1. Introduction

X-ray phase-contrast and dark-field imaging have emerged as promising extensions to conventional, attenuation-based radiography and computed tomography [1–14]. In grating-based phase-contrast imaging, the attenuation, differential phase-shift and dark-field signals are measured simultaneously using three gratings with periods on the micrometer scale. While phase-contrast imaging features superior soft-tissue contrast, the dark-field signal enables the detection of features below the resolution limit of the imaging system [15,16].

In recent years, several potential medical applications of X-ray grating-based dark-field imaging have been explored [17,18], including diagnosis and staging of pulmonary emphysema [19], assessment of osteoporosis, improved differentiation of kidney stones, and enhanced sensitivity and specificity in mammography [20]. Furthermore, the first in-vivo dark-field radiography and computed tomography of mice have been demonstrated. Recently, polychromatic far-field interferometers have been introduced that feature significantly higher sensitivity and could further push grating-based imaging towards clinical feasibility [21,22]. Additionally, there are potential applications of X-ray [23] and neutron dark-field imaging [24] in the field of material science.

Considering a potential future clinical implementation, strict requirements with respect to low radiation doses will have to be satisfied. However, dose reduction in dark-field imaging could prove difficult as it has been shown that the signal cannot be retrieved in the limit of low photon counts [25] when using the conventional signal extraction scheme, the so-called phase-stepping procedure [7]. In this approach, one of the gratings is translated perpendicular to the grating bars while a series of images is recorded. The measured intensities for each detector pixel can be described by

In contrast to conventional attenuation-based imaging, the measurement of the visibility follows a Rician distribution [25]. This is due to the fact that the visibility is calculated as the magnitude of a complex number, whose real and imaginary part are Gaussian distributed [28]. The Rician distribution can only be approximated by a Gaussian distribution in scans with high signal-to-noise ratio (SNR). There, the expectation value of the measured visibility is E(v) = v. For scans with lower statistics, the Rician distribution becomes skew and its mean shifts towards higher values. In the low-SNR limit, it can be approximated by a Rayleigh distribution. In this case, the expectation value of the measured visibility is $E(v)=\sigma \sqrt{\pi }$. The noise level σ is defined as $\sigma ={\sqrt{a_{0}N}}^{-1}$ with N being the number of phase steps.

Note that this expectation value is independent of the visibility v, which is the quantity that we want to measure, and is instead only a function of the noise level. This implies that the transition from Gaussian to Rayleigh distributions, which is present in scans with poor statistics, renders the extraction of a meaningful visibility signal impossible. Consequently, there is a lower limit for the number of photons needed to successfully retrieve the dark-field signal using the conventional phase-stepping technique.

2. Two-shot dark-field imaging

In this article, we investigate a different approach to extract the dark-field and attenuation signals in grating interferometry (GI) using only two phase steps [29]. This technique was inspired by similar approaches for analyser-based imaging (ABI) [30]. ABI and GI are sensitive to the same physical quantities and thus share parallels in some aspect of measurement and signal extraction [31,32].

Technically, this “two-shot” method can only accurately retrieve the dark-field signal in pixels that do not show differential phase-contrast signal. However, the accompanied error is minor for small differential phase-shifts, which is often satisfied for biological samples in typical Talbot-Lau interferometers.

For samples imposing no differential phase-shifts φ, the phase-stepping curve has turning points at x/p = 0 and 1/2. At those working points, the measured intensity is only changed slightly by a phase shift but is influenced most strongly by a change in the visibility, i.e., the dark-field signal. The corresponding grating positions are from here on called x₁ and x₂. A Taylor expansion of the phase stepping curve at the working points x_1,2 results in

Only considering the first order of the expansion, the zeroth and first order Fourier coefficients can be extracted from these two measured intensities by addition or subtraction, respectively. Correspondingly, the attenuation of the sample is given as

The major advantage of our method is the fact that no retrieval of the phase-stepping coefficients through Fourier analysis (complex values) is required. Thus the dark-field values are expected to be Gaussian- rather than Rician-distributed. This would make correct retrieval of the dark-field signal possible even for very short exposure times, i.e., low dose.

In current experimental setups the differential phase of the reference scan is not uniform over the field of view, which is due to grating imperfections and beam divergence. This implies that sampling the stepping curve at the two extrema for each pixel is not feasible by only recording two images. If the stepping curve is not sampled exactly at its minimum and maximum, but at offsets Δx₁ and Δx₂ from these points, the intensities I′₁ and I′₂ are measured (cf. Fig. 1). Then, additional corrections have to be applied to extract the correct visibility and attenuation signal. In order to obtain the corrected values, the measured values a′₀ and a′₁ have to be subtracted and divided by

 

Fig. 1 Principle of the two-shot dark-field acquisition scheme. The reference stepping curve is recorded with high statistics and a sufficient number of phase steps. The dark-field signal is directly retrieved from the two measured intensities I_1,2, rather than by performing a fitting procedure.

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3. Experimental results

In the following, the performance of the two-shot dark-field acquisition and the conventional dark-field retrieval technique are compared on an experimental basis. The Talbot-Lau interferometer used for this measurement consists of three gold-plated silicon gratings with periods of 5.4μm. The second grating is designed to induce a phase shift of π for X-rays of 27keV. A rotating anode tube operated at 40kV and 70mA was used to generate X-rays. The X-rays were detected by a Dectris PILATUS II 100k single-photon counting detector with a 1mm thick silicon sensor and 487 × 195 pixels. The pixel pitch was 172 × 172 μm². Additional details about the experimental setup can be found elsewhere [33].

3.1. Radiography of a scattering sample

We measured a sample made of layers of common printer paper using a range of different exposure times. The average visibility in the reference projections was 30.3%. The phase-stepping procedure was used to obtain phase-stepping curves with 11 phase steps. The exposure time per phase step ranged from 0.1ms to 26.2s. In a sample free area, this equates to around 0.5 or 120000 photons per pixel or counts, respectively. A weighted least-squares fit was used to obtain the three image signals. Additionally, the novel two-shot method was used to obtain the attenuation and the dark-field signal. For this analysis, the reference scan was used to select the two steps closest to the turning points of the stepping curve for each pixel separately. The reason for this procedure is the aforementioned non-uniformity of the reference phase. In addition, Eqs. (5) and (6) were used to correct the retrieved values.

The results of the experiment are displayed in Fig. 2. The scan with the highest statistics, i.e., longest exposure time, is shown in the left column. The transmission and dark-field images that were retrieved using the phase-stepping procedure with subsequent least-squares fitting are shown in the top rows. In the high statistics scan, the five different areas of the sample show distinct values of the dark-field signal. In the low statistics scan, however, the retrieved dark-field signals are higher than the values measured previously. Further, the dark-field signals in the areas of 11, 5 and 3 sheets of paper are of similar value, rendering a distinction of the different material thickness impossible.

 

Fig. 2 Transmission and dark-field projections of air and stacks of paper (number of layers: 1, 3, 5, and 11) with high and low SNR obtained using the conventional and the novel two-shot method. Using the phase-stepping approach, the dark-field values rise in all areas of the sample when the exposure time is decreased. In comparison, when using the two-shot method the correct values are maintained even for low statistics. The mean count numbers refer to the number of photons per pixel and were measured in a sample free area.

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The dark-field projections that were obtained with the proposed two-shot method are shown in the bottom row. Here, the average dark-field signal does not deviate from the correct value when the exposure time is decreased. The corresponding differential phase-contrast projection is shown in appendix B.

The mean dark-field values for three regions of interest in the sample are visualized in dependence on exposure time, respectively noise level, in Fig. 3. Again, it can clearly be seen that the retrieved dark-field signal deviates from the correct mean for the conventional signal extraction method. In contrast, the proposed method is able to correctly retrieve the dark-field signal even for scans with very high noise levels.

 

Fig. 3 Dependency of the measured mean dark-field values on the noise levels in the attenuation channel for the three ROIs marked in Fig. 2. In the case of least-squares fitting (lsq-fit), which is the conventional signal extraction method, the measured values deviate from the correct values for high noise levels, indicating a failure of signal retrieval. In contrast, using the novel two-shot approach the image results remain still consistent for highly increased noise levels.

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3.2. Radiography of a strongly absorbing, non-scattering sample

We also successfully applied the method to a highly absorbing sample. We measured a sample consisting of aluminum blocks with thicknesses of 1.95mm, 4.95mm and 8.95mm and exposure times ranging from 0.1ms to 26.2s at close to 5000 photons per second per pixel in a sample-free area.

Generally, the dark-field image arises from (ultra) small-angle scattering at microstructures of the investigated material, however it may also be related to so-called visibility hardening. Here, the absorption of low energy photons leads to a change in overall visibility, i.e. a generic dark-field [34,35]. To compensate for this effect, the dark-field signal of all materials was normalized to unity using the appropriate values from the high statistics scans.

The results of the experiment are displayed in Fig. 4. The scan with the highest statistics, i.e., longest exposure time, is shown in the left column. The transmission and dark-field images that were retrieved using the phase-stepping procedure with subsequent least-squares fitting are shown in the top rows. The average transmission in the indicated regions of interest was unity for air and 0.08 for 8.95mm of aluminum in all scans.

 

Fig. 4 Transmission and dark-field projections of air and aluminum (thickness 1.95mm, 4.95mm and 8.95mm) with high and low SNR obtained using the conventional and the novel two-shot method. Visibility hardening was corrected, which results in a mean dark-field of unity for all materials in the high-SNR scan. Using the phase-stepping approach, regions with lower transmission exhibit a dark-field signal exceeding the correct value of unity with decreasing exposure time. In comparison, when using the two-shot method the correct value is maintained even for low staticsts. The dark-field images were scaled from [0 − 2σ, 2+2σ], with σ being the standard deviation in a region of interest in the dark-field projection of 8.95mm aluminum.

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In the high statistics scan, the dark-field signal is unity for all four uniform parts of the sample after correction for visibility hardening. In the low statistics scan, the retrieved dark-field signal is higher than unity for all materials. Since this effect depends on the attenuation of the sample, it is most strongly visible in the region of the thickest aluminum plate. The dark-field projections that were obtained with the proposed two-shot method are shown in the bottom row. Here, the average dark-field signal does not deviate from the correct value when the exposure time is decreased.

The mean dark-field values for air and aluminum in dependence on exposure time, respectively noise level, are visualized in Fig. 5. Again, it can clearly be seen that the retrieved dark-field signal deviates from the correct mean for the conventional signal extraction method. In contrast, the proposed method is able to correctly retrieve the dark-field signal even for scans with very high noise levels.

 

Fig. 5 Dependency of the measured mean dark-field values on the noise levels in the attenuation channel for two materials with vastly differing attenuation properties. In the case of least-squares fitting, which is the conventional signal extraction method, the measured values deviate from the expected value of unity for high noise levels, indicating a failure of signal retrieval. Using the novel two-shot approach, the image results remain still consistent for highly increased noise levels. The mean values were measured in the regions of interest as displayed in Fig. 4

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3.3. Tomographic two-shot dark-field imaging

Unlike in radiographic applications where only a few exposures are taken, the applied radiation dose has to be split up between many exposures in tomographic imaging. Thus, the aforementioned problem of failing signal retrieval is of particular relevance there.

In the following, we will demonstrate that the two-shot method is also applicable to tomographic imaging. A foam ear plug with a diameter of roughly 10mm was measured in a tomographic scan with 2400 projections, each comprising 11 phase steps. Additionally, a reference projection was recorded every 15 projections. To avoid ring artifacts, the sample was shifted randomly in the range of 10 pixels in x- and y-direction after each projection.

Exemplary transmission and differential phase-contrast projections of this sample are shown in Fig. 6. The transmission of the sample is close to unity due to its weak absorption. The sample’s differential phase-contrast signal is centred around zero and shows substantial image noise. This is due to the fact that the corresponding dark-field signal is very strong which leads to high image noise in the differential phase-contrast projections [36]. The number of photons per pixel were around 96800 for this high statistics scan, which corresponds to a total exposure time of 33s, i.e. 3s per phase step. The average flux at one detector pixel was roughly 2933 photons per second.

 

Fig. 6 Transmission, differential phase-contrast and dark-field projections of the foam ear plug. The attenuation of the sample is very weak, leading to transmission values close to unity. The sample has a very noisy differential phase-contrast signal, due to its strong scattering (dark-field) signal. The arrow marks the rotation axis of the tomographic measurement.

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A statistical iterative reconstruction (SIR) algorithm was used to reconstruct maps of the linear attenuation coefficient and the dark-field signal [27, 37–39]. We assumed a Gaussian distribution for the dark-field signal for both the two-shot and the least-squares method. This assumption was incorporated into the data model of the SIR. While a Rician instead of the Gaussian distribution could lead to improved reconstructions, it would imply performing a joint iterative reconstruction of the attenuation and dark-field signals, which is beyond the scope of this work. A Huber potential function was used for regularization. For better comparability, the regularization strength was chosen to be rather weak and kept constant for all reconstructions.

The results of the tomographic reconstruction are displayed in Fig. 7. The inner structure of the foam plug is clearly visible in the high-SNR reconstructions of both techniques and their quantitative values are comparable. However, the two methods respond differently to a decrease in exposure time. In the phase-stepping approach, the signal of the ear plug starts to vanish already at counts of around 300 photons per pixel and projection, which corresponds to an exposure time of around 0.1s. When further reducing the number of counts, the ear plug is no longer distinguishable from the background. This can be explained by similar noise levels in all areas of the projection as the foam sample is barely absorbing (see Fig. 6). Since the extracted dark-field signal is only dependent on the noise level in the low-SNR case, the projections and hence the reconstructions do not contain any information about the true dark-field signal of the sample any more.

 

Fig. 7 Axial slices of the reconstructed dark-field tomography of a foam earplug. The count numbers correspond to one projection and pixel. The reconstructions of the phase-stepping projections show a vanishing signal for scans with low counts. In contrast, the two-shot method is able to visualize the sample even in low-SNR scenarios.

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In contrast, the signal does not vanish in the reconstructions of the two-shot method when reducing the exposure time. The sample is still visible in the scan with the shortest exposure time of 6ms, which corresponds to only 18 photons per pixel and projection. Even though the noise level is increased significantly, it is still possible to distinguish a difference in signal intensity between the inner and the outer part of the sample.

3.4. Dose-optimized two-shot dark-field tomography using projection weights

Up to this point, we always recorded a complete phase-stepping dataset and retrospectively selected the two optimal phase steps for each pixel separately. We then used these steps for signal extraction in the two-shot method. Consequently, the total required dose for imaging is significantly higher than needed, since a majority of the acquired phase steps is not used. Clearly, this is not the optimal procedure although we have shown that it still can be useful to extract the dark-field signal in cases where the conventional retrieval algorithms fail.

In the following, we propose a dose optimized variant of the aforementioned approach where only a total of two phase steps are recorded. These are spaced roughly half the grating period apart. Using Eqs. (5) and (6), accurate retrieval of the dark-field signal is possible with these two phase steps. However, in pixels where the two measured points lie too close to the zero crossings of the phase-stepping curve, the retrieved values will be dominated by noise.

We re-evaluate the data from the tomographic experiment described previously, this time without selecting the optimal phase steps for each pixel separately. Instead, we globally select the 1^st and 6^th step of the 11 phase steps that were recorded, i.e., the same phase steps for each pixel. The method is first demonstrated for the scan with the highest statistics with an exposure time of 3 seconds per phase step, i.e., 6 seconds per projection. It can be seen from the dark-field projection shown in Fig. 8(a) that the signal retrieval works well for most pixels. Note that the pixels for which the signal extraction failed form a pattern that matches the shape of the phase in the reference scan.

 

Fig. 8 Results of two-shot dark-field tomography with globally fixed steps. (a) two-shot dark-field projection showing areas of unsuccessful signal extraction, (b) corresponding weights used for reconstruction, (c) reconstruction without weights showing severe artifacts, (d) artifact-free reconstruction of the same projections using the weights shown above, (e) successful reconstruction of low-SNR scan (59 counts) using weights.

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The reference phase can change over the course of a long scan due to setup instabilities and temperature changes. These changes introduce artefacts in conventional phase-stepping acquisitions, which is why reference projections are recorded in a regular interval between projections. The reference projection recorded closest in time to the sample projection is used for a reference correction to achieve artefact free images. For the two-shot dark-field method, we use the same reference projections to calculate the projection weights. No additional reference projections have to be recorded compared to a conventional phase-stepping acquisition.

The reconstructions of the two-shot dark-field projections are shown in Fig. 8(c). The noisy pixels present in the projection result in severe artifacts in the reconstructed slice. However, the overall shape and the quantitative values match the reconstructed slice of the phase-stepping projections shown in Fig. 7.

With the goal of mitigating the apparent reconstruction artifacts, the iterative reconstruction was combined with pixel-wise weights that regulate the influence of each pixel on the reconstruction result. These weights contain information about the accuracy of the two-shot retrieval in each pixel, which depends on the reference phase of each pixel. In particular, the weights are set as the square of the correction factor c₁.

The weights determined from the projection shown in Fig. 8(a) are presented in Fig. 8(b). The regions where the weights have the lowest values correspond to the regions where the signal retrieval failed. These weights are used for a reconstruction using the SIR algorithm with the same parameters as described earlier. The result of this reconstruction is shown in Fig. 8(d). It is apparent that this reconstruction is far superior compared to the one without weights (cf. Fig. 8(c)) and comparable to the reconstructions shown in Fig. 7. We also applied this approach to a low-dose scan and a successful reconstruction is shown in Fig. 8(e). Finally, note that the reconstruction benefits from a random translation of the sample between projections, since pixels are moved over both good and poor weights.

The iterative tomographic reconstruction of the two-shot dark-field acquisition with fixed steps relies on the projection weights. These weights vary considerably throughout the image. Therefore it is important to inspect the reconstruction quality in the complete volume. A variety of slices through the volume is displayed in Fig. 9. The panels (a–f) show the high statistics scan also displayed in Fig. 8(d). The bottom row of Fig. 9 shows the low statistics scan, cf. Fig. 8(e). Note that Figs. 9(d)–9(f) and Figs. 9(g)–9(i) show the same coronal slices. It is evident from these results that the image quality is consistent over the whole volume.

 

Fig. 9 Additional slices of the two-shot dark-field tomography with globally fixed steps shown in Fig. 8(d–e). The top two rows (a–f) show the high statistics scan that was previously displayed in Fig. 8(d). The bottom row shows the low statistics scan (59 photons per pixel and projection) of Fig. 8(e). All slices through the volumes are 50 pixels apart. The slices are in (a–c) axial and (d–i) coronal direction. The image quality is consistent in all slices, which illustrates the effectiveness of the weighted reconstruction scheme.

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4. Conclusion

In conclusion, we have demonstrated an alternative procedure to acquire the dark-field signal in X-ray grating-based phase-contrast radiography. We have further shown that it is possible to successfully reconstruct tomographic dark-field scans from only two phase steps by combining the two-shot procedure with an appropriately weighted iterative reconstruction. We demonstrated that the noise properties of this algorithm are superior to the conventional approach for scans with low photon counts. Thus, it enables the correct retrieval of the dark-field signal even for scans with very low photon counts, where the conventional signal extraction procedure fails. The disadvantages of this method are its inability to simultaneously extract the phase-contrast signal as well as its inapplicability to samples with a strong differential phase shift. Since the current medical investigations of grating-based imaging focus on the dark-field signal (e.g. lung imaging, mammography, kidney stones) we consider this technique a milestone towards the clinical feasibility of grating-based imaging.

Appendix A: Derivation of the correction parameters c₀ and c₁

The correction factors are used when the stepping curve is not sampled exactly its turning points. In this case, the measured intensities are denoted as I′₁ and I′₂ and the extracted Fourier coefficients as a′₀ and a′₁. The correction factors derived below can be used to relate the measured coefficients a′_0,1 with the correct coefficients a₀ and a₁.

(7)$$a_1^{\prime }=\frac{1}{2}\left(I_1^{\prime }-I_2^{\prime }\right)=\frac{1}{2}\left\{\left[a_0+a_1\text{cos}(2\pi \mathrm{\Delta }{x}_1)\right]-\left[a_0-a_1\text{cos}(2\pi \mathrm{\Delta }{x}_2)\right]\right\}=$$

(8)$$=\frac{a_1}{2}\left[\text{cos}(2\pi \mathrm{\Delta }{x}_1)+\text{cos}(2\pi \mathrm{\Delta }{x}_2)\right]\equiv a_1{c}_1$$

(9)$$a_0^{\prime }=\frac{1}{2}\left(I_1^{\prime }+I_2^{\prime }\right)=\frac{1}{2}\left\{\left[a_0+a_1\text{cos}(2\pi \mathrm{\Delta }{x}_1)\right]+\left[a_0-a_1\text{cos}(2\pi \mathrm{\Delta }{x}_2)\right]\right\}=$$

(10)$$=a_0+\frac{a_1}{2}\left[\text{cos}(2\pi \mathrm{\Delta }{x}_1)-\text{cos}(2\pi \mathrm{\Delta }{x}_2)\right]\equiv a_0+c_0$$

Appendix B: Differential phase-contrast projection of the paper sample

Figure 10 shows a differential phase-contrast projection of the paper sample that is described in the main article. Even though paper is refracting and scattering, the values of the differential phase-contrast signal are small and centered around zero. This is an important prerequisite for the applicability of the two-shot dark-field acquisition technique. In the area in the middle of the top part of the projection (11 sheets of paper, cf. Fig. 2), the differential phase-contrast signal shows higher image noise. This is a result of the decreased visibility due to scattering (i.e. dark-field signal) in this region.

 

Fig. 10 Differential phase-contrast projection of the paper sample, obtained using the phase-stepping approach with subsequent least-squares fitting and an exposure time of 26.2s.

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Funding

European Research Council (ERC) (H2020, AdG 695045); German Research Foundation (DFG) Cluster of Excellence Munich-Centre for Advanced Photonics (MAP); German Research Foundation (DFG) Gottfried Wilhelm Leibniz program.

Acknowledgments

We acknowledge the support of the TUM Institute for Advanced Study, funded by the German Excellence Initiative. This work was carried out with the support of the Karlsruhe Nano Micro Facility (KNMF, www.kit.edu/knmf), a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT).

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