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Abstract
Advances in x-ray imaging have enabled breakthroughs in medicine, biology, materials science, and cultural heritage studies. Three-dimensional visualizations of micrometer-scale structures inside soft tissues and dense specimens have been made successful by state-of-the-art methods based on hard x-ray phase-contrast imaging. Nevertheless, it remains challenging to achieve high spatial resolutions without increasing radiation dose and risking damage to specimens. A promising approach to reduce dose and still achieve high spatial resolution is x-ray ghost imaging, which uses single-pixel, yet efficient direct x-ray detectors made of high-density materials. However, all currently existing x-ray ghost imaging protocols are unable to achieve phase contrast and suffer from low-quality image reconstructions. Here, we present an efficient approach which exploits structured-detection single-pixel imaging in order to produce an x-ray ghost image with phase contrast, accuracy, and high fidelity. The resulting x-ray phase-contrast ghost images provide accurate information regarding density variations in the sample and visibly render edges that are otherwise invisible with x-ray attenuation contrast. This demonstration of phase-contrast ghost imaging with x-rays has the potential to promote x-ray ghost imaging from a niche technique to a routinely applied method.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Hard x-ray phase-contrast imaging [1–4] renders visible both weakly absorbing objects (e.g., soft tissues [5]) and structures inside dense objects (e.g., fossils [6]) that are otherwise invisible in conventional x-ray attenuation-contrast imaging. Using the propagation-based approach [3,7,8], phase retrieval combined with computed tomography yields orders of magnitude gain in image contrast-to-noise ratio that may be traded for reductions in x-ray radiation dose to the specimen [9–15]. Indirect x-ray detection using thin scintillator screens has been the key technology to achieve up to sub-micrometer spatial resolutions [16], albeit inefficiently and hence at the expense of increased radiation dose to the specimen. Indirect x-ray detection efficiency reduces with increasing x-ray energy, and hence resolutions of around ten micrometers at best have been achieved with indirect x-ray detectors employed at photon energies above 100 keV [6].
Recently, ghost imaging [17] was demonstrated with hard x-rays [18,19]. Ghost imaging gained attention in the x-ray domain [18–27] because the primary detector, which collects the photons that interact with the object, is a large-area, single-pixel detector called a "bucket" detector. Since single-pixel, high-Z (atomic number), direct x-ray detectors are significantly more efficient than indirect x-ray detectors [16] commonly used in x-ray phase-contrast imaging, the potential reduction of the radiation dose to the specimens is highly appealing [19,23,25,27]. For example, direct x-ray detectors made with the high-Z material CdTe have a quantum efficiency close to 100% even above 100 keV. Moreover, since the spatial resolution of a ghost image is determined by the size of the speckles in the illuminating field [21,28] and not by the detector, the ghost-imaging principle may ultimately achieve the much-coveted high spatial resolution with high x-ray energies.
The term “ghost imaging” highlights the property that in this modality an image can be obtained even though neither of the detectors used in the measurement yields an image of the object. In classical realizations of ghost imaging (the other realizations being quantum in nature), a ghost image is retrieved from intensity correlations between (1) a series of speckled fields that illuminate an object and (2) the total integrated intensities transmitted by the object. A ghost-imaging setup is composed of a beam path, called a reference arm, in which the speckled fields are resolved by a 2D detector, and a beam path called an object arm in which the speckled fields illuminate an object. The object’s integrated transmitted intensities are detected by a bucket detector.
In computational ghost imaging [29–31], the so-called reference images are pre-determined or pre-characterized spatial light-intensity modulations. Computational ghost imaging is closely related to single-pixel-camera imaging [32–34], insofar as the latter technique also uses single-element detectors and spatial light modulators to produce an image. In essence, both classical ghost imaging and single-pixel-camera imaging express the spatial distribution of an object’s transmission function as a linear combination of basis patterns [35,36]. Each such pattern is considered as a linearly independent function-space basis vector from which a computational image may be synthesized. These basis patterns can be speckles or random patterns [30–32], or deterministic patterns such as Hadamard [33] and Fourier [34] basis patterns.
Fig. 1. Computational x-ray ghost imaging and single-pixel camera configurations. (a) A computational x-ray ghost imaging setup (structured illumination approach). An x-ray beam illuminates a mask that is scanned in the transverse plane to generate a series of speckle fields that illuminate a sample. For each illuminating speckle field, the sample’s total transmitted intensity is collected by a bucket detector. Intensity correlations (${\textbf{G}}$) between the measured bucket signals (${b_1},{b_2}, \ldots ,{b_N}$) and the known illuminating speckle fields (${I_1},{I_2}, \ldots ,{I_N}$) are utilized to synthesize an attenuation-contrast x-ray ghost image. (b) Our experimental x-ray phase-contrast ghost imaging with a single-pixel camera setup (structured detection approach) using 1D gratings as masks in combination with a 1D bucket detector, which is a collection of “mailbox” detectors. The ghost image reconstruction formula is applied for each mailbox detector, $l = 1\;{\rm mm}$ and $h = 6.5\;\unicode{x00B5}{\rm m}$. By interchanging the mask and the sample in the sequence, the bucket signal is sensitive to the effect of Fresnel diffraction, and a propagation-induced x-ray phase-contrast ghost image can be synthesized.
Classical x-ray ghost imaging has been realized with speckled illumination. This has been done by using intrinsic photon noise of a synchrotron x-ray source [19] and speckle patterns generated by phase-contrast or attenuation-contrast masks [18,20,21,23]. Recently, a realization of computational x-ray ghost imaging using a set of Hadamard masks has also been reported [26]. The attenuation-contrast component of an object’s x-ray transmission function has been retrieved with classical x-ray ghost imaging [19–23,26], but to date no x-ray ghost imaging experiment has achieved sensitivity to x-ray phase contrast.
In this paper, we show that the key to achieving phase contrast in x-ray ghost imaging is to reverse the sequence of sample and mask: this is a structured detection approach akin to single-pixel cameras [32,33] instead of a structured illumination approach used in computational ghost imaging [29]. We emphasize that while phase-contrast ghost imaging has been achieved with visible light [33,37,38], our work reports what we believe to be the first demonstration of phase-contrast ghost imaging with x-rays. The main advantage of phase-contrast ghost imaging with x-rays compared to traditional x-ray ghost imaging is given by superior image contrast. By enabling phase contrast in x-ray ghost imaging, a significant contrast-to-noise boost becomes possible: weakly absorbing objects and structures inside dense objects, that are otherwise invisible without phase contrast, become visible.
2. PRINCIPLE
Figure 1(a) shows a computational x-ray ghost imaging setup employing speckle-generating masks. Let ${I_{{\rm in}}}$ be a uniform, incident illumination intensity and ${\hat T_{M,j}}({{\textbf{r}}_ \bot},z = {R_{M}})$ be the transmission function of the $ j $th speckle mask (M) at an x-ray beam propagation distance ${R_{M}}$ from the mask to the sample. Here, ${{\textbf{r}}_ \bot} = (x,y)$ denotes spatial coordinates in planes perpendicular to the optical axis $z$. The $j$th illumination intensity onto the sample is ${I_j}({{\textbf{r}}_ \bot}) = {\hat T_{M,j}}({{\textbf{r}}_ \bot},z = {R_{M}}){I_{{\rm in}}}$. Let ${\hat T_{S}}({{\textbf{r}}_ \bot},z = {R_{S}})$ be the transmission function of the sample (S) at an x-ray beam propagation distance ${R_{S}}$ from the sample to the bucket detector; then the signal collected by the bucket detector may be written as
Using an ensemble of $N$ speckle illuminations, an x-ray ghost image may be synthesized via the cross correlation [30,31]
The inability of existing x-ray ghost-imaging protocols to retrieve a phase-contrast image is due to the fact that by conservation of energy, the average intensity of a phase-contrast image or Fresnel image [10,42] over an area that contains the sample, at any x-ray beam propagation distance $R$ from the object, is equal to that of the contact image ($R = 0$). The same conclusion may be drawn from the unitarity of the Fresnel diffraction operator ${\cal U}$ for paraxial coherent scalar fields $\Psi$, since, in Dirac notation, the unitarity of ${\cal U}$ implies that $\langle {\cal U}\Psi |{\cal U}\Psi \rangle = \langle {{\cal U}^\dagger}{\cal U}\Psi |\Psi \rangle = \langle \Psi |\Psi \rangle$. This means that the bucket signals are not sensitive to the propagation-induced phase-contrast component of ${\hat T_{S}}$ when the masks are upstream of the sample. Any transverse rearrangement of intensity downstream of the sample that is induced by propagation does not alter the bucket signals. With these bucket signals, only the attenuation-contrast component can be recovered in a ghost image. This conclusion remains unchanged if the relative position of the sample, which for now is taken to remain in the space anywhere between the mask and the bucket detector, is varied in Fig. 1(a). The bucket detector is insensitive to the phase shifts of the sample and, provided that no phase-sensitive filtration of the beam downstream of the sample is performed, it will remain insensitive to transparent structures in the sample. This leads to the question of how a phase-sensitive filtration can be performed in the space between the sample and the detector.
The key to achieving phase contrast in x-ray ghost imaging is to reverse the sequence of the sample and the set of masks in the beam, and place the masks at a desired Fresnel image plane after the sample in the beam. This crucial order of the sample and masks can be emphasized in the order of operation of the transmission function in the bucket signal equation, which we now write as
3. EXPERIMENTAL SETUP
Our experimental setup for x-ray phase-contrast ghost imaging using a structured detection approach is depicted in Fig. 1(b). The setup consists of a partially spatially coherent illuminating x-ray beam, an object, a set of 1D masks that are placed at a desired Fresnel image plane, and a 1D bucket detector. The combination of a set of masks and a bucket detector placed after the sample comprises a single-pixel camera. Similar to standard propagation-based phase-contrast imaging [8], x-ray phase-contrast ghost imaging only requires sufficient spatial coherence via a small source size but essentially no temporal coherence.
Instead of using speckle patterns, which form a non-orthogonal basis and require $N \gg p$ measurements in order to synthesize a ghost image consisting of $p$ pixels, we employed a linear combination of periodic fields with varying frequencies, which form a nearly orthogonal set of linearly independent basis patterns. We implemented this using transmission gratings, which is practical because the fabrication technologies for such gratings for hard x-rays are well established. For example, a high aspect ratio grating made of gold with 2.4 µm line width and 160 µm thickness used for up to 180 keV x-ray photon energy has been reported [43]. Other concepts that have been employed to reduce $N$ include compressive sensing [30], orthogonalization of speckle fields [21] and iterative refinement [22], as well as the use of orthogonal deterministic basis patterns such as Hadamard [33] and Fourier basis patterns [34].
We opted not to use two-dimensional (2D) gratings. Instead, we considered that a combination of 1D gratings with a 1D bucket detector is equivalent to using 2D gratings with a single-pixel (0D) detector. A 1D bucket detector constitutes a set of what we call “mailbox detectors”. Recently, high-Z, 2D direct detectors such as the EIGER2 X CdTe (Dectris Ltd., Switzerland) with pixel size of 75 µm have become commercially available. Line arrays or even pixels of such 2D detectors may be used as mailbox detectors. The ghost image reconstruction formula was applied for each mailbox detector. The advantages were two-fold: (1) $N$ was reduced by a factor equal to the number of mailbox detectors used; and (2) the spatial resolution could be tuned independently in two directions, one with the smallest grating line width ${w_N}$ and the other with the mailbox detector height $h$.
The experiment was carried out at beamline ID19 of The European Synchrotron—ESRF (Grenoble, France). A U-17.6 type undulator was used, with the gap tuned to generate 19 keV pink x-ray radiation. The vertical and horizontal x-ray source sizes (root mean square) were 3.4 and 50 µm, respectively. The sample was located 140 m from the source, while the mask and the detector were located 13 m from the sample. The sample was a metal foam (Mayser GmbH & Co. KG, Germany) made of 99.7% aluminum and with average pore size of 2.5 mm. An off-the-shelf x-ray test pattern (Type 23, Hüttner, Germany) composed of 0.5 mm thick lead patterns on 1 mm plexiglass was used for the gratings. The x-ray transmissions through the plexiglass and the lead were 96% and 2%, respectively. The 10 grating patterns used have 1 to 10 lines per mm. The set of 10 reference images [Fig. 2(a)] was obtained by scanning the test pattern in steps of 65 µm parallel with the middle grating line. This scanning was repeated for the measurements with the sample. An indirect x-ray image detector composed of an sCMOS camera (pco.edge; pixel size: 6.5 µm, PCO AG, Germany) coupled with a 100 µm thick LuAG:Ce scintillator using a tandem of lenses (Hasselblad, Sweden) with 100 mm focal lengths (${1} \times$ magnification) was used. The mailbox detectors used were line arrays of 1000 µm length and 6.5 µm height (of the same indirect x-ray detector). While an indirect detector was used to collect the bucket signal for the purposes of the present proof-of-concept experiment, a direct detector could be employed in future developments of our protocol. For each line array, a simple MATLAB implementation of Eq. (2) was used to synthesize the ghost images with (${G_{S}}$) and without (${G_{{\rm flat}\,{\rm field}}}$) the sample. A total of 1700 line arrays were used in the reconstructed ghost image. The 11 mm horizontal field of view was achieved by stitching 21 images, with 500 µm width each, that were cropped from 1 mm width ghost images. Both the scanning and stitching would not have been necessary if we had a large rectangular grating.
Fig. 2. Reference x-ray images and calculated point-spread-function. (a) Experimentally obtained reference x-ray images of $N = 10$ grating patterns with 1 to 10 lines per mm. The smallest grating width ${w_{10}} = 100\;\unicode{x00B5}{\rm m}$. (b) Calculated point-spread-function ${\rm PSF}(x,x^\prime)$ of the x-ray ghost image without the sample for one of the mailbox detectors ($N = 10$, $l = 1\;{\rm mm}$, $h = 6.5\;\unicode{x00B5}{\rm m}$). Note that $x$ and $x^\prime $ run over the mailbox detector length $l$.
Fig. 3. X-ray transmission images of interconnected aluminum lamellae cut from a sponge sample. (a) Phase-contrast x-ray image (PC-XI): calculated ${I_{S}}/{I_{{\rm flat}\,{\rm field}}}$ from radiographs directly recorded using a 2D imaging detector. (b) Phase-contrast x-ray ghost image (PC-XGI): calculated ${G_{S}}/{G_{{\rm flat}\,{\rm field}}}$ from synthesized ghost images both with (${G_{S}}$) and without (${G_{{\rm flat}\,{\rm field}}}$) the sample. Note that the masks’ stripes are vertical and the detector’s stripes are horizontal, as illustrated in the inset showing the setup diagram. Scale bars in (a) and (b), 1 mm. (c) Horizontal line profiles $T(x)$. (d) Vertical line profiles $T(y)$. (e), (f) Magnified views of the insets in (a) and (b) showing phase-contrast enhancement at the edges of representative thick and thin regions of the lamellae. Scale bars in (e) and (f), 250 µm.
4. RESULTS
In order to check whether a set of gratings constitutes a complete set of basis elements, we calculated the position-dependent x-ray ghost imaging point-spread function [21,28]:
By way of example, we show the calculated point-spread function ${\rm PSF}(x,x^\prime)$ [Fig. 2(b); Eq. (4)] for mailbox detector length $l = 1\;{\rm mm}$ and $N = 10$ is a near-diagonal matrix, proving that the set of gratings [Fig. 2(a)] is nearly orthogonal up to a resolution given by the width of the diagonal. The measured average full-width-at-half-maximum of ${\rm PSF}(x,x^\prime)$, which represents the spatial resolution of the system, was approximately 100 µm. As expected, this was equal to the smallest grating width ${w_{N = 10}}$. With $N = p$, where $p = l/{w_{10}}$, the resulting periodic illuminating fields indeed constitute a nearly orthogonal set. Note that neither orthogonalization nor compressive sensing methods were applied prior to calculation of the PSF.
Figure 3 shows a comparison of x-ray transmission images of interconnected lamellae cut from an aluminum sponge sample. The x-ray phase-contrast image (${I_{S}}/{I_{{\rm flat}\,{\rm field}}}$) in Fig. 3(a) was calculated from images directly recorded by a 2D imaging detector with (${I_{S}}$) and without (${I_{{\rm flat}\,{\rm field}}}$) the sample. Note that in terms of the mechanism for image formation, there is no difference between propagation-based x-ray phase-contrast imaging and x-ray phase-contrast ghost imaging. An x-ray phase-contrast image or Fresnel image is formed downstream of the sample, given that the x-ray beam has a sufficient degree of spatial coherence via a small source size. The difference between the two images in Figs. 3(a) and 3(b) is in image detection. The x-ray phase-contrast ghost image in Fig. 3(b) was calculated from synthesized ghost images with (${G_{S}}$) and without (${G_{{\rm flat}\,{\rm field}}}$) the sample. The x-ray phase-contrast ghost image shows accurate x-ray transmission values similar to the “direct” x-ray phase-contrast image as illustrated in the line profiles shown in Figs. 3(c) and 3(d). The transmission $T$ is unity at the sponge pores, greater than unity at the material edges (phase contrast), and decreases at regions with increasing material density (attenuation contrast). Strong contrast, which is expected at a thick edge where a large x-ray phase gradient occurs, can be resolved along the horizontal direction [Fig. 3(e)]. Due to the better spatial resolution along the vertical direction, a fringe pattern at the thin edge can also be resolved [Fig. 3(f)]. The white–black–white contrast is attributed to Fresnel diffraction [7]. We emphasize that with attenuation contrast alone, thin edges like this would have been invisible, as the x-ray transmissions by the air and near the thin edge are essentially unity [see line profile in Fig. 3(f)].
The accurate x-ray transmission values that were achieved by our approach are a consequence of using a near-complete, near-orthogonal set of basis patterns versus a non-orthogonal set. It is important to note that this accuracy was achieved despite the fact that the resolution along the horizontal direction is an order of magnitude less than along the vertical direction (${w_N} = 100\;\unicode{x00B5}{\rm m}$ and $h = 6.5\;\unicode{x00B5}{\rm m}$). This ensures that the same high fidelity and accuracy can be achieved with large-pixel direct detectors when combined with micrometer-pitch gratings. For example, a combination of ${w_N} = 6.5\;\unicode{x00B5}{\rm m}$ and $h = 100\;\unicode{x00B5}{\rm m}$ should achieve a similar ghost image.
5. DISCUSSION AND CONCLUSION
In this work, we introduced an imaging method for obtaining x-ray phase-contrast ghost images using a single-pixel camera. We showed that the structured detection approach is the key to successfully producing an x-ray ghost image with phase contrast. By using a near-complete, near-orthogonal set of periodic structures for such structured detection, high fidelity images with accurate x-ray transmission values were obtained.
While ghost imaging has been previously demonstrated in the x-ray regime without phase contrast [18–23], and phase contrast has been achieved in the x-ray regime without employing ghost imaging [3,7,8,44], the work of the present paper unites these previously disparate modalities for x-ray imaging. We believe this to be a useful advance since it opens the possibility for the distinct advantages of each modality—namely x-ray ghost imaging’s promise of significant dose reduction and x-ray phase contrast’s ability to see the otherwise invisible refractive effects of x-ray-transparent structures [3,7,8,44]—to be achieved simultaneously.
Here, we list potential avenues which could be explored in future. The spatial resolution of the presented method, which was determined by the point spread function (Eq. (4)), was limited by the smallest grating structure width. With mature fabrication technologies for x-ray optics, high aspect ratio gratings made of gold can be implemented. For example, LIGA technology [45] can provide gratings having line widths down to 2.4 µm and gold thicknesses up to 160 µm that can be used above 100 keV [43]. With silicon-based technology, it has been demonstrated that micro-fabrication of gratings by deep reactive ion etching [46], Au electroplating [47], and metal assisted chemical etching (MacEtch) [48] can achieve aspect ratios with structures down to the sub-micrometer range, opening new routes of nano-fabrication for high x-ray energies. Here, we demonstrated the technique with 1D gratings and a 1D bucket detector. Avenues for fabricating high aspect ratio speckle masks and Hadamard masks can be explored to implement the technique with a true bucket detector and achieve a desired spatial resolution in two transverse directions. Also, while we have employed a base-level correlative algorithm in Eq. (2), it would be interesting to investigate more sophisticated algorithms in our phase-contrast setting, such as those based on the Moore–Penrose inverse [49] or Landweber iteration [21,22,25]. Furthermore, while bucket detectors have no spatial resolution, they do have energy resolution, and hence it would be interesting to extend our work to the multi-energy regime.
The potential of the proposed method for reduced-dose imaging can be evaluated by employing direct detectors, and would be another avenue for future work. The method can be tested against conventional x-ray phase-contrast imaging, which uses indirect x-ray detectors. X-ray phase-contrast ghost imaging can also be implemented and combined with phase retrieval [10] and computed tomography [22,25]. For example, for the case of a sample that is composed of a single material, applying the phase-retrieval method of Paganin et al. [10] to images such as those in Fig. 3(b), would be a means of decoding the Fresnel-fringe data that is present in such images in order to recover quantitative information regarding the projected thickness of the sample. Our method of synthesis of ghost imaging with phase contrast may thus have three significant factors contributing to dose reduction. The first factor is the previously mentioned near-100% quantum efficiency possible with ghost-imaging bucket detectors, compared to the lower detection efficiency for position-sensitive detectors. The second factor is the significant signal-to-noise boosts that become possible by enabling x-ray ghost imaging with phase contrast [9–15]. The third factor is the possibility that suitable leveraging of a priori knowledge, e.g., via compressive sensing in a ghost-imaging setting [27,35,49], may enable reduced dose. One could altogether bypass the need to reconstruct an image if one is instead interested only in particular parameters regarding the sample under study, such as porosity or fractal dimension [27].
With our structured detection approach, an x-ray phase-contrast ghost image in the Fresnel (near-field) or Fraunhofer (far-field) regime can be synthesized, depending on the sample-to-mask distance. The ghost image synthesis can even be extended to other x-ray phase-contrast imaging approaches such as Talbot interferometry [4,50], and near-field speckle-tracking [51,52]. Simulations have indeed shown that our detection approach is a means to achieve analyzer-based x-ray phase-contrast ghost imaging [36]. Our approach is also compatible with ghost imaging combined with x-ray diffraction topography and crystallography. Finally, our x-ray phase-contrast ghost imaging approach is in principle applicable to other probes such as neutrons [53], electrons [54], ions [55], atoms [56], molecular beams, alpha rays, and muons.
Acknowledgment
M.P.O. acknowledges Daniele Pelliccia (Instruments & Data Tools Pty Ltd., Australia) for introducing x-ray ghost imaging to her, Harald Reichert (ESRF, France) for his valuable advice which improved the manuscript, and Lucia Romano (PSI, Switzerland) for discussions on state of the art in grating fabrication. D.M.P. acknowledges useful discussions with David Ceddia, Timur Gureyev, Andrew Kingston, Glenn Myers, and Daniele Pelliccia, as well as ESRF for a visiting fellowship.
Disclosures
The authors declare no conflicts of interest.
Data Availability
The data and the MATLAB code supporting the results reported in this paper are available on request from the corresponding author.
REFERENCES
1. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996). [CrossRef]
2. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995). [CrossRef]
3. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x–ray phase contrast microimaging by coherent high–energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995). [CrossRef]
4. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003). [CrossRef]
5. A. Khimchenko, C. Bikis, A. Pacureanu, S. E. Hieber, P. Thalmann, H. Deyhle, G. Schweighauser, J. Hench, S. Frank, M. Müller-Gerbl, G. Schulz, P. Cloetens, and B. Müller, “Hard X-ray nanoholotomography: large-scale, label-free, 3D neuroimaging beyond optical limit,” Adv. Sci. 5, 1700694 (2018). [CrossRef]
6. I. Nengo, P. Tafforeau, C. C. Gilbert, J. G. Fleagle, E. R. Miller, C. Feibel, D. L. Fox, J. Feinberg, K. D. Pugh, C. Berruyer, S. Mana, Z. Engle, and F. Spoor, “New infant cranium from the African Miocene sheds light on ape evolution,” Nature 548, 169–174 (2017). [CrossRef]
7. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D 29, 133–146 (1996). [CrossRef]
8. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996). [CrossRef]
9. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. 75, 2912–2914 (1999). [CrossRef]
10. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002). [CrossRef]
11. M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D x-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18, 6423–6436 (2010). [CrossRef]
12. M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011). [CrossRef]
13. Y. I. Nesterets and T. E. Gureyev, “Noise propagation in x-ray phase-contrast imaging and computed tomography,” J. Phys. D 47, 105402 (2014). [CrossRef]
14. T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D 47, 365401 (2014). [CrossRef]
15. M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using x-ray phase contrast,” Sci. Rep. 7, 15953 (2017). [CrossRef]
16. A. Koch, C. Raven, P. Spanne, and A. Snigirev, “X-ray imaging with submicrometer resolution employing transparent luminescent screens,” J. Opt. Soc. Am. A 15, 1940–1951 (1998). [CrossRef]
17. B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010). [CrossRef]
18. H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117, 113901 (2016). [CrossRef]
19. D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016). [CrossRef]
20. A. Schori and S. Schwartz, “X-ray ghost imaging with a laboratory source,” Opt. Express 25, 14822–14828 (2017). [CrossRef]
21. D. Pelliccia, M. P. Olbinado, A. Rack, A. M. Kingston, G. R. Myers, and D. M. Paganin, “Towards a practical implementation of x-ray ghost imaging with synchrotron light,” IUCrJ 5, 428–438 (2018). [CrossRef]
22. A. M. Kingston, D. Pelliccia, A. Rack, M. P. Olbinado, Y. Cheng, G. R. Myers, and D. M. Paganin, “Ghost tomography,” Optica 5, 1516–1520 (2018). [CrossRef]
23. A.-X. Zhang, Y.-H. He, L.-A. Wu, L.-M. Chen, and B.-B. Wang, “Table-top x-ray ghost imaging with ultra-low radiation,” Optica 5, 374–377 (2018). [CrossRef]
24. A. Schori, D. Borodin, K. Tamasaku, and S. Shwartz, “Ghost imaging with paired x-ray photons,” Phys. Rev. A 97, 063804 (2018). [CrossRef]
25. A. M. Kingston, G. R. Myers, D. Pelliccia, I. D. Svalbe, and D. M. Paganin, “X-ray ghost-tomography: artefacts, dose distribution, and mask considerations,” IEEE Trans. Comput. Imaging 5, 136–149 (2019). [CrossRef]
26. Y.-H. He, A.-X. Zhang, M.-F. Li, Y.-Y. Huang, B.-G. Quan, D.-Z. Li, L.-A. Wu, and L.-M. Chen, “High-resolution sub-sampling incoherent x-ray imaging with a single-pixel detector,” APL Photon. 5, 056102 (2020). [CrossRef]
27. A. M. Kingston, W. K. Fullagar, G. R. Myers, D. Adams, D. Pelliccia, and D. M. Paganin, “Inherent dose-reduction potential of classical ghost imaging,” Phys. Rev. A 103, 033503 (2021). [CrossRef]
28. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010). [CrossRef]
29. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008). [CrossRef]
30. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009). [CrossRef]
31. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009). [CrossRef]
32. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008). [CrossRef]
33. P. Clemente, V. Durán, E. Tajahuerce, P. Andrés, V. Climent, and J. Lancis, “Compressive holography with a single-pixel detector,” Opt. Lett. 38, 2524–2527 (2013). [CrossRef]
34. Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6, 6225 (2015). [CrossRef]
35. T. E. Gureyev, D. M. Paganin, A. Kozlov, Y. I. Nesterets, and H. M. Quiney, “On the efficiency of computational imaging with structured illumination,” Phys. Rev. A 97, 053819 (2018). [CrossRef]
36. D. Ceddia and D. M. Paganin, “Random-matrix bases, ghost imaging, and x-ray phase contrast computational ghost imaging,” Phys. Rev. A 97, 062119 (2018). [CrossRef]
37. W. Gong and S. Han, “Phase-retrieval ghost imaging of complex-valued objects,” Phys. Rev. A 82, 023828 (2010). [CrossRef]
38. D.-J. Zhang, Q. Tang, T.-F. Wu, H.-C. Qiu, D.-Q. Xu, H.-G. Li, H.-B. Wang, J. Xiong, and K. Wang, “Lensless ghost imaging of a phase object with pseudo-thermal light,” Appl. Phys. Lett. 104, 121113 (2014). [CrossRef]
39. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). [CrossRef]
40. R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956). [CrossRef]
41. L. Wang, S. Qamar, S.-Y. Zhu, and M. Zubairy, “Hanbury Brown–Twiss effect and thermal light ghost imaging: a unified approach,” Phys. Rev. A 79, 033835 (2009). [CrossRef]
42. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983). [CrossRef]
43. M. Ruiz-Yaniz, F. Koch, I. Zanette, A. Rack, P. Meyer, D. Kunka, A. Hipp, J. Mohr, and F. Pfeiffer, “X-ray grating interferometry at photon energies over 180 keV,” Appl. Phys. Lett. 106, 151105 (2015). [CrossRef]
44. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996). [CrossRef]
45. E. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plastic moulding (LIGA process),” Microelectron. Eng. 4, 35–56 (1986). [CrossRef]
46. K. Jefimovs, J. Vila-Comamala, C. Arboleda, Z. Wang, L. Romano, Z. Shi, M. Kagias, and M. Stampanoni, “Fabrication of x-ray gratings for interferometric imaging by conformal seedless gold electroplating,” Micromachines 12, 517 (2021). [CrossRef]
47. D. Josell, Z. Shi, K. Jefimovs, L. Romano, J. Vila-Comamala, and T. P. Moffat, “Pushing the limits of bottom-up gold filling for x-ray grating interferometry,” J. Electrochem. Soc. 167, 132504 (2020). [CrossRef]
48. L. Romano, M. Kagias, J. Vila-Comamala, K. Jefimovs, L.-T. Tseng, V. A. Guzenko, and M. Stampanoni, “Metal assisted chemical etching of silicon in the gas phase: a nanofabrication platform for x-ray optics,” Nano. Horiz. 5, 869–879 (2020). [CrossRef]
49. T. J. Lane and D. Ratner, “What are the advantages of ghost imaging? Multiplexing for x-ray and electron imaging,” Opt. Express 28, 5898–5918 (2020). [CrossRef]
50. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006). [CrossRef]
51. S. Bérujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional x-ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012). [CrossRef]
52. 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). [CrossRef]
53. A. M. Kingston, G. R. Myers, D. Pelliccia, F. Salvemini, J. J. Bevitt, U. Garbe, and D. M. Paganin, “Neutron ghost imaging,” Phys. Rev. A 101, 053844 (2020). [CrossRef]
54. S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron ghost imaging,” Phys. Rev. Lett. 121, 114801 (2018). [CrossRef]
55. A. Trimeche, C. Lopez, D. Comparat, and Y. J. Picard, “Ion and electron ghost imaging,” Phys. Rev. Res. 2, 043295 (2020). [CrossRef]
56. R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. G. Dall, K. G. H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540, 100–103 (2016). [CrossRef]
