shows the profile plot in several different energy windows. From these results we can see the imaging system works under a wide spectrum. We also notice the decrease in dark-field signal with higher x-ray energies, which is within our expectation since x-ray refraction is weaker with higher x-ray energy.

[0138] Fig. 10 showcases the retrieved images of a dried fish sample using a single shot with the single-mask DF configuration. The attenuation image (Fig. 10a) provides an overall view of the sample’s structure, whereas the dark-field image (Fig. 10b) reveals some special features that are not visible in the attenuation image. The combined image (Fig. 10c) merges information from both modalities, offering a comprehensive view of the sample, highlighting both structural and microstructural details. These images show the efficacy of dark-field imaging in capturing detailed structural information that is otherwise undetectable with conventional absorption imaging.

[0139] Figs, lla-llc collectively show the retrieved images of a dried fish sample using a single shot with the single-mask DF configuration, under an embodiment. Fig. Ila shows the attenuation image which provides an overall view of the sample’s structure, under an embodiment. Fig. lib shows the darkfield image which reveals some special features that are not visible in the attenuation image, under an embodiment. Fig. 11c shows the combined image which merges information from the attenuation and darkfield signals, offering a comprehensive view of the sample, highlighting both structural and microstructural details, under an embodiment. These images showed the efficacy of dark-field imaging in capturing detailed structural information that is otherwise undetectable with conventional absorption imaging.

[0140] The single-mask phase-contrast and dark-field imaging approach offers practical advantages in terms of system simplicity and flexibility, while also presenting unique challenges and trade-offs.

[0141] One of the core challenges in XPCI is to detect micron-scale refraction and USAXS signals. Grating-based and edge-illumination techniques use phase stepping, requiring multiple exposures with sub-pixel shifts for signal extraction. Speckle-based, beam-tracking, and mesh methods use high-resolution detectors or large magnification to directly resolve deformations in reference patterns. In contrast, the method described herein adopts a third strategy based on mask-to-pixel alignment. By precisely aligning the mask pattern with the detector pixels, the system detects sub-pixel distortions in beamlet intensity distributions without requiring high-resolution detectors or multiple exposures. This detection approach offers a practical balance between sensitivity, resolution, and system simplicity.

[0142] In general, the performance of phase-contrast and dark-field imaging systems depends critically on the spatial coherence of the X-ray beam, which in turn is influenced by the source focal spot size. Our experiments demonstrate robust performance at focal spot sizes up to 50 pm, the largest available on our system. This tolerance arises from the geometric configuration of our single-mask setup. However, we acknowledge that at much larger spot sizes (e.g., in millimeter level typical of clinical radiography), the beamlet profiles would broaden significantly, blurring the mask pattern on detector plane and ultimately the system’s ability to produce usable contrast. In such cases, introducing a source mask, similar to the GO grating in Talbot-Lau interferometry, could restore spatial coherence. Our results provide a practical reference: an aperture width of ~50 pm remains compatible with contrast preservation in this configuration.

[0143] The experimental results presented herein demonstrate that with our current configuration, the single-mask system is capable of detecting USAXS signals from a range of microstructures, from sub-micrometer level to 80 pm level, which is the largest diamond powder sample available in our lab. We show herein that using a simple setup, in a single shot, without high resolution detectors, we can extract dark-field signal from structures of similar size ranges shown by existing techniques (See, e.g., References 42-47).

[0144] Theoretically, a smaller mask or grating pitch would improve phase and dark-field sensitivity of both grating-based, beam-tracking, and our method.

[0145] When comparing between the three configurations, our results (Figs.3, 5, and 7) show that the DF-DPC setup produced slightly higher contrast in the DPC and DF images, due to its longer effective propagation distance. However, it should also be noted that the DF-DPC configuration may exhibit reduced sensitivity to weakly scattering materials, because the distance between the beamlets and the dark pixels is larger in this configuration. As a result, scattered beamlets must deviate by a greater angle to reach adjacent dark pixels, which slightly raises the detection threshold. This angular acceptance effect is a trade-off of this configuration and worth further investigation in future studies.

[0146] Another aspect worth noting is that, for all XPCI systems utilizing a reference pattern to extract phase and dark-field signals, the effective image resolution is limited not only by the detector pixel size but also by the spatial period of the reference pattern, whichever is larger. In our case, the mask period corresponds to two pixels for the DPC or DF configurations, and three pixels for the DF-DPC configuration, which results in a slight reduction in spatial resolution compared with the detector’s native sampling, a common characteristic of this type of technique.

[0147] An advantage of our imaging system is its compatibility with polychromatic X-ray spectra, unlike grating-based methods that rely on the energy-dependent Talbot effect. In such systems, fringe visibility degrades under broad-spectrum illumination with typical visibility around 20% even after optimization (See, e.g., Reference 48). In our setup, beamlet formation is governed primarily by geometric projection rather than interference, avoiding visibility loss under a broad energy range. As a result, it can achieve higher visibility (near 50%), with a gradual decline at higher energies due to the reduced attenuation of gold.

[0148] Although the forward model (based on Fokker-Planck equation) is energy-dependent, the polychromatic measurements can be considered as effective-energy-weighted integrations, which is sufficient for many imaging tasks.

[0149] Translating this technique to the scale of clinical imaging (like chest radiography) requires consideration of tolerance to geometric misalignments. While the approach described herein requires alignment of the mask with the detector at the pixel level, fortunately, the alignment tolerance in our system is on the order of the aperture width (~10 pm), which we believe is achievable with modern manufacturing and assembly techniques. Additionally, our method uses a relatively larger mask pitch, and the number of required mask elements is lower than grating-based systems, which may simplify large-area fabrication and alignment. Future work will explore tile-based mask designs and automated calibration procedures to enable deployment in large-area detector arrays.

[0150] The versatility of the novel single-mask method and configuration described herein has broader application / use in many areas including but not limited to clinical or medical diagnostics by making advanced X-ray imaging systems and methods more accessible and cost-effective. Clinical applications include but are not limited to imaging nanoparticle contrast agents in a biologic body or entity and imaging used during drug delivery and drug discovery applications. Further applications comprise using the single-mask method to reveal the microstructure of materials in two and three dimensions by analyzing how X-rays are absorbed, diffracted, or refracted by a sample and using the results to map internal structures, defects, and phase distributions with microscopic or even nanoscopic resolution. The single-mask method and configuration described herein has additional application in partial angle or complete computed tomography (CT) systems by acquiring intensities at multiple angles and retrieving differential phase and dark field images for each angle and then performing tomographic reconstruction of these entities as well as attenuation.

[0151] The single-mask method and configuration herein further includes industrial applications. For example, an application includes imaging in the food industry to inspect, analyze, monitor, and enhance food quality, safety, and production processes. Additional uses of the advanced imaging methods and configurations described herein include baggage scanning or screening systems to inspect the contents of luggage, packages, and parcels for security purposes to detect prohibited items while allowing safe passage of everyday belongings.

[0152] The imaging system and methods described herein also have application in nanostructure imaging involving the visualization and characterization of materials or biological structures at the nanoscale (e g., 1-100 nm), especially within thick or deep objects. This is especially useful for fields like materials science, nanotechnology, biology, and medicine, where conventional imaging methods fall short due to the diffraction limit (~200 nm).

[0153] The single-mask method and configuration described herein has application in dynamic imaging systems. While spanning multiple fields, including medical imaging, computer vision, photography, and scientific visualization, dynamic imaging systems include techniques and technologies that capture, process, or display visual data in motion or over time and / or under changing conditions (e.g., temperature, pressure, chemical, environmental conditions, etc.). As an example, dynamic imaging includes porosity imaging which refers to the use of advanced imaging techniques like those described in detail herein to visualize, quantify, and analyze the pore structures (void spaces) within materials such as rocks, metals, ceramics, biomaterials, and composites. Porosity, which is the fraction of void volume to total volume, affects material properties like permeability, strength, durability, and fluid flow. Unlike traditional methods, the single-mask imaging techniques described herein enable non-destructive, 3D insights into pore size, distribution, connectivity, and evolution over time or under stress or changing conditions. Porosity imaging is used in fields including but not limited to petroleum engineering (for reservoir rocks), materials science (for castings and foams), geology (for soil and biomass), and biomedical engineering (for bone analysis).

[0154] The single-mask X-ray imaging setup described herein represents a significant advancement in multi -contrast imaging, providing a practical and efficient method for simultaneously capturing attenuation, differential phase contrast (DPC), and dark-field images in a single exposure. By utilizing the specially designed mask-pixel alignment, this novel technique avoids the need for ultra-high-resolution detectors, phase stepping, or intricately fabricated gratings, which would significantly enhance the field of X-ray phasecontrast imaging with simple, low-dose and practical methods.

[0155] The three configurations of the technique described herein provide exceptional flexibility for different specific applications: whether emphasizing phase-contrast, dark-field signals, or a combination of both. Notably, transitioning between these variations requires no replacement of equipment; only a simple adjustment in the positioning of optical elements is needed. This adaptability enables the achievement of all three imaging modalities within a single system, enhancing its versatility and utility across diverse applications.

[0156] Preliminary investigations involving the system and methods described herein demonstrate a single-shot dark field retrieval from a range of structure sizes from 250 nm to 80 pm with a relatively low-resolution detector of 55 pm. System configuration can be optimized to get dark field images from much smaller or larger sizes based on the application. The light-transport model developed for these three variations provides a clear and intuitive framework for understanding signal formation and offers robust retrieval methods for each contrast type. The model also elucidates the influence of system parameters on the sensitivity of each imaging modality, guiding further optimization and application.

[0157] Overall, embodiments described herein lay the groundwork for more accessible and cost-effective X-ray imaging solutions with broad potential applications in clinical diagnostics and industrial inspection. The innovations presented here promise to enhance the capabilities of X-ray imaging systems, making advanced multi-contrast imaging a viable option for a wider range of settings and users.

[0158] References

[0159] Following are References referred to in the description herein.

[0160] 1. R. Fitzgerald, Phase-Sensitive X-Ray Imaging. Phys. Today 53, 23-26 (2000). 2. R. A. Lewis, Medical phase contrast x-ray imaging: Current status and future prospects. Phys. Medicine & Biol. 49, 3573 (2004).

[0161] 3. S. D. Auweter, et al., X-ray phase-contrast imaging of the breast — advances towards clinical implementation. The Br. J. Radiol. 87, 20130606 (2014).

[0162] 4. S. Tao, C He, X Hao, C Kuang, X Liu, Principles of Different X-ray Phase-Contrast Imaging: A Review. Appl. Sci. 11, 2971 (2021).

[0163] 5. L. Quenot, S. Bohic, E. Brun, X-ray Phase Contrast Imaging from Synchrotron to Conventional Sources: A Review of the Existing Techniques for Biological Applications. Appl. Sci. 12, 9539 (2022).

[0164] 6. S. K. Lynch, et al., Interpretation of dark-field contrast and particle-size selectivity in grating interferometers. Appl. Opt. 50, 4310-4319 (2011).

[0165] 7. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, Phasecontrast imaging using polychromatic hard X-rays. Nature 384, 335-338 (1996).

[0166] 8. Y. Suzuki, N. Yagi, K. Uesugi, X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object. J. Synchrotron Radiat. 9, 160-165 (2002).

[0167] 9. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. J. Microsc. 206, 33-40 (2002).

[0168] 10. A. Yan, X. Wu, H. Liu, An attenuation-partition based iterative phase retrieval algorithm for in-line phase-contrast imaging. Opt. Express 16, 13330 (2008).

[0169] 11. D. Gursoy, M. Das, Single-step absorption and phase retrieval with polychromatic x rays using a spectral detector. Opt. Lett. 38, 1461 (2013).

[0170] 12. T. Weitkamp, et al., X-ray phase imaging with a grating interferometer. Opt. Express 13, 6296-6304 (2005). 13. F. Pfeiffer, T. Weitkamp, O. Bunk, C. David, Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources. Nat. Phys. 2, 258-261 (2006).

[0171] 14. K. Taphorn, F. De Marco, J. Andrejewski, T. Sellerer, F. Pfeiffer, and J. Herzen, “Grating-based spectral X-ray dark-field imaging for correlation with structural size measures,” Sci. Reports 10, 13195 (2020).

[0172] 15. X. Zhu, C. Wu, H. Gao, X. Zhang, and L. Zhang, “Visibility of Polychromatic Grating-Based X-Ray Imaging Systems Based on Wave-Optical Theories and Frequency Domain Analysis,” Front. Phys. 9 (2021).

[0173] 16. G. Pelzer, A. Zang, G. Anton, F. Bayer, F. Hom, M. Kraus, J. Rieger, A. Ritter, J. Wandner, T. Weber, A. Fauler, M. Fiederle, W. S. Wong, M. Campbell, J. Meiser, P. Meyer, J. Mohr, and T. Michel, “Energy weighted x-ray dark-field imaging,” Opt. Express 22, 24507-24515 (2014).

[0174] 17. K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100, 124102 (2012).

[0175] 18. M. C. Zdora, “State of the Art of X-ray Speckle-Based Phase-Contrast and Dark-Field Imaging,” J. Imaging 4, 60 (2018).

[0176] 19. F. A. Vittoria, G. K. N. Kallon, D. Basta, P. C. Diemoz, I. K. Robinson, A. Olivo, and M. Endrizzi, “Beam tracking approach for single-shot retrieval of absorption, refraction, and dark-field signals with laboratory x-ray sources,” Appl. Phys. Lett. 106, 224102 (2015).

[0177] 20. F. A. Vittoria, M. Endrizzi, G. K. Kallon, C. K. Hagen, F. lacoviello, P. De Coppi, and A. Olivo, “Multimodal Phase-Based X-Ray Microtomography with Nonmicrofocal Laboratory Sources,” Phys. Rev. Appl. 8, 064009 (2017).

[0178] 21. A. Pil-Ali, S. Q. Z. Shi, S. Adnani, P. B. Noel, and K. S. Karim, “Multimodal single-shot x-ray phase-contrast imaging using a high-resolution micropillarbased grating,” in Medical Imaging 2023: Physics of Medical Imaging, vol. 12463 (SPIE, San Diego, CA, USA, 2023), pp. 37-46.

[0179] 22. J. C. Petruccelli, D. Hayden, S. Starr-Baier, S. Tahir, M. U. Rehman, L. Hassan, and C. A. MacDonald, “Phase and coherent scatter imaging for improved discrimination of low-density materials,” in Anomaly Detection and Imaging with X-Rays (ADIX), vol. 9847 (SPIE, Baltimore, MD, United States, 2016), pp. 47-55. 23. M. A. Beltran, D. M. Paganin, M. K. Croughan, and K. S. Morgan, “Fast implicit diffusive dark-field retrieval for single-exposure, single-mask x-ray imaging,” Optica 10, 422-429 (2023).

[0180] 24. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).

[0181] 25. A. Olivo, “Edge-illumination x-ray phase-contrast imaging,” J. Physics:

[0182] Condens. Matter 33, 363002 (2021).

[0183] 26. M. Das and Z. Liang, “Spectral x-ray phase contrast imaging for single-shot retrieval of absorption, phase, and differential -phase imagery,” Opt. Lett. 39, 6343 (2014).

[0184] 27. J. Yuan and M. Das, “Transport-of-intensity model for single-mask x-ray differential phase contrast imaging,” Optica 11, 478-484 (2024).

[0185] 28. F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector,” Rev. Sci. Instruments 81, 113702 (2010).

[0186] 29. D. Vavrik, J. Jakubek, I. Jandejsek, F. Krejci, I. Kumpova, and J. Zemlicka, “Visualization of delamination in composite materials utilizing advanced X-ray imaging techniques,” J. Instrumentation 10, C04012 (2015).

[0187] 30. J. Yuan, I. Harmon, and M. Das, “Comparing SNR benefits of single-mask edgeillumination and free-space propagation x-ray phase contrast imaging,” in Medical Imaging 2022: Physics of Medical Imaging, vol. PC12031 (SPIE, San Diego, CA, USA, 2022), p. PC120310X.

[0188] 31. J. Yuan, I. Harmon, J. C. R. Luna, and M. Das, “Single-mask phase imaging micro-computed tomography with multi -contrast images and single-shot per projection angle,” in Medical Imaging 2024: Physics of Medical Imaging, vol. 12925 (SPIE, San Diego, CA, USA, 2024), pp. 719-725.

[0189] 32. J. Yuan, J. C. R. Luna, and M. Das, “Single-shot dark-field and phase contrast micro-CT with single-mask phase contrast imaging,” in Medical Imaging 2025: Physics of Medical Imaging, vol. 13405 (SPIE, San Diego, CA, USA, 2025), p. 134051H.

[0190] 33. D. M. Paganin and K. S. Morgan, “X-ray Fokker-Planck equation for paraxial imaging,” Sci. Reports 9, 17537 (2019). 34. P. C. Diemoz, F. A. Vittoria, C K. Hagen, M. Endrizzi, P. Coan, A. Bravin, U. H. Wagner, C. Rau, I. K. Robinson, and A. Olivo, “A single-image retrieval method for edge illumination X-ray phase-contrast imaging: Application and noise analysis,” Phys. Medica 32, 1759-1764 (2016).

[0191] 35. K. M. Pavlov, H. T. Li, D M. Paganin, S. Berujon, H. Rouge-Labriet, and E. Brun, “Single-Shot X-Ray Speckle-Based Imaging of a Single-Material Object,” Phys. Rev. Appl. 13, 054023 (2020).

[0192] 36. D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, “Single-image geometricflow x-ray speckle tracking,” Phys. Rev. A 98, 053813 (2018).

[0193] 37. L. Quenot, H. Rouge-Labriet, S. Bohic, S. Berujon, and E. Brun, “Implicit tracking approach for X-ray phase-contrast imaging with a random mask and a conventional system,” Optica 8, 1412-1415 (2021).

[0194] 38. C. Zuo, J. Li, J. Sun, Y. Fan, J. Zhang, L. Lu, R. Zhang, B. Wang, L. Huang, and Q. Chen, “Transport of intensity equation: A tutorial,” Opt. Lasers Eng. 135, 106187 (2020).

[0195] 39. R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, S. Procz, L. Tlustos, and W. Wong, “The Medipix3RX: A high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Instrumentation 8, C02016 (2013).

[0196] 40. M. Das, B. Kandel, C. S. Park, and Z. Liang, “Energy calibration of photon counting detectors using x-ray tube potential as a reference for material decomposition applications,” in Medical Imaging 2015: Physics of Medical Imaging, vol. 9412 (SPIE, Orlando, Florida, United States, 2015), pp. 263-268.

[0197] 41. S. Vespucci, C. S. Park, R. Torrico, and M. Das, “Robust Energy Calibration Technique for Photon Counting Spectral Detectors,” IEEE Transactions on Med. Imaging 38, 968-978 (2019).

[0198] 42. M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110, 014105 (2017).

[0199] 43. A. Pandeshwar, M. Kagias, Z. Wang, and M. Stampanoni, “Modeling of beam hardening effects in a dual-phase X-ray grating interferometer for quantitative dark-field imaging,” Opt. Express 28, 19187-19204 (2020). 44. S. Lee, O. Oh, Y. Kim, D. Kim, J. Won, and S. W. Lee, “Study on dark-field imaging with a laboratory x-ray source: Random stress variation analysis based on x-ray grating interferometry,” Rev. Sci. Instruments 92, 015103 (2021).

[0200] 45. S. Spindler, D. Etter, M. Rawlik, M. Polikarpov, L. Romano, Z. Shi, K. Jefimovs, Z. Wang, and M Stampanoni, “The choice of an autocorrelation length in dark-field lung imaging,” Sci. Reports 13, 2731 (2023).

[0201] 46. S. Meyer, S. Z. Shi, N. Shapira, A. D. A. Maidment, and P. B. Noel, “Quantitative analysis of speckle-based X-ray dark-field imaging using numerical waveoptics simulations,” Sci. Reports 11, 16113 (2021).

[0202] 47. Y. Y. How, D. M. Paganin, and K. S. Morgan, “On the quantification of sample microstructure using single-exposure x-ray dark-field imaging via a single-grid setup,” Sci. Reports 13, 11001 (2023).

[0203] 48. M. Viermetz, N. Gustschin, C. Schmid, J. Haeusele, M. von Teuffenbach, P. Meyer, F. Bergner, T. Lasser, R. Proksa, T. Koehler, and F. Pfeiffer, “Dark-field computed tomography reaches the human scale,” Proc. Natl. Acad. Sci. 119, e2118799119 (2022).

[0204] Supplemental Information

[0205] 1. Derivation of the Physics Model for the Single-Mask Imaging System

[0206] A. General Form of Equation. The formulation for the single-mask imaging system is based on the approximated Fokker-Planck equation:

[0207]

[0208] When applying it to single-mask configuration, the transmitted intensity at the object plane is contributed by the mask and the sample, hence:

[0209] I(r,0) = T(r)M(x)

[0210] [2] where T(r,z) and M(x) represents the transmission function of the sample and the mask, respectively. Since the period of the mask has the similar scale level compared with the detector resolution, then we can assume T(r) varies much more slowly with position compared with M(x). Therefore,

[0211]

[0212]

[0213] For simplicity, here we focus on a single detector row, the X-ray intensity measured by a certain detector pixel can be written as the integration of the Fokker-Planck equation within the pixel region:

[0214]

[0215] where Inrepresents the intensity measured by the nthpixel of this row. The variable xnand xn+ 1 is the x coordinate of the left and right boundary of that pixel and can be written as xn= np, where p is the detector pixel size. Then we have:

[0216]

[0217] We can assume that the attenuation T(x), phase φ(x), and dark field S(x) distribution varies slowly within the range of one pixel, then the equation becomes:

[0218]

[0219] [6] This is the general form of the equation for the single-mask method. Inside the integrals, only the mask’s transmission function M(x) and its derivatives remain, which depend on the mask’s dimensions and how it is aligned with the detector pixels. The specific formulations for the three configurations can be derived by calculating these integrals.

[0220] B. Single-mask DPC Configuration. For this configuration, the period of the projected mask pattern is twice the detector pixel size p, and the transparent strips of the mask are aligned at pixel boundaries, as illustrated in Fig. 2a. Thus, the transmission function of the mask can be written as the following Fourier series with a period of 2p

[0221]

[0222] Then we can calculate each integration in Equation [6]:

[0223]

[0224] Then Equation [6] becomes:

[0225]

[0226] where:

[0227]

[0228] which represents the average attenuation, differential phase, and Laplacian phase across the corresponding pixel, respectively.

[0229] One can see from this equation that the signal is a combination of two distinct effects. The first term in Equation [9], which we refer to as the attenuation (or propagation-based (PB)) part, mainly contributed by attenuation of the object together with Laplacian phase edge enhancement, sharing the same form as the propagation-based imaging. The second term, referred to as the differential phase contrast (DPC) term, gives rise to the characteristic bright and dark fringes within the image, as demonstrated in the example depicted in Fig. 2(b). This is because it contains the factor (— 1)”, where n denotes the pixel column index. The magnitude of these fringes is directly proportional to the DPC signal Dn.

[0230] Also, the two parts of the signal are multiplied by two mask-related coefficients weand a, respectively. The coefficients’ values are given by:

[0231]

[0232] According to this equation, the coefficient werepresents the integration of the masktransmission function within a pixel, which corresponds to the average transmission of the mask. Thus, it can be interpreted as the effective transparent width or aperture size.

[0233] Coefficient a is a unit-less coefficient that depends on the odd Fourier coefficients of the mask’s transmission function. It can be understood as the attenuation contrast between the blocked and transmitted region of the mask.

[0234] The two coefficients can be interpreted as separate filters for the attenuation part and the DPC part independently. In comparison with the PB method, the intensity of the attenuation signal in the single-mask method is reduced by the coefficient we- This implies that the mask selectively reduces the x-ray intensity that contributes to the attenuation part, thereby allowing for a reduction in x-ray radiation dose to the sample without affecting the signal intensity of the DPC part. The second coefficient, a, which represents the contrast of the mask, determines the efficiency of obtaining the DPC signal.

[0235] C. Single-mask DF Configuration. For this configuration, the period of the mask pattern is still twice the detector pixel size p. The transmission function of the mask is similar to that of the DPC configuration but shifted for half of a pixel (See Fig. 4a).

[0236] Therefore, it can be written with the same form as Equation [7], with the only difference that it’s the sum of sines instead of cosines:

[0237]

[0238] Then we can calculate each integration in Equation [6]:

[0239]

[0240]

[0241]

[0014] where:

[0242]

[0243] Here Tnand Lnrepresent the same signals as in the DPC configuration, and Snrepresents the average dark-field signal at the corresponding pixel.

[0244] The coefficients in the equation are given by:

[0245]

[0246] Similar to the DPC configuration, these coefficients are parameters related to the mask. The coefficient 'W'eis still the equivalent transparency of the mask. Similar to coefficient a in Equation [9], coefficients

[0247]

[0248] and a3are related to the contrast of the mask, but with a multiplier in each term of the Fourier series.

[0249] From Equation

[0014] , we observe that, in the absence of a sample (Tn= 1, Ln= 0,and Sn= 0), the intensity is given by In= r£re+ a1(—l)n, which corresponds to an image displaying a high-contrast mask pattern with a period of two pixels. This pattern is characterized by the effective transparency w-eand contrast a. When a sample is

[0250] introduced into the beam path, the intensity captured by each pixel can be divided into two parts, corresponding to the two terms in the equation.

[0251] The first term represents the attenuation component, where the original intensity pattern created by the mask is reduced by the X-ray attenuation from the sample. The second term, related to the dark-field signal Sn, reduces the local contrast of the mask pattern due to ultra-small-angle scattering (USAXS).

[0252] We can also observe that the effective transparency only affects the first term. For instance, if the mask has lower transmission, it reduces the intensity of the first part (attenuation), without impacting the intensity of the dark-field part. This indicates that, similar to the DPC configuration, the mask can selectively block the X-ray photons that contribute to the attenuation term without affecting the dark-field signal.

[0253] D. Single-mask DF-DPC Configuration. For this configuration, the period of the projected mask pattern is three times the detector pixel size p. Then the transmission function of the mask can be written as a Fourier series with a period of 3p

[0254]

[0255] To make the equation easier to understand, we assume the nthpixel is the bright pixel on the right of each beam-let, as illustrated in Fig. 6a. We can calculate the integrals in Equation [6] separately for the three pixels in one period.

[0256] the integrals are given by:

[0257]

[0258] For In, the integrals are given by:

[0259]

[0260] For In+1, the integrals are given by:

[0261]

[0262] RO] According to the integration results, the intensity measured by the corresponding pixels can be written as:

[0263]

[0264]

[0021] where:

[0265]

[0266] which represents attenuation, differential phase, Laplacian phase, and dark field signal, respectively. The coefficients in the equation are given by:

[0267]

[0268] Similar to the other two configurations, the coefficients in the equation (we, a±, a2, and a3) are parameters related to the mask. The coefficient

[0269]

[0270] is still the equivalent transparency of the mask. Coefficients a. <z2, and a3are related to the contrast of the mask, similar to a, a1and a3in Equation [9] and Equation

[0014] ,

[0271] From Equation

[0021] we observe that, without the sample (Tn= 1, Ln= 0, Dn= 0 and Sn= 0), this special mask alignment creates a pattern with a period of 3p. Within each period, there are two bright pixels ( / n_i and / n) with equal intensity of (e+

[0272]

[0273] and one dark pixel ( / „ + 1) with an intensity of (e— a^. After the sample is introduced into the beam path, the signal is composed of three components.

[0274] The first component is the attenuation term, where the original intensity pattern created by the mask is reduced due to X-ray attenuation by the sample. This term also includes a small contribution from the Laplacian phase signal, which manifests as edge enhancement. The second component, referred to as the DPC term, contributes to the intensity differences between the two bright pixels. This difference is proportional to the differential phase contrast (DPC) signal. The third component, referred to as the dark-field term, reduces the contrast between the bright and dark pixels due to ultra-small-angle scattering (USAXS).

[0275] Similarly, as in the other two configurations, the effective transmission of the mask only influences the attenuation term, without affecting the DPC or dark-field signals. This feature is particularly important because it allows for a reduction in the X-ray dose to the sample without compromising the differential phase or dark-field signals. . Derivation of Differential Phase and Dark-field Image Retrieval.

[0276] In an experimental realization, a single image is taken with the object and the mask in the beam path. This image (represented as / „ ) can be compared with the image with only the mask (also called flat-field, noted as

[0277]

[0278] We define Tnto be the flat-field corrected (or normalized image):

[0279]

[0280] A. Single-mask DPC Configuration. The sample image and the mask image of the DPC configuration can be written as the following according to Equation [9]:

[0281]

[0282] Then the flat-field corrected image is given by:

[0283] a

[0284] In= Tn(l — Ln) - (—l)nTnDn

[0285] We

[0286]

[0026] Thus, we can write the corrected intensity for nthand (n + 1)thpixels in the same row:

[0287]

[0288] If we assume Tn, Lnand Dnvaries slowly within the two-pixel range, we can separate the attenuation and DPC signals by adding and subtracting the intensity values on nthand (n + 1)thpixels in each row:

[0289]

[0290] From the first equation, we can easily have the retrieval of the attenuation (or propagationbased) part. In order to retrieve Dnwe can consider the intensity of the Laplacian of phase to be relatively weak compared with 1, since the Laplacian phase is only noticeable along sharp edges. Then we can apply the approximation of 1 — Ln« 1 when solving differential phase Dn. Then we can arrive at the retrieval formula for attenuation and differential phase:

[0291]

[0292]

[0029] The right-hand side of the equations can be regarded as retrieved attenuation (enhanced by Laplacian phase) and DPC image, respectively. Also, we note the intensity of the DPC image equals the differential phase term Dnmultiplied by a / we, which can be regarded as the DPC sensitivity of the imaging system.

[0293] B. Single-mask DF Configuration. According to Equation

[0014] , the intensity captured by two neighboring pixels can be written as the following equations. With sample and mask together:

[0294]

[0295] With only the mask:

[0296] =Wp + ai(-l)n

[0297] d =we - ai(-l)n

[0298]

[0031] If we assume Tn, Lnand Snvary slowly within the two pixels, the retrieval method can be derived by the following steps. We also applied the same assumption that Tn(l — Ln) ~ Tn, which has been explained before.

[0299]

[0300]

[0301] Therefore, the retrieval method can be written as:

[0302]

[0303]

[0036] Similar to the previous section, the right-hand side of the equations is retrieved attenuation and dark field image, respectively. The dark field intensity ranges from zero to one, indication the portion of photons scattered to neighboring pixels. In this configuration, the dark field sensitivity of the imaging system is a3 / a1, related to the masks’ transmission function.

[0304] C. Single-mask DF-DPC Configuration. According to Equation

[0021] , the intensity measured by the three pixels within one period of the mask pattern can be written as the equation following below. Again, we note that the nthpixel corresponds to the pixel shown in Fig. 6a. With sample and mask together:

[0305]

[0306] With mask only:

[0307]

[0308] If we assume Tn, Dn, Lnand Snvaries slowly within the three pixels, the retrieval method can be derived by the following steps. We also applied the same assumption that Tn(1 — £n) ~ Tn, which has been explained before.

[0309] The attenuation image is given by the average intensity of the three pixels in one period:

[0310]

[0311] The DPC image can be retrieved based on the intensity differences between the two bright pixels:

[0312]

[0313] The dark-field image is based on comparing the contrast between bright and dark pixels with and without the sample:

[0314]

[0315] Therefore, the retrieval method can be written as:

[0316]

[0317] The right-hand side of the equations is retrieved attenuation, DPC, and dark field image, respectively. The DPC and dark field sensitivity of the imaging system is written as

[0318] respectively.

[0319]

Claims

CLAIMSWhat is claimed is:

1. A method for multi-contrast imaging, the method comprising:positioning a mask between a signal source and a sample, wherein the mask comprises a periodic absorption X-ray mask and the signal source comprises an X-ray;configuring the mask using each of a plurality of mask alignments; simultaneously capturing attenuation, differential phase contrast (DPC), and darkfield images of the sample at the detector in a single exposure of the mask with the signal.

2. The method of claim 1, wherein the simultaneously capturing comprises using the plurality of mask alignments to generate a plurality of dimensions of the signal to simultaneously capture at the detector multiple contrast features including the attenuation, the differential phase contrast (DPC), and the dark-field during the single exposure, wherein the detector comprises a low-resolution detector including pixel sizes larger than approximately 20 micrometers.

3. The method of claim 1, comprising optimizing each of the plurality of mask alignments for different types of contrast to provide more dimensions of the signal at the detector for material classification.

4. The method of claim 1, wherein the configuring of the mask comprises configuring the mask using a first mask alignment of the plurality of mask alignments, wherein the first mask alignment comprises positioning the mask such that a period of a projected mask pattern is twice a detector pixel size and a center of each beamlet strip produced by the mask is aligned with a boundary of every other pixel, wherein each beamlet strip is generated by the mask.

5. The method of claim 4, comprising retrieving attenuation and differential phase images from the single exposure of the sample.

6. The method of claim 4, wherein,when a sample is not present in a beam path between the signal source and the detector, the first mask alignment creates approximately uniform illumination at each detector pixel and a mask pattern is invisible;when a sample is present in the beam path, the beamlet strips are shifted due to refraction by the sample, resulting in intensity variations at corresponding detector pixels manifesting as bright and dark fringes that reveal the mask pattern, wherein the refraction angle or differential phase signal can be detected from the intensity variations.

7. The method of claim 1, wherein the configuring of the mask comprises configuring the mask using a second mask alignment of the plurality of mask alignments, wherein the second mask alignment comprises positioning the mask such that the period of the projected mask pattern is twice the detector pixel size and the center of each beamlet strip is aligned with a center of every other pixel.

8. The method of claim 7, comprising retrieving attenuation and dark-field images from the single exposure of the sample.

9. The method of claim 7, wherein,when a sample is not present in the beam path, the second mask alignment produces an image showing a high-contrast mask pattern with a period of two pixels, characterized by an effective transparency and contrast;when a sample is present in the beam path, photons of the signal scattered by the sample strike dark pixels of the detector and decrease the contrast of the mask pattern locally, wherein a dark-field signal is generated by comparing the intensity contrast between bright and dark pixels with and without the sample.

10. The method of claim 1, wherein the configuring of the mask comprises configuring the mask using a third mask alignment of the plurality of mask alignments, wherein the third mask alignment comprises positioning the mask such that a period of a projected maskpattern is three times the detector pixel size and the center of each beamlet strip is aligned with one in every three pixel boundaries.

11. The method of claim 10, comprising retrieving attenuation, differential phase, and dark-field images from the single exposure of the sample.

12. The method of claim 10, wherein, when a sample is not present in the beam path a mask-only image exhibits a pattern with a period of three pixels, where two pixels are bright pixels (equally illuminated) and one pixel remains dark, wherein differential phase signals are extracted from the bright pixels and a dark-field signal is extracted from the dark pixel.

13. An imaging system for multi -contrast imaging, the system comprising:a signal source comprising an X-ray;a mask comprising a periodic absorption X-ray mask; anda low-resolution detector;wherein the mask is positioned between the signal source and a sample; wherein the mask is configured using each of a plurality of mask alignments; and wherein the mask is configured to simultaneously capture attenuation, differential phase contrast (DPC), and dark-field images of the sample at the detector in a single exposure of the mask with the signal.

14. The system of claim 13, wherein the mask is configured to simultaneously capture the images using the plurality of mask alignments to generate a plurality of dimensions of the signal to simultaneously capture at the detector multiple contrast features including the attenuation, the differential phase contrast (DPC), and the dark-field during the single exposure.

15. The system of claim 14, wherein each of the plurality of mask alignments are optimized for different types of contrast to provide more dimensions of the signal at the detector for material classification.

16. The system of claim 13, wherein the mask is configured using a first mask alignment of the plurality of mask alignments, wherein the first mask alignment is configured to position the mask such that a period of a projected mask pattern is twice a detector pixel size and a center of each beamlet strip produced by the mask is aligned with a boundary of every other pixel, wherein each beamlet strip is generated by the mask.

17. The system of claim 13, wherein the mask is configured using a second mask alignment of the plurality of mask alignments, wherein the second mask alignment is configured to position the mask such that the period of the projected mask pattern is twice the detector pixel size and the center of each beamlet strip is aligned with a center of every other pixel.

18. The system of claim 13, wherein the mask is configured using a third mask alignment of the plurality of mask alignments, wherein the third mask alignment is configured to position the mask such that a period of a projected mask pattern is three times the detector pixel size and the center of each beamlet strip is aligned with one in every three-pixel boundaries.

19. The system of claim 13, wherein the low-resolution detector comprises a detector comprising pixel sizes larger than approximately 20 micrometers.

Images