A three-image method to extract absorption, refraction and scattering information for hard x-ray grating interferometry is presented. The method comprises a post-processing approach alternative to the conventional phase stepping procedure and is inspired by a similar three-image technique developed for analyzer-based x-ray imaging. Results obtained with this algorithm are quantitatively comparable with phase-stepping. This method can be further extended to samples with negligible scattering, where only two images are needed to separate absorption and refraction signal. Thanks to the limited number of images required, this technique is a viable route to bio-compatible imaging with x-ray grating interferometer. In addition our method elucidates and strengthens the formal and practical analogies between grating interferometry and the (non-interferometric) diffraction enhanced imaging technique.
© 2013 OSA
Hard x-ray phase-sensitive methods are becoming increasingly used . Their main advantages over standard absorption radiography are the ability to detect weakly absorbing details and to produce contrast also between features with similar index of refraction. The first approach to x-ray phase-sensitive methods dates back to 1965, when Bonse and Hart  proposed a crystal-based interferometer. Remarkably, just 30 years later this technique was systematically used for imaging, developing specific algorithms to quantitatively extract the phase shift caused by the object . More recently, a new interferometric approach based on the use of gratings has been introduced [4–6], overcoming many of the limitations of crystal-based methods, including their high requirements in terms of beam coherence and mechanical stability. Grating interferometry (GI) is based on a pair of gratings spaced by a fractional Talbot distance. The first grating produces a Talbot pattern that is analyzed by the second grating. The data acquisition and analysis methods mostly employed with GI are essentially derived from the algorithms developed for crystal interferometry: in particular, the so-called phase stepping (PS) technique  used in GI bears a strong resemblance to the fringe-scanning method used in crystal interferometry . More in detail, PS exploits the periodicity of the transmission function of the gratings pair: several images are acquired in different positions along one or more oscillation periods. A first-order approximation of the Fourier series of the transmission function is applied to extract three different parametric images, usually referred to as absorption, differential phase and dark field images. The latter actually depends on the scattering properties of the sample and it is measured as the decrement of the amplitude of the periodic transmission function with respect to its value in absence of the sample .
Among the other phase-sensitive methods, a non-interferometric technique, analyzer-based imaging (ABI) has been extremely successful. ABI is based on an analyzer crystal, placed between the sample and the detector. The high angular sensitivity of the transmission function of the crystal (rocking curve) allows one to detect small deviation of the x-ray beam after its interaction with a sample. The first and most popular data analysis method for ABI is the Diffraction-Enhanced Imaging (DEI) algorithm, introduced in : by acquiring two images on the opposite flanks of the rocking curve, the quantitative imaging of both absorption and refraction angle can be derived. The method works under the approximation of small angular deviation compared to the width of the rocking curve. Of note is that the DEI technique has been subsequently generalized [8, 9] to extract absorption, differential phase and scattering signal from three input images acquired at different angles along the rocking curve. Thus, by inserting an optical system with high angular sensitivity between sample and detector, both DEI (with a crystal rocking curve) and GI (with a periodic transmission function) are sensitive to the same physical parameters of the interaction between x-rays and matter [10, 11].
In this paper we apply the three-image method, introduced for ABI  to a single period of the transmission function of the GI to image samples featuring absorption, refraction and scattering. We show that the novel method can be used alternatively to conventional PS even though it relies on different approximations. In both cases at least three images are required to measure absorption, differential phase and scattering width maps. Moreover, in the case of samples with negligible scattering, we show that our approach – unlike PS – can be additionally simplified in analogy with the original DEI approach . In this case only two images are sufficient to extract absorption and refraction information, shortening the required measurement time and possibly reducing the dose.
The paper is organized as follows: in Sec. 2 the theoretical basis of the method is reviewed with reference to the previous works. The measurement procedure is described in Sec. 3 where the differences – in both data acquisition and analysis – between conventional PS and our method are explained. Sec. 4 is devoted to the presentation of the experimental results and the comparison of results obtained with our method and with PS. Finally such results are discussed in Sec. 5.
The x-ray beam passing through the sample can be absorbed, refracted or scattered. Refraction is characterized by an angular deviation while the deviation due to the ultra-small angle scattering is correspondingly indicated with. The latter accounts for deviations that cannot be resolved by the detector and are therefore convolved with the point spread function of the detection system [8, 12]. While the refraction angleis assumed to have a well-defined value at each object plane position, the ultra-small-angle scattering anglederives from a stochastic process, represented by a certain probability density function. Moreover by collecting a large number of photons, each pixel effectively integrates over this statistical distribution .
In the case of ABI, the angular sensitivity of a crystal rocked about its Bragg peak is used in  to decouple the refraction and scattering information from the absorption image. If the measurement is performed with a GI instead, for a certain distance L between the sample and the second grating, we can define refraction and scattering displacement at the position of the second grating as and respectively. When the sample is placed right before the first grating, L is the just the chosen fractional Talbot distance. Furthermore, let us indicate with the probability density that a photon would be scattered byin the horizontal plane, i.e. in the plane perpendicular to the grating lines. Therefore, adapting Eq. (1) in  to the present case, we can model the intensity transmitted through a GI as:Eq. (1) T is the transmission function of the GI and is the relative displacement of the two gratings. The function T is periodical with the period p of the Talbot pattern. Equation (1) is formally identical to the corresponding expression used in ABI . Nonetheless the difference in the transmission function between ABI and GI must be taken into account to correctly interpret this formula.
In the ABI setup, the transmission function of the analyzer crystal (i.e. the rocking curve) is generally a sharp peak in angular space. This characteristic has been used in  to simplify Eq. (1) in the approximation of small refraction and scattering displacements with respect to the width of the rocking curve. In this approximation the transmission function has been written as a second order Taylor expansion about some position θ = θ0 of the analyzer crystal rocking curve. On the other hand the angular dependence of a GI is related to the relative displacement by and the Taylor expansion of Eq. (1), considered about some position ξ = ξ0, can be formally written as:Eq. (2) represents the first derivative of T with respect to calculated in and accordingly stands for the second derivative. The variance of the scattering distribution is given by8, 12], the first moment is actually vanishing:13]. Here we extend the formalism to the second order. Such expansion is valid under certain approximations. Formally one needs to have both and to be much smaller than the period (for the first-order expansion in  the authors assume the scattering to be negligible and). In practice, for hard x-rays, whereas the approximation of small refraction shift compared to the period is generally valid, the tails of the scattering distribution can extend for over a period of the transmission function. Nonetheless it is important that the standard deviation is small compared to p. Thus the expansion written in Eq. (2) can still be retained (up to the fourth order of sigma) considering the symmetric nature of the scattering distribution.
Numerous studies both in DEI [12–17] and GI [6, 18–24] investigated the nature and the measurement of the small and ultra-small angle scattering signal in different configurations. In these contexts authors refer to scattering as the signal originated by photons that undergo single or multiple refraction by structures beyond the resolution limit of the detector [6, 12]. In the ABI case, the rocking curve of a crystal is a narrow function in angular space, while the transmission function of a GI is a periodic function. Therefore, in ABI photons scattered at larger angles would undergo extinction in the crystal, while such photons would still contribute, up to a certain degree, to the transmission function of the GI. This effect has been modeled as a convolution of the reference transmission function with a Gaussian distribution [18–20] assuming mutually incoherent scattering contributions within each detector pixel. In [19, 20] such distribution has been identified with an ultra-small-angle scattering distribution whose expectation value is related to the refraction shift, its integral value to the conventional absorption contrast and its variance to the dark field signal. The theoretical approach used here, as in , is completely equivalent to the convolution approach [19, 20]. This is manifested if we introduce the coordinate change in Eq. (1) obtaining:
The symbol denotes the convolution operator and the last equality holds by defining. Therefore, as in [19, 20], the function g is centered on the value ∆xR of the refraction shift while the scattering contribution produces a decrease in the amplitude of the oscillation of T, corresponding to the standard deviation of the distribution. Hence, while the refraction shift of the oscillatory transmission function is directly measurable, the sole information accessible for scattering is the standard deviation of its distribution, i.e. formally the there is no explicit dependence on η of in Eq. (5). The scattering displacement is effectively treated as a random number that is ensemble-averaged in the convolution process and thereby, assuming that the distribution is symmetric, does not induce a net shift of the transmission function. This concept was already noted in  where it is stated that positive and negative scattering contributions “cancel out” when the second derivative of the transmission function is zero. As we notice here, in general those contributions do not cancel out but convolute with the transmission function without shifting its center of mass. Accordingly, even when the extension of the tails of the scattering distribution is comparable to the period, one can safely assume Eq. (2) to be approximately valid, when the refraction shift is small compared to the period of T.
Therefore, we can follow the treatment shown in [8, 9]. The key idea is that, by acquiring three images at different displacement of the grating, the three unknown quantities in Eq. (2) viz., and can be obtained. The latter quantities have been introduced to deal with the angular deviation and angular scattered distribution respectively, which are independent of the chosen fractional Talbot distance. The solution is:Eq. (6) εijk is the totally antisymmetric tensor and all quantities are to be considered function of the coordinates, i.e. and so forth, i.e. these relations are applied on a pixel-per-pixel basis.
Equations (6) represent an alternative and independent method to measure absorption, refraction and scattering distribution of a sample. In particular the measurement of the variance of the angular scattering distribution is related to the dark field signal [18–20], i.e. the normalized visibility map as measured by the PS technique, via:
As concluding remarks we point out that Eq. (6) can be further simplified when the sample scattering is negligible. This has been discussed in  noting that in this case Eq. (6) reduces to the original DEI equations . This simplified approach uses only two images and thus it can be extended to the GI as well. A similar result has been derived in  in the context of post-processing of x-ray phase tomography. There the authors demonstrate a phase retrieval algorithm requiring two images corresponding to the projections of a sample rotated by an angle ϕ and ϕ + π respectively, while keeping the analyzing system (a GI in ) in a constant position. Here we derive an equivalent result as a special case of a more general approach, and with a different experimental procedure consisting in changing the relative coordinate ξ of the GI, without rotating the sample, i.e. without the need of a tomography setup. It is worth noting that the minimum number of images required for the phase stepping to work is three, therefore the simplified approach described here could be useful for radiation-sensitive samples to reduce the number of required images. As noted in previous works [25, 26] the possibility of reducing the number of required exposures (and eventually eliminate the need of the phase stepping procedure) is a major step forward to implement low-dose applications. Obviously the delivered dose depends on the total exposure time, therefore the dose reduction is accomplished by keeping constant the exposure time of a single image and reducing the number of required images, as in [25, 26].
3. Measurement procedure
The novel algorithm presented here uses a similar experimental approach as the PS technique, while introducing a qualitative change in the post-processing approach. The GI setup is made of two gratings: a first (phase) grating used to produce a Talbot pattern downstream of it, reproducing a periodic intensity distribution of period p  and a second (absorption) grating used to analyze such pattern with and without the sample. The measurement procedure consists in scanning one of the gratings with sub-period steps. In this way the absorption grating is continuously moved with respect to the Talbot pattern (or vice versa) causing a periodical variation of the transmitted intensity, with period p. The transmitted intensity in any given pixel can be written as a Fourier series:Eq. (8) as a cosine graph (i.e. neglecting the terms in the Fourier expansion) and determine the parameters, and via a Fourier analysis of the transmitted intensity in each pixel. In practice these quantities are measured with respect their corresponding value measured in absence of the sample, viz., and. The superscripts s and r refer to the measurement with the sample and the reference beam (i.e. without the sample) respectively. is the absorption image of the sample, while the refraction and the visibility images can be computed from and as:Eqs. (8) and (9). The same physical quantities can be retrieved with the novel method via Eqs. (6) and (7). For both approaches the minimum number of steps required to find the three parameters is three. Both post-processing approaches are able to access the same physical quantities and can be implemented using the same experimental setup but rely on different approximations.
PS procedure works under the approximation of periodic transmission function and — in the specific case of three-image PS —under the more stringent condition that the transmission function is well described by a cosine term only. On the contrary, the method proposed here does not require a perfect cosine graph, while it needs a very precise knowledge of the reference intensity (without the sample) to guarantee sufficient stability in the numerical computation of the derivatives. These considerations lead to a slightly different experimental approach. For the PS one needs to acquire scans with the same number of steps either with or without the sample. In our alternative method instead the reference intensity must be acquired with a very large number of steps. Then only three-images at specific locations are to be acquired with the sample in the beam.
We performed GI scans and compared the results obtained with our method and with the PS method on the same samples. In Fig. 1 the measured reference intensity for a typical GI scan is shown. The total number of steps in one period is 48 in this case. The blue crosses mark the positions where measurements with the sample have been taken to apply Eq. (6). For an honest comparison between GI and PS the photon statistics was kept constant: the respective PS positions used are indicated by the open red circles.
Measurements have been performed on the SYRMEP beamline at the Synchrotron light source Elettra (Trieste, Italy). The incident x-ray beam was monochromatized at the energy of 25 keV by a double-bounce Si 111 monochromator. The beam size at the sample was 3.2 mm vertical and 27 mm horizontal. The GI was composed of two gratings (phase and absorption) separated by a distance L = 58 mm, corresponding to the first fractional Talbot distance for the working parameters. The phase grating had a period of 2.36 μm and Ni structures, while the absorption grating had period 2.4 μm and Au absorbing lines. The period mismatch has been designed to compensate for the beam divergence. All images were recorded using a 16-bit CCD detector (Photonic Science Ltd, Roberts-bridge, UK) with pixel size of 14x14 μm2 placed 10 cm downstream of the absorption grating.
4. Experimental results
In Fig. 2 absorption and refraction images of a bovine bone sample are shown. A very good agreement is clearly found in the absorption maps obtained with the two methods, shown in the panels (a) and (c), respectively. Refraction images are shown in the panels (b) and (d). The diagram in Fig. 2(e) shows the line profiles of the refraction maps at the position indicated by the arrows. The black solid line corresponds to the PS approach while the red line is obtained using the novel method. The agreement of the two line profiles is quantitatively very good, confirmed by the value of the calculated correlation coefficient R = 0.96. A minor difference is noticeable: while the image in 2(d) possesses higher dynamics, its background noise is higher. This is visually appreciable from the panels 2(f) and 2(g) showing the histograms taken in a region in the middle of the sample and in the background region respectively. As before the black line corresponds to PS and the red line to the novel method. Both distributions obtained with the novel method are consistently wider, indicating larger dynamics and larger fluctuations associated with the background region, (panel 2(g)). This effect is a direct consequence of the application of Eq. (6): the computation of the local derivatives produces higher dynamics with respect to the “non-local” Fourier analysis. On the other hand the numerical evaluation of the derivatives is also more sensitive to noise, resulting in images with noisier background. It is important to reiterate though that the differences are actually minor and the results obtained with the two methods are largely consistent.
The maps of, obtained from the same data set of previous picture, are shown in Fig. 3. In the PS post-processing, Fig. 3(a), the scattering information is obtained using the left-hand side of Eq. (7) on the normalized visibility map. The novel method instead produces a contrast associated with the variance of the scattering distribution, as shown in Fig. 3(b). To test the accuracy of our system, the difference between the two independent maps in 3(b) and 3(a), respectively, is reported in Fig. 3(c). Data show a substantial equivalence of the two results, with the novel method rendering a wider distribution in the region where the sample is less absorbing. On the other hand, in the central and highly absorbing part of the sample the PS post-processing yields a wider distribution. This effect is currently under study and has to be verified through further measurements.
Another quantification of the similarity of the two results is given by the histograms of the counts, from the retrieved maps in Figs. 3(a) and 3(b). The histograms are superimposed in Fig. 3(d) where the black line is the histogram of Fig. 3(a) and the red line of Fig. 3(b). The histograms represent the frequency of pixels counting within a certain range. The peaks around the zero come from the background noise, while the broad peak on the right is the scattering signal. The map obtained with the novel algorithm shows a wider distribution that, from Fig. 3(c), it is seen to correspond to the low absorbing part of the sample. Despite the small differences the results obtained with both methods are consistent, thus confirming the validity of the new approach.
It is worth remarking that the possibility of measuring the variance (and higher moments) of the scattering distribution using a GI without PS has been already shown in [19, 20]. In that case the scattering information has been obtained by deconvolution of the transmission function measured with the sample by the same curve obtained in absence of the sample. When more than three steps are used, a map for each position along the PS curve could be measured and higher order momenta of the scattering distribution become accessible. As we have shown in Sec. 2, the method presented here can be viewed as a special case of the one in [19, 20], with the important advantage of simple and numerically stable post-processing and low dose because of the need of only three images.
A very important consequence of the novel approach is the possibility of using only two images for the reconstruction of the absorption and refraction images, as in the original DEI algorithm. Mathematically this corresponds to neglecting (negligible scattering) in Eq. (2) and therefore solving it for the absorption and refraction only. To test this procedure we post-processed the bovine bone images with this simplified method. Results are displayed in Figs. 4(a) and 4(b) for the apparent absorption and refraction respectively. While the absorption image appears to be little affected by the reduced statistics, the refraction map shows distortions, especially in the region close to the middle of the sample where absorption and scattering are stronger.
On the other hand, an example of the same method applied to a sample producing negligible scattering – a nylon wire, with diameter 0.8 mm – is shown in Figs. 4(c)–4(f). In this case the approximation leading to neglect the second derivative is well justified. Absorption and refraction images obtained with only two steps (Figs. 4(c) and 4(d)) are compatible with the corresponding images obtained by the PS method (Figs. 4(e) and 4(f)) taking into account the better statistics of the latter.
PS method is based on the periodicity of the transmission function of the GI. The principle of the method presented here, initially introduced as a generalization of the DEI algorithm , stands simply on a variable transmission function such as a crystal’s rocking curve. More specifically one needs to find three points where the transmission function T(x) and its derivatives take distinct values, to have the system in Eq. (6) well posed. An ideal GI with square grating profile and fully coherent illumination is characterized by a triangular transmission function. Deviations of the gratings shape from the nominal profile, along with partially coherent illumination contribute in “smoothing” the triangular function making it well approximated with a first order Fourier series. It is worth pointing out that in our experimental case the GI transmission function was actually well approximated by a simple cosine therefore the PS approach with three steps was well suited.
However, the post-processing method introduced here is qualitatively different from PS as it relies on a Taylor expansion of the transmitted intensity. From a general point of view the experimental demonstration of the applicability of the novel algorithm to the GI setup provides a very interesting bridge between the two widely used methods of GI and ABI. The result itself is not surprising as both techniques are sensitive to the same physical quantities. Despite slightly different theoretical considerations required to use of Eq. (6) with GI, the results are quantitatively very similar to PS technique. This would potentially offer a way to quantitatively compare ABI and GI in a way that has not been done thus far.
Considering GI alone, one might envisage specific experimental situations in which one of the two post-processing methods can be preferable. For instance the application of Fourier analysis for PS requires the working positions to be equally spaced along the oscillation of the transmission function. This requirement is dropped in the new method that can perform also when the spacing between the points is not equal. Furthermore the presence of higher order harmonics in the transmission function may lead to artifact in the PS method , especially when only three steps are used. This constraint is removed in the novel method while, on the other hand, the numerical computation of the local derivatives of the transmission function may increase the noise of the computed images.
It is worth remarking that the measurements, for both, our method and the PS, are always taken with reference to the measurement without the sample. This means that when measuring with the sample, the actual location of the measurement positions along the curve is not crucial, while the important parameter is the relative spacing between working points. Nevertheless two main differences do exist: As already noticed in Sec. 3, the reference scan in the novel method must be acquired with a large number of steps (unlike PS) to attain a reliable numerical estimation of the derivatives needed to apply Eq. (6). Furthermore since quantities obtained with PS are always calculated relatively to the corresponding quantities measured without the sample, the two scans (with and without the sample) should be in phase. However if a small fixed phase existed between the sample scan and the reference scan this would only cause a small constant offset to (see Eqs. (8)–(9)). On the other hand the restriction to acquire the sample scan exactly in phase with the reference scan must be enforced for the novel method. In other words the position along the transmission function where the three images are acquired has to be known with sufficiently good accuracy with respect to the reference scan to avoid reconstruction errors. Therefore particular care must be taken that both, the reference scan and the three-step scan used in the phase retrieval actually have the same starting position for an accurate estimation of the refraction shift.
The general methodology of using three or two images allows for a sensible reduction of the radiation dose, especially extending the methodology to tomography [25, 26]. Therefore our approach paves the way for using GI at bio-compatible doses retaining the key advantages of this system, namely the robustness against partial spatial and spectral coherence and mechanical issues.
As alternative to conventional phase stepping we have presented a new method to record absorption, refraction and scattering from a sample using a hard x-ray grating interferometer. It utilizes three images acquired at different position along the transmission function of the interferometer. Differences in data acquisition and analysis between the novel method and the phase stepping are presented. The results obtained with the two methods are quantitatively compared and a very good agreement is found. A further approximation of the method, using only two images is presented and discussed with the potential of obtaining quantitative absorption and refraction imaging with bio compatible radiation dose. The post-processing method makes use of the same equations already derived for DEI, thus providing a good experimental verification of the formal and practical analogies between grating interferometry and analyzer-based imaging.
The authors acknowledge the support from the European Research Infrastructure EUMINAFAB Grant No. FP7-226460. D. Kunka and J. Mohr are acknowledged for the fabrication of the gratings. S. Lagomarsino and M. Fratini for providing the sample, D. Dreossi, N. Sodini and F. Scarinci for technical assistance. D.P. acknowledges travel funding provided by the International Synchrotron Access Program ISP4372 managed by the Australian Synchrotron and funded by the Australian Government. The authors acknowledge the INFN of Italy (Istituto di Fisica Nucleare) for partial financial support.
References and links
2. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965). [CrossRef]
3. A. Momose, “Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer,” Nucl. Instrum. Meth. A 352(3), 622–628 (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(Part 2, No. 7B), L866–L868 (2003). [CrossRef]
5. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]
6. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef] [PubMed]
7. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. H. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef] [PubMed]
8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]
9. L. Rigon, F. Arfelli, and R.-H. Menk, “Generalized diffraction enhanced imaging to retrieve absorption, refraction and scattering effects,” J. Phys. D Appl. Phys. 40(10), 3077–3089 (2007). [CrossRef]
10. P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: Propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express 20(3), 2789–2805 (2012). [CrossRef] [PubMed]
11. F. Arfelli, D. Pelliccia, A. Cedola, A. Astolfo, I. Bukreeva, P. Cardarelli, D. Dreossi, S. Lagomarsino, R. Longo, L. Rigon, N. Sodini, and R. H. Menk, “Recent developments on techniques for differential phase imaging at the medical beamline of ELETTRA,” J. Instrum. 8(06), 06001 (2013). [CrossRef]
12. L. Rigon, H.-J. Besch, F. Arfelli, R.-H. Menk, G. Heitner, and H. Plothow-Besch, “A new DEI algorithm capable of investigating sub-pixel structures,” J. Phys. D Appl. Phys. 36(10A), A107–A112 (2003). [CrossRef]
13. P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011). [CrossRef] [PubMed]
14. M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48(23), 3875–3895 (2003). [CrossRef] [PubMed]
15. G. Khelashvili, J. G. Brankov, D. Chapman, M. A. Anastasio, Y. Yang, Z. Zhong, and M. N. Wernick, “A physical model of multiple-image radiography,” Phys. Med. Biol. 51(2), 221–236 (2006). [CrossRef] [PubMed]
17. Ya. I. Nesterets, “On the origin of decoherence and extinction contrast in phase-contrast imaging,” Opt. Commun. 281(4), 533–542 (2008). [CrossRef]
18. Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95(9), 094105 (2009). [CrossRef]
19. P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the Ultrasmall-Angle X-Ray Scattering Distribution with Grating Interferometry,” Phys. Rev. Lett. 108(4), 048101 (2012). [CrossRef] [PubMed]
20. F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle X-ray scattering,” Phys. Med. (2013), doi:. [CrossRef] [PubMed]
23. S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Appl. Opt. 50(22), 4310–4319 (2011). [CrossRef] [PubMed]
25. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based X-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. U.S.A. 107(31), 13576–13581 (2010). [CrossRef] [PubMed]
26. I. Zanette, M. Bech, A. Rack, G. Le Duc, P. Tafforeau, C. David, J. Mohr, F. Pfeiffer, and T. Weitkamp, “Trimodal low-dose X-ray tomography,” Proc. Natl. Acad. Sci. U.S.A. 109(26), 10199–10204 (2012). [CrossRef] [PubMed]
27. A. Momose, W. Yashiro, and Y. Takeda, “X-Ray Phase Imaging with Talbot Interferometry” in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning, and Inverse Problems, Y. Censor, M. Jiang, G. Wang, ed. (Medical Physics Publishing, 2009).