1 Introduction

Analyzer-based imaging (ABI) is an important X-ray phase-contrast imaging (XPCI) method that has higher contrast and resolution than conventional X-ray absorption-contrast imaging (XACI), particularly for materials made of light elements. Therefore, the ABI method has great potential in medical diagnosis [1,2,3,4,5], biology [6,7,8,9], material science [10, 11], and others. Thanks to progress in synchrotron radiation (SR) sources, ABI has been developed and applied extensively. In parallel with the advances in synchrotron-based ABI methods, there has been a continual development of laboratory source technology [12,13,14,15] that aims to translate the ABI method into practical applications. ABI has the higher signal-to-noise ratio (SNR) [21] than other well-known XPCI methods, such as propagation-based imaging (PBI) [16, 17] and grating interferometry (GI) [18,19,20], especially when high X-ray energies are employed. This can effectively reduce the absorbed dose to the objects and present a big advantage for those applications in which thick or high-Z materials have to be investigated (such as imaging of whole and large human organs) [13, 21]. Furthermore, the effects of beam hardening and scattering in ABI can be negligible due to the monochromaticity of the X-rays and the angular selectivity of the analyzer.

Several types of information, i.e., absorption, refraction, and ultra-small-angle X-ray scattering (USAXS), can be retrieved by ABI using appropriate extraction methods, including diffraction-enhanced imaging (DEI) [22], extended diffraction-enhanced imaging (E-DEI) [23, 24], generalized diffraction-enhanced imaging (G-DEI) [25], multiple-image radiography (MIR) [26,27,28], and Gaussian curve fitting (GCF) [29]. Two images are acquired in DEI and E-DEI to extract absorption and refraction information. Image contrast is poor in samples with a subpixel structure—namely with a strong USAXS signal, such as the lung. This is because no USAXS is considered [30]. Further, G-DEI, MIR, and GCF can extract absorption, refraction, and USAXS simultaneously. Only three images are collected in G-DEI; however, the extraction results are limited by the width of the rocking curve, which is greatly affected by the noise [30, 31]. MIR and GCF are both statistical methods that acquire a set of images by rocking the analyzer; thus, the extraction results are more accurate and have better anti-noise performance. As Gaussian fitting is performed for each pixel, the computational time of the GCF increases significantly [29, 30]. In general, MIR is a better information extraction method, especially when the radiation dose and acquisition time are not critical. The MIR method was separately proposed by Oltulu et al. [26], Wernick et al. [28], and Pagot et al. [27] in 2003, and it has been extensively studied due to its robustness, scattering rejection, and high extraction accuracy. Previously, research and improvements have been carried out on the characteristics of the MIR method. Wernick et al. [32, 33] evaluated the effects of the number of acquired images on the accuracy of extraction results using the Cramér–Rao lower bound (CRLB). Hu et al. [34] extracted three kinds of information using MIR and fused the contrast images in one image, while a comparison of refraction information extraction methods in analyzer-based imaging was carried out [24]. Huang et al. [35, 36] and Zhu and his coworkers [37, 38] discussed the limitations of MIR and proposed some improved information extraction methods. Diemoz et al. [21, 30] compared the extraction results of several ABI information extraction methods in different conditions and theoretically compared the SNR and FOM of three XPCI techniques of ABI, PBI, and GI. Li et al. [31, 39] improved the MIR method to extract more accurate refraction information and compared the refraction angle extraction results of the DEI, G-DEI, and MIR methods with different sampling strategies.

However, MIR has been the subject of extensive research in recent years, particularly when using SR sources. However, many obstacles remain to use conventional X-ray sources. Poisson noise has a significant effect on extraction results due to the low intensity of the source [32, 33]. Here, the selection of the sampling strategies becomes more complicated. In addition, previous reports have focused on refraction information [24, 31]. In fact, absorption, refraction, and USAXS dominate these different cases. Absorption is dominant for samples with high absorption coefficients and/or large dimensions—USAXS is prominent for samples with a subpixel structure, such as lung, powders, and paper.

In this paper, the effects of Poisson noise on extraction results were studied to guide the application of MIR in low-intensity sources (such as X-ray tube sources), so that an optimized radiation dose and imaging time can be employed. The effects of different sampling strategies (including range and interval) on the extraction results are studied while considering the Poisson noise. This provides references for the selection of sampling strategies in practical applications. Furthermore, the mutual effects of the three kinds of information in MIR were analyzed, and its manifestations, causes, and trends are given below. These can be used to optimize information extraction and lay the foundation for correcting MIR extraction results.

2 Principles of the MIR method

A sketch of the ABI system is shown in Fig. 1. It is mainly composed of the monochromator and analyzer crystals. The M1 and M2 are monochromator crystals, which are mostly fabricated of perfect Si or Ge crystals. The X-rays become monochromatic and are collimated after reflecting from M2 according to Bragg’s law. The analyzer is fabricated from the same monochromator crystals and is placed between the sample and detector. The X-rays will be recorded by the detector after being reflected by the analyzer when the X-rays are incident to the analyzer at the Bragg angle (θB) with a narrow range (typically on the order of a few microradians). Normally, the analyzer parallels with the monochromator crystals, and the initial incident angle of the X-ray is equal to θB. The analyzer rocks around the z-axis near θB, and the rocking angle is expressed as θ. Note that clockwise is defined as the “positive” direction. The relationship of reflectivity (R) and θ is the rocking curve (RC). The shape of the intrinsic RC is determined by the crystals and the X-ray energy.

Fig. 1
figure 1

(Color online) Diagram of the ABI setup. Here, (x, y, z) refer to the coordinate system fixed in an imaging system

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The sample/intrinsic RC is obtained by acquiring multiple images (≥ 3) at different analyzer positions (corresponding to θ) with/without sample. Figure 2 shows that the sample RC is reduced, shifted, and broadened compared with the intrinsic RC.

Fig. 2
figure 2

(Color online) The rocking curves of Si(333) at 59.31 keV with (sample RC)/without (intrinsic RC) objects. FWHM is the full width at half maximum of the intrinsic RC and is about 1 μrad. The Δθp and Δθc are the peak and centroid offsets between the two RCs, respectively. Here, Δθc is smaller than Δθp because parts of the sample RC are out of the sampling range

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Information on absorption, refraction, and USAXS can be extracted by comparing the sample RC and the intrinsic RC. For each pixel, the zeroth-, first-, and second-order moments are calculated with the following [27]:

$$ M_{i = 0,1,2} = \sum\limits_{j = 1 \ldots N}^{{}} {(\theta_{j} )^{i} \cdot I(\theta_{j} ),} $$
(1)

where I(θ j ) refers to the X-ray intensity in a pixel for the angular setting θ j , and N is the number of acquired images. The integrated intensity (I0), the center of mass (θ0), and the variance of the RC (\(\sigma_{0}^{2}\)) can be expressed as

$$ \left\{ {\begin{array}{*{20}l} {I_{0} = M_{0} } \hfill \\ {\theta_{0} = M_{1} /M_{0} } \hfill \\ {\sigma_{0}^{2} = M_{2} /M_{0} - \theta_{0}^{2} } \hfill \\ \end{array} ;} \right. $$
(2)

by measuring the intrinsic RC and the sample RC, the information of absorption, refraction, and USAXS can be obtained

$$ \left\{ \begin{aligned} P = \ln (I_{{0,{\text{ref}}}} /I_{{ 0 , {\text{obj}}}} ) \hfill \\ \Delta \theta = \theta_{{0 , {\text{obj}}}} - \theta_{{0 , {\text{ref}}}} \hfill \\ \sigma^{2} = \sigma_{{0 , {\text{obj}}}}^{2} - \sigma_{{0 , {\text{ref}}}}^{2} \hfill \\ \end{aligned} \right.. $$
(3)

Here, P is the projection of the absorption coefficient, Δθ is the refraction angle, and σ2 is the variance of USAXS. Here, (I0,ref, θ0,ref, \(\sigma_{0, ref}^{2}\)) and (I0,obj, θ0,obj, \(\sigma_{0, obj}^{2}\)) correspond to the intrinsic RC and the sample RC, respectively, and can be calculated by Eq. (2).

3 Simulation conditions

The simulation conditions are as follows: The sample used in the simulation experiment is a cylinder—a common structure in biomedical and materials science. The cylinder is made of polystyrene (ρ = 1.06 g/cm3), and its density is similar to soft tissue. The X-ray energy is 59.31 keV, i.e., the Kα1 characteristic energy of tungsten, and tungsten is the most applied target materials. Since the higher-energy X-rays have the stronger penetrating ability, the radiation dose delivered to the sample can be significantly reduced. For instance, a 59.31-keV X-ray passes through 5 cm water with a transmittance of 35%, while only 0.65% of an 18 keV X-rays is transmitted [13]. The absorption coefficient of polystyrene is 0.1982 cm−1, and the refractive index decrement δ is ~ 6.7239 × 10−8 at 59.31 keV. Both the monochromator and the analyzer are Si(333) crystals because of the high angular sensitivity of Si(333) [13, 22]. Figure 2 shows the intrinsic RC of Si(333) at 59.31 keV. Its full width at half maximum (FWHM) is ~ 1 μrad. The transverse and longitudinal sections of the sample are displayed in Fig. 3a, b, respectively. The inside and outside diameters of the cylinder are 5 and 10 mm, respectively, and the height is 10 mm. The pixel unit of the detector is 20 μm × 20 μm.

Fig. 3
figure 3

(Color online) a Transverse and b longitudinal section schematic diagram of the model. Different USAXS levels are defined and indicated as A, B, C, D, and E

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The absorption, refraction, and USAXS are all simultaneously considered in this simulation by assuming the cylinder has only absorption and refraction, while also considering that the levels of USAXS defined by σ2 can be controlled by setting the thickness of paper (no absorption and refraction) behind the sample [28, 40]. Figure 3b shows five regions (12 mm × 2 mm) marked as A, B, C, D, and E that correspond to the five USAXS levels, i.e., σ2 = 0, 0.5, 1, 2, and 5 μrad2, respectively. Poisson noise is also added to study the effects of photon noise [28, 41, 42]. Poisson noise level is negatively related to the number of photons incident on the detector—it can be defined by the average number of photons n irradiating each pixel and can be described in terms of PN(n) when no sample is present. The initial counts of each pixel will fluctuate around n. After n is defined, Poisson noise can be generated by the noise-generated function of poissrnd (n, p, q) in MATLAB. This function can generate Poisson random numbers with mean n, where scalars p and q are the row and column dimensions of the image; each Poisson random number corresponds to one pixel. The effects of Poisson noise become more complicated when the sample is inserted because of the sample’s absorption, refraction, and USAXS of X-rays. Note that each incident photon is assumed to produce one count [12, 21].

4 Results and discussion

4.1 Effects of Poisson noise

Different PN(n)s were used to investigate the effects of Poisson noise on the extraction results of absorption, refraction, and USAXS, where n = 10x and x = 1: 0.1: 4. The sampling range was set as − FWHM to FWHM, i.e., − 1 to 1 μrad, and an interval of 0.1 μrad was adopted. To compare the results extracted under different Poisson noise levels (test images) and noise free (reference image), we define the coefficient of similarity ε as

$$ \varepsilon = \frac{{\sum\nolimits_{i = 1}^{N} {(f_{i} - f_{\text{a}} )(g_{i} - g_{\text{a}} )} }}{{\left[ {\sum\nolimits_{i = 1}^{N} {(f_{i} - f_{\text{a}} )^{2} \sum\nolimits_{i = 1}^{N} {(g_{i} - g_{\text{a}} )^{2} } } } \right]}}, $$
(4)

where f i and g i are the gray values of the ith pixel in the reference image and the test image, respectively. The fa and ga denote the average gray values of the reference image and the test image. The two images are exactly the same when ε = 1.

Figure 4 shows the theoretical refraction angles Δθt of the sample in the simulation experiments. The maximum Δθt, i.e., Δθt_max, is ~ 1.5 μrad. This is observed at the edge of the sample and is comparable to the FWHM (~ 1 μrad). Two different orders of theoretical refraction angles (Δθt and 0.1Δθt) are employed to compare the effects of Poisson noise. The 0.1Δθt_max is smaller than FWHM by one order of magnitude. The extracted contrast images (P, Δθ, and σ2) with different Poisson noise values (PN(10), PN(100), PN(1000), PN(10000), and noise free) when the theoretical refraction angles are set as Δθt and 0.1Δθt are shown in Fig. 5a, b, respectively. The images are degraded and blurred with an increasing Poisson noise. The ε of the images at different levels of Poisson noise is shown in Fig. 6. The degree of image degradation caused by Poisson noise is Δθ > σ2 > P. Obviously, refraction is more sensitive to Poisson noise when the theoretical refraction angles are small (0.1Δθ t ). This agrees with the results presented in Fig. 5.

Fig. 4
figure 4

(Color online) The theoretical refraction angles of the sample in the simulation experiment. The refraction angle is determined by the gradient of the phase shift, which is obvious at the edge of the objects

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Fig. 5
figure 5

Extraction results with different Poisson noise levels. a, b Correspond to the results when the theoretical refraction angles are Δθt and 0.1Δθt, respectively. P is the projection of absorption coefficient, Δθ is the extracted refraction angle, and σ2 is the variance of the USAXS. The extraction results are shown as the images of P, Δθ, and σ2

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Fig. 6
figure 6

(Color online) The relations between the coefficient of similarity, ε, and different Poisson noise levels when the theoretical refraction angles are a Δθt and b 0.1Δθt

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Figure 6 shows that ε approaches 1 when the Poisson noise level is less than PN(1000), indicating that the effects of Poisson noise decrease when the detector counts are more than 1000 ph/pixel. For larger theoretical refraction angles (Δθt), acceptable results can still be extracted at higher Poisson noise levels, e.g., PN(100). As a result, the detector counts should be more than 1000 ph/pixel in practical applications to confirm that the effects of Poisson noise are negligible. Under these circumstances, a photon flux of 2.5 × 106 ph mm−1 is required when the pixel unit of 20 μm × 20 μm is exploited. The required photon flux can be reduced with the same Poisson noise level when a larger pixel unit is used—increasing the pixel unit would also decrease the spatial resolution so that some appropriate trade-offs are seen in each specific case. For the SR sources, the photon flux is generally up to 107 ph s−1 mm−2 [28]. Thus, Poisson noise levels better than PN(10000) are easily implemented. However, the photon flux of conventional, low-intensity, X-ray sources is only about 104 ph s−1 mm−2 [12, 13]; thus, an additional exposure time is needed for this case. In general, counts of 1000 are an ideal reference value. When the theoretical refraction angles are large (equivalent to FWHM), counts of 100 can reduce the radiation dose and exposure time with acceptable image quality (ε > 0.95; Fig. 6a). These results confirm prior conclusions in the literature [28]. Images can still be well identified with a count of 50. Of note, the dark current should be considered when the photon flux is too small—several temperature control modules can be exploited to reduce its impact [12].

4.2 Effects of the sampling strategies

An appropriate sampling strategy (including range and interval) can improve the extraction results and reduce the radiation dose. Here, the sampling strategy is defined in terms of (− A: s: A) and is abbreviated as A_s, which refers to the sampling range of − A to A μrad with an interval of s μrad. Consequently, the number of acquired images N is (2A/s + 1). In our previous study [31], only the refraction information was considered, and the effects of Poisson noise were not considered. Here, the effects of the sampling strategies on extraction results are studied under two Poisson noise levels of PN (10000) and PN(10) while considering all three kinds of information. Three symmetric sampling strategies (1_0.1, 2_0.2, and 1_0.2) were defined by referring to the FWHM. The values of ε were calculated by comparing the theoretical images (reference images) and the extracted images (test images). The extraction results with PN(10000) and PN(10) are shown in Fig. 7a, b, respectively. The values of ε are listed in Table 1.

Fig. 7
figure 7

Comparison of the results extracted by different sampling strategies (2_0.2, 1_0.1, and 1_0.2) at two levels of Poisson noise a PN(10000) and b PN(10). The extraction results are shown as the images of P, Δθ, and σ2

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Table 1 The coefficient of similarity with different sampling strategies and Poisson noise
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At a low Poisson noise (PN(10000)), better extraction results are obtained using a wider sampling range, 2_0.2, and an appropriate decrease in the sampling interval, 1_0.2, barely impacts the extraction results compared with 1_0.1. However, the extraction results are more affected when the wider sampling range 2_0.2 is used at a high Poisson noise level (PN(10)); image degradation is serious, as shown in Fig. 7b. In addition, the ε of 2_0.2 has a more substantial reduction than that of 1_0.1 as shown in Table 1; this agrees with Fig. 7. The deviation and broadening of the sample RC is produced by refraction and USAXS, respectively. Incomplete results will be obtained when a limited sampling range is employed (Fig. 2).

The degree of incompleteness increased with enhanced refraction and USAXS and can generally be reduced by using a wider sampling range. However, since the wider sampling range (2_0.2) acquires data at low points of the sample RC, the number of photons incident on the detector is small due to the low reflectivity at these points. This corresponds to the high Poisson noise. The mass center of the sample RC will be strongly affected by these points because they match with large rocking angles θ. However, for the narrow sampling range (1_0.1 and 1_0.2), more photons are incident on the detector due to the relatively high reflectivity at the sampling points. This implies lower Poisson noise, particularly for the small theoretical values of refraction and USAXS. Of course, the effects of Poisson noise will increase if the number of acquired images, N, is lower (1_0.2). However, the effects of Poisson noise are negligible in the case of the Poisson noise level of PN(10000).

The extraction results of refraction angle Δθ at the edge of the sample (the 51st column) are calculated via two noise levels, PN(10000) and PN(10), with theoretical refraction angles of Δθt and 0.1Δθt. The results extracted by 2_0.2 are better than that of 1_0.1 and 1_0.2 despite the larger data fluctuation when the theoretical refraction angles are Δθt with Δθt_max of 1.5 μrad (Fig. 8). As illustrated in Fig. 9, when the theoretical refraction angles are 0.1Δθt with 0.1Δθt_max of 0.15 μrad—which are smaller than FWHM by one order of magnitude—the values extracted by these three sampling strategies are comparable without USAXS. Meanwhile, 1_0.1 has a better SNR than 2_0.2. Of note, the data fluctuation enlarges with the increase in USAXS, while the extracted values are reduced by USAXS.

Fig. 8
figure 8

The results of Δθ extracted at the edge of the sample using different sampling strategies (2_0.2, 1_0.1, and 1_0.2) at two different Poisson noises: a PN(10000) and b PN(10). The theoretical refraction angles are Δθt

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Fig. 9
figure 9

The results of Δθ extracted at the edge of the sample using different sampling strategies (2_0.2, 1_0.1, and 1_0.2) at two different Poisson noises: a PN(10000), b PN(10). The theoretical refraction angles are 0.1Δθt

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4.3 Interaction of the three kinds of contrast information

Deriving from the principle of the MIR method, the contrast information of absorption, refraction, and USAXS is affected by each other. Some manifestations are in Sects. 4.1 and 4.2; herein, a more systematic study is performed. To study the intrinsic shortcomings of the MIR method, the effects of Poisson noise are not considered. Section 4.2 and the conclusions from the literature [33] suggest a sampling strategy of 1_0.2, where 11 images should be acquired in total.

4.3.1 Refraction

The deviation of sample RC produced by refraction will affect the extraction results within a certain and limited sampling range. To study the effects under different theoretical refraction angles, the parameter of kr = Δθt_max/A characterizes the degree of refraction, where A is 1 μrad at a sampling strategy of 1_0.2. The initial P is calculated based on the simulation conditions mentioned in Sect. 3 with σ2 = 0. The extraction results of P with kr = 1 and kr = 0 are comparable far from the edge of the sample (Δθt is small), but the extraction results of P when kr = 1 are much larger than that when kr = 0 near the edge of the sample (Δθt is large) (Fig. 10a).

Fig. 10
figure 10

(Color online) The effects of refraction on the extraction results of a P, b Δθ, and c σ2 (the 250th row). Here, kr characterizes the degree of refraction

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This will illuminate the edges of the details (Fig. 7a). The cylinder edges are obviously illuminated. The illumination partially enhances the contours of the sample, but the intrinsic image contrast significantly declined. Thus, wrong quantitative information will be extracted, which agrees with the literature [30]. The extraction results of Δθ when k r  = 0.5 and kr = 1 are shown in Fig. 10b. These are marked as Δθ_0.5 and Δθ_1, respectively. At the edge of the sample, Δθ_1 < 2Δθ_0.5, indicating that the accuracy of the Δθ declines with increasing Δθt. Besides, as Fig. 10c shows, several additional USAXS signals are extracted due to the effect of refraction and presented at the edges of the sample as “negative peaks”. This leads to the dark lines in the USAXS image in Fig. 7a.

The results in Fig. 11 show the dependence of the extraction results on kr, wherein the extraction value corresponds to the 51st pixel in Fig. 10. Figure 11 shows that P remains mostly constant until kr ~ 0.5. It then increases with kr. The Δθ increases linearly until kr ~ 0.6. When kr increases further, the Δθ gradually becomes smooth and tends to 1 μrad, which is equal to A (the sampling range of the sampling strategy of 1_0.2). This indicates that the extracted maximum value of Δθ is related to the sampling range. Here, σ2 decreases with kr, and the changes in σ2 are not obvious if kr < 0.15. These results show that as kr increases (denoting the degree of refraction), the effects on the extraction results due to refraction behave as σ2 > P > Δθ within a certain and limited sampling range.

Fig. 11
figure 11

(Color online) The relationships between refraction and the extraction results of P, Δθ, and σ2

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4.3.2 USAXS

The results are affected by the broadening of sample RC caused by USAXS. Analogous to Sect. 4.3.1, the parameter ks = 2.36σ/A is defined, where σ is the standard deviation of USAXS. The initial P is calculated based on the simulation conditions mentioned in Sect. 3. The theoretical refraction angles are set to 0.1Δθt, and in this case, the effects of refraction are negligible. A comparison of the extraction results when ks = 0 and ks = 1 is shown in Fig. 12. The values of P increased overall because of the effect of USAXS, and the degree of the increase depends on the level of USAXS. This will cause layering, contrast reduction, and image distortion in the absorption image (Fig. 7a). In Fig. 12b, USAXS will make Δθ smaller than the theoretical one—especially at the edges of the sample. Figure 7 shows that Δθ decreased significantly with strong USAXS. Figure 12c indicates that the extraction result of σ2 (0.06 μrad2) is less than the theoretical value (0.18 μrad2) that is due to the limitations of the sampling range. The effects of refraction can be neglected if the theoretical refraction angles are 0.1Δθt and correspond to kr = 0.15 (Fig. 12).

Fig. 12
figure 12

(Color online) The effects of USAXS on the extraction results of a P, b Δθ, and c σ2. Here, ks characterizes the degree of USAXS

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The extraction results as a function of ks are shown in Fig. 13, wherein the extraction value corresponds to the 51st pixel in Fig. 12. The results in Fig. 13 show that P increases with ks, and the increment is negligible if ks < 0.5. Here, Δθ decreases slowly when ks < 0.4, and it then obviously decreases with increasing ks. It finally becomes constant at 0 when ks is larger than 3. The σ2 changes inversely as Δθ, and the maximum value extracted, approaches 0.18 μrad2 due to the limitation of the sampling range. We can infer that the USAXS leads to layering and distortion in the P and Δθ image (Fig. 7). The extraction results of σ2 are inaccurate if the USAXS is strong.

Fig. 13
figure 13

(Color online) The relationship between USAXS and the extraction results of P, Δθ, and σ2

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4.3.3 Absorption

Ideally, the sample absorption of X-ray does not affect the extraction results of refraction and USAXS because of the information extraction characteristics of the MIR method. However, samples that have strong absorption of X-rays will cause the Poisson noise level to increase due to the large reduction in the number of photons incident on the detector. Thus, additional exposure is needed. In general, phase-contrast imaging techniques are applied in materials made of light elements, where a low contrast is obtained via absorption-contrast imaging. Here, the absorption is weak—especially when using high X-ray energy (59.31 keV). For instance, only 18% of the X-rays can be absorbed by 10-mm polystyrene with an absorption coefficient of 0.1982 cm−1. As a consequence, the effects of absorption on the extraction results can be somewhat neglected.

5 Conclusion

In this paper, the characteristics of MIR information extraction were studied via a simulation experiment. The results suggest that Poisson noise has significant effects on information extraction. As the Poisson noise increases, the results obviously degraded, and the image blurs. The effects of the Poisson noise on refraction, USAXS, and absorption are sequentially increased. The effects of Poisson noise are insignificant when the Poisson noise level is better than PN(1000). When the theoretical refraction angle is large, the results are slightly affected by Poisson noise. Acceptable results can still be obtained even if the Poisson noise level is as high as PN(100). Here, the requirements of radiation dose and imaging time are reduced.

For SR sources, the effects due to Poisson noise are negligible at high photon flux: The noise level is normally better than (PN(10000)). On the contrary, X-ray tube sources have Poisson noise levels that are high due to their low intensity. Thus, they require additional exposure and better system stability and temperature control. When Poisson noise is negligible (e.g., PN(10000)), the sampling interval barely impacts the information extraction results. More accurate results can be achieved by increasing the sampling range. The interaction of the three kinds of information can also be reduced if a wide sampling range is employed. However, in the case of high Poisson noise level (e.g., PN(10)), the results extracted with a narrow sampling range have better SNR. Consequently, the selection of a sampling strategy should be based on the Poisson noise level and the theoretical value to optimize the extraction results and reduce imaging time.

The three kinds of information will affect each other during extraction. Across a certain and limited sampling range, the refraction will cause bright and dark edges in the absorption image and USAXS image, respectively. There are incomplete results for the refraction image. This will decrease image contrast, and the resulting quantitative results will be incorrect. The sampling range and the degree of refraction determine the refraction effects. The maximum extracted value is related to the sampling range. The effects of the extraction results due to refraction can be neglected when the theoretical value of the refraction angle is smaller than the sampling range by an order of magnitude (e.g., kr = 0.15). The USAXS of X-rays will lead to the layering of the absorption image and a decrease in the image contrast, as well as a reduction in the refraction results with incomplete USAXS results. The effects of USAXS can be ignored when the standard deviation of USAXS σ < 0.4(A/2.36), i.e., ks < 0.4. In general, it is unnecessary to consider the effects of sample absorption on the results of information extraction; however, an appropriate increase in radiation exposure should be performed to reduce the Poisson noise caused by the absorption to X-rays when the sample is made of high-Z elements and/or is large.