1. Introduction

Grating-based x-ray phase contrast imaging (GPCI) has been developed as a promising technology in x-ray phase contrast imaging field. Considering that the traditional x-ray imaging is based on the attenuation index, which is three orders of magnitude lower than the refractive index decrement for materials comprised of light (low-Z) elements like soft tissues under the low energy range of x-ray (15-60 keV) [1], GPCI utilizing the phase shift as the contrast has been proposed and developed rapidly in the last decade. Comparing with other x-ray phase contrast imaging methods such as crystal interferometry [2–4], analyzer-based imaging [5,6] and the propagation-based imaging [7,8], GPCI [9,10] and edge-illumination imaging [11,12] can overcome the strict limitations by the coherence of x-ray sources, and therefore can be implemented using a conventional x-ray source nowadays, which makes it possible for wide applications. In addition to the attenuation contrast (ATC) and the differential phase-contrast (DPC), GPCI can obtain the dark-field contrast (DFC) simultaneously [10,13], which delivers the small-angle x-ray scattering (SAXS) information of the sample on sub-pixel scales. Therefore, GPCI is becoming an advanced tool for characterization of materials and has many applications in studies of a material porosity [14,15] and orientation distribution [16], while the latter is further extended by the vector radiography [17] and tensor tomography [18,19]. Recently, GPCI also shows great potentials for clinical diagnosis such as lung imaging [20–22] and breast imaging [23–25].

From the perspective of imaging principles, GPCI systems can be categorized into two types: the coherent systems (such as the Talbot [26] and Talbot-Lau interferometry [9]), and the incoherent systems (such as the geometrical projection system [27]). The principle of all GPCI systems is to measure the subtle difference due to the refraction and scattering caused by the sample. In practical imaging configuration of all GPCI systems, the phase-stepping strategy is the most commonly used approach to acquire the periodic pattern of the intensity signal at each pixel. As illustrated in Fig. 1, in the phase-stepping approach, one of the gratings (usually the last grating G2) is moved along the transverse direction perpendicular to the grating lines step by step over one grating period and the detector acquires an image at each step to obtain a so-called phase-stepping curve (PSC) at each pixel. After two PSCs with and without the sample object are acquired, the multiple contrasts such as ATC, DPC and DFC can be retrieved using an information retrieval algorithm, which is a critical data processing and has been actively investigated in the field of GPCI.

 

Fig. 1. The schematic diagram of three-gratings GPCI systems like the Talbot-Lau interferometry and the geometrical-projection system (a) and its phase-stepping curves acquired with (the sample PSC) and without (the flat-field PSC) the specimen in a pixel (b).

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So far, there are mainly two types of information retrieval methods in GPCI. One is the traditional Fourier component analysis (FCA), which is usually recognized as a gold standard and most commonly used since the beginning of GPCI [9,28]. According to wave optical derivations, PSCs in GPCI can be considered as first-order cosine functions [9,10,26,29], and therefore, the contrasts can be calculated from PSCs’ Fourier coefficients. The other is the developing multi-order moment analysis, referred to as MMA in this paper. Totally different from FCA, Modregger et al. regarded the typical contrasts as moments of the underlying SAXS distribution [30]. The physical foundation of MMA is the well-established convolution relationship that the sample PSC can be considered as the convolution of the flat-field PSC (without the sample) and the SAXS distribution [31]. Based on the convolution relationship, the original MMA-type method obtains the underlying SAXS distribution by the Lucy-Richardson deconvolution, thus referred to as deconvolution-based MMA or DB-MMA in this paper. Recently, Modregger et al. proposed a direct MMA form (referred as D-MMA) for edge-illumination imaging, which provides almost equivalent contrasts with DB-MMA while speeding up data analysis by almost three orders of magnitude [32]. Unfortunately, it is not directly applicable to GPCI, whose PSCs have the cosine nature, not satisfying the linear convolution assumption in D-MMA. After some data pre-processing operations, D-MMA can be extended to GPCI and used under some conditions [33].

From all these reported results, it can be seen that the moments calculated by MMA-type methods are highly similar to the typical contrasts retrieved by FCA, in spite of totally different theoretical foundations. So is there any inherent connection between FCA and MMA? In the literature, Li et. al utilized an specific cosine assumption of the SAXS distribution to explore an answer of this question, and found that the first two orders of moments retrieved by MMA depend on both DPC and DFC signals [34]. The motivation of our work is also to answer the question further in theory, and clarify the quantitative connection between FCA and MMA.

2. Methods

2.1 Two main information retrieval approaches: FCA and MMA

As introduced in the previous section, there are mainly two types of information retrieval approaches in GPCI, i.e. FCA and MMA. FCA is established on the cosine model for both the flat-field PSC and the sample PSC [9,10,28], i.e.,

On the other hand, MMA is based on the well-established convolution relationship that the sample PSC can be considered as the convolution of the flat-field PSC and the underlying SAXS distribution [31], i.e.,

To calculate the moments of the scattering distribution $g(\phi )$, one usually adopt the DB-MMA method, and firstly obtain the $g(\phi )$ by an iterative deconvolution, where the deconvolution method and the stopping criterion are needed to be pre-determined. Compared with FCA, MMA extended the number of complementary contrasts from three to potentially as many as needed, and some higher order moments like the skewness $\tilde {M}_3(g)$ [30] and the kurtosis ($\tilde {M}_4/\tilde {M}_2^2$) [38] has been demonstrated to be useful in practice. However, from our experience, the higher order moments for most samples we scanned provide little extra information when compared with the first two moments, unlike the powder materials whose higher order moments can be quite different as they feature a significant spread of cross-sectional particle sizes. Moreover, the iterative process is usually difficult to reach convergence and has to be stopped at fixed point empirically determined, which may bring uncertainties in the retrieved contrasts for wide applications. And the computationally time-consuming deconvolution process makes DB-MMA less-favorable for time-sensitive applications.

2.2 Fourier-based interpretation of the multi-order moment analysis

The physical model of MMA is the particle transport theory and directly relative to the SAXS distribution of the imaging object, while that of FCA is the cosine model derived from wave optics and directly dependent on the acquired PSCs. Therefore, before presenting a Fourier-based interpretation of MMA, we attempt to establish an analytic form of MMA with the PSCs, rather than the SAXS distribution.

According to the convolution relationship in Eq. (6) and the convolution theorem, one can get the equivalent multiplication equation, i.e.,

In GPCI, both PSCs are $2\pi$-periodic functions and therefore, both PSCs and the SAXS distribution can be written with their Fourier series in $\phi \in [-\pi , \pi ]$, i.e.,

Taking Eq. (14) and Eq. (15) into the "positive" and "negative" forms of Eq. (10), one can obtain the relationship between these Fourier coefficients as,

By adding and subtracting Eq. (16) and Eq. (17), one can obtain the $n$-th order Fourier coefficients of the SAXS distribution as:

To further give the Fourier-based interpretation of MMA, one can express the multi-order moments of $g(\phi )$ with its Fourier coefficients. First, the raw $m$-th order moment of $g(\phi )$ can be written as:

Also, the $m$-th order power function, $\phi ^m$, can be expanded with Fourier series in $\phi \in [-\pi , \pi ]$, i.e.,

Also, $M_m(s)$ and $M_m(f)$ have same expressions with Eq. (22). It can be seen that the raw $m$-th order moment of a function is actually the sum of its Fourier coefficients multiplied with constant Fourier coefficients of the $m$-th order power function, which can be also derived by the Parseval’s theorem. The $m$-th order normalized and centralized moment can be calculated from the $m$-th and lower order raw moments, such as $\bar {M}_1(g)=M_1(g)/M_0(g)$ in Eq. (8) and $\tilde {M}_2(g) = M_2(g)/M_0(g) - \bar {M}_1^2(g)$ in Eq. (9).

Moreover, it is known that all odd order power functions are odd functions, and all even order ones are even functions, which means all odd order moments of $g(\phi )$ are the sum of $\beta _n^m b_n^g$, while all even order moments are the sum of $\alpha _n^m a_n^g$. For instance, Fourier series for the first four power functions are given as:

Taking Eqs. (18)–(19) and Eqs. (23)–(26) into Eq. (22), one can obtain the analytic forms of first three moments defined in Eqs. (7)–(9) of DB-MMA, i.e.,

Until here, Eqs. (27)–(29) establish the Fourier-based interpretation of the typical three moments retrieved by MMA. For simplicity, we put the corresponding third and fourth order expressions into the appendix, and other higher order moments can be written with similar derivations if needed. However, one cannot calculate infinite terms in Eqs. (28) and (29), and hence it should be truncated at the maximum order $N_{\textrm {max}}$ in practice. So how many terms do we need to calculate? It depends on the shape of PSCs. For instance, for Gaussian-shaped PSCs in analyzer-based imaging and edge illumination, one need to choose $N_{\textrm {max}}$ such that it is a few times larger than the number of points in the full width at half maximum (FWHM) of the flat-field PSC, which is usually smaller than $10$. Due to GPCI being the focus in this work, its truncated form is discussed in detail in Section 2.3.

From Eqs. (27)–(29), it can be seen that multi-order moments calculated by MMA are actually weighted compositions of Fourier coefficients of both PSCs, which means MMA delivers similar physical information as FCA. To compare the contrasts retrieved by FCA and MMA more clearly, one can transform the sine-cosine form of Fourier series in Eqs. (11)–(13) to the amplitude-phase form like Eqs. (1)–(2), taking the sample PSC as an example,

Comparing Eq. (3) and Eq. (31), it can be seen that the ATC retrieved by FCA is exactly the negative logarithm $M_0(g)$ calculated by MMA. From Eq. (32) and Eq. (33), it is straightforward to find that $\bar {M}_1(g)$ and $\tilde {M}_2(g)$ retrieved by MMA are the combinations of the $n$-th order of phase-shift ($\varphi _n=\varphi _n^s-\varphi _n^f$) and visibility reduction ($V_n=V_n^s/V_n^f$), whose first-order forms are just the main part of DPC and DFC retrieved by FCA in Eq. (4) and Eq. (5). With similar derivations, one can imagine that higher order moments of the SAXS distribution are also the combinations of these Fourier components, which are omitted for simplicity.

Specifically for GPCI, it is supported by many literature [30,39,40] that the typical contrasts (ATC, DPC and DFC) retrieved by FCA and ($M_0$, $\bar {M}_1$, and $\tilde {M}_2$) retrieved by DB-MMA are very similar, but there still exist some differences between them. To answer the question, we carried out further analyses in the following subsection.

2.3 Truncated analytic multi-order moment analysis (TA-MMA)

Based on the wave propagation derivations in [29], the $n$-th order visibility of the flat-field PSC in a Talbot system (without the first $G_0$ grating) can be written as:

 

Fig. 2. (a) The $n$-th order visibility of the flat-field PSC as a function of order $n$ and demagnified source size in a Talbot system with a Gaussian shaped x-ray source. (b) The histogram curves of the first-order and third-order amplitudes of the sample PSC and the flat-field PSC in a experimental data acquired at a Talbot system, which indicates the first-order amplitudes are much greater than the third-order ones.

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For Talbot-Lau systems with a large source size, a source grating $G_0$ is introduced to produce interference patterns, and one can obtain the similar conclusion by replacing the Gaussian-shaped source with the squared-shaped source, which supports that the visibility $V_n^f$ also decays rapidly with the increasing order $n$. This decay law is the main theoretical basis of FCA, using a first-order cosine model in Eq. (1) to describe the acquired flat-field PSC in Eq. (12). Moreover, the corresponding visibility $V_n^s$ of the sample PSC is usually smaller than $V_n^f$, and therefore it decays rapidly, such that the sample PSC in Eq. (11) can be also modeled with Eq. (2).

Although the $n$-th order visibilities $V_n^f$ and $V_n^s$ ($n\geq 2$) are much smaller than the first order visibility $V_1^f$ and $V_1^s$, the visibility reduction term, $V_n^s/V_n^f$, cannot be determined due to the divison of two small quantities. In order to evaluate the visibility reduction $V_n^s/V_n^f$ as a function of $n$, in general, the SAXS distribution can be approximated as a Gaussian distribution [31,41], i.e.,

According to Eq. (36), for $n=1$, Wang et al. established the quantitative relationship (i.e. Eq. (5)) between the second moment of the SAXS distribution and the visibility reduction of the PSCs for Gaussian scatters in [31]. From Eq. (36), it can also be seen that the visibility reduction decays exponentially as the increasing $n^2$. Figure 2(b) shows the histogram curves of the first-order and third-order amplitudes of the sample PSC and the flat-field PSC in a experimental data acquired at a Talbot system, which indicates that the first-order amplitudes are much greater than the third-order ones. Moreover, due to the noises in the detected signal, the third and higher orders of amplitudes are likely close to their noise levels, and therefore we can obtain very little useful information from these orders of amplitude images in GPCI.

With the analytic MMA described in Eqs. (27)–(29) and our specific analyses for GPCI above, one can obtain a truncated analytic MMA form as,

Equations (27), (37) and (38) constitute a stable, efficient and analytic information retrieval method (i.e. TA-MMA) that can obtain multi-order moments of the SAXS distribution directly from raw PSCs, without the deconvolution process in DB-MMA, leading to hundreds of times faster in computation than the original DB-MMA. It is worth noting that $\bar {M}_1^{\textrm {TA}}(g)$ and $\tilde {M}_2^{\textrm {TA}}(g)$ are only involved of zero-order and first-order components in Eqs. (25) and (26), and can be also expressed with the amplitude-phase form to connect contrasts retrieved by FCA and MMA,

From Eqs. (39) and (40), it can be seen that both $\bar {M}_1^{\textrm {TA}}(g)$ and $\tilde {M}_2^{\textrm {TA}}(g)$ are hybrid representations of the original DPC ($\varphi _1^s-\varphi _1^f$) and DFC ($V_1^s/V_1^f$) retrieved by FCA. More interestingly, $\bar {M}_1^{TA}$ is just the product of DPC and DFC under the small angle assumption (i.e. $\sin (\varphi _1^s-\varphi _1^f) \approx \varphi _1^s-\varphi _1^f$), and therefore $\bar {M}_1^{TA}$ can be considered as a new physical fused contrast, which is called as the hybrid-field contrast (HFC) in this work. Compared with other existing image fusion methods in GPCI, our proposed HFC combines the DPC and DFC in a natural physical way, rather than in an image processing point of view such as [42–46]. Another advantage of HFC compared with the original DPC is to reduce some phase-wrapping artifacts, which can be explained by the operation $\sin (\varphi _1^s-\varphi _1^f)$ to smooth the phase shift $(\varphi _1^s-\varphi _1^f)$ when it jumps around $\pi$ or $-\pi$. As shown in Fig. 3, the schematic diagrams of FCA, DB-MMA and our proposed TA-MMA reflect the advantages of TA-MMA both in higher computation efficiency compared with DB-MMA, more compact image representation compared with FCA, and clearer physical relationship compared with other image fusion methods.

 

Fig. 3. The schematic diagrams of the conventional FCA, DB-MMA and our proposed TA-MMA.

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As we know, there have been many studies conducted to retrieve the real space correlation function of the sample in the last decade, including changing the coherence length by moving the sample along the optical axis [47–50], changing the incident spectrum or detecting energy ranges [51], and changing the optical elements [52] or setups [53]. These studies help us understand the so-called dark-field signal from physical and optical aspects, and guide us how to utilize the dark-field scattering signal to distinguish different samples or materials under different conditions. Our study shows that it is possible to retrieve more detailed information for a sample scanned under a fixed condition with a fixed coherence length or a fixed system setup, and therefore if needed it can be integrated with these studies to further combines the reciprocal and real space information of the sample.

2.4 Noise analysis of multi-order moments retrieved by TA-MMA

The noise behavior of multi-contrasts retrieved by FCA in GPCI has been demonstrated in many studies [54–56]. For a general cosine function, i.e. $f(x) = A_0 + A_1 \cos (x+\varphi )$, the covariance matrix of the parameter vector $(A_0, A_1, \varphi )$ can be given as

According to the Fourier-based expressions of moments retrieved by TA-MMA in Eq. (39) and Eq. (40), one can find their main parts consist of $V_1\sin \varphi _1$ and $V_1\cos \varphi _1$. Therefore, we can firstly calculate the Jacobian matrix for the transformation $t: (\varphi _1, V_1) \rightarrow (V_1\sin \varphi _1, V_1\cos \varphi _1)$ as:

With the help of this Jacobian matrix and the error propagation formula, we can calculate the covariance matrix for $(V_1\sin \varphi _1, V_1\cos \varphi _1)$, i.e.,

Hence, the covariance matrix is calculated as:

From Eq. (46), it turns out that the noise variances of multi-order moments calculated by TA-MMA are also weighted compositions of noise variances of DPC and DFC calculated by FCA. Taking the HFC (i.e., $\bar {M}_1^{\textrm {TA}}$) as an example, its variance is still dependent on the overall number of photons $NA_0^f$ and the system’s visibility $V_1^f$, but weighted by the sample’s DPC and DFC signal values, i.e. $V_1$ and $\varphi _1$. Although $\sigma ^2_{\textrm {HFC}}=4(V_1^2 \sigma ^2_{\varphi _1} \cos ^2\varphi _1 + \sigma ^2_{V_1} \sin ^2\varphi _1)$ seems to be larger than pure $\sigma ^2_{\varphi _1}$ mostly, our analysis above provides a theoretical noise model of multi-order moments retrieved by MMA, which can be utilized to further reduce their noises by some model-based post-processing denoising methods. Moreover, considering that the HFC signal, i.e. $2V_1\sin \varphi _1$, may also be larger than the original DPC, the signal-to-noise ratio (SNR) of HFC, i.e.

3. Results and discussion

To validate our analyses above, simulations and physical experiments were conducted. Firstly, we carried out numerical simulations to prove that the derived Fourier-based expressions of MMA, i.e. Equations (27)–(29), are valid under ideal conditions. As we know, the experimental GPCI system can only measure the first-order term of phase-stepping curves, and therefore we cannot directly validate our proposed methods for systems with high-order terms such as edge-illumination and analyzer-based imaging. Therefore, we used the simulated Gaussian-shaped phase-stepping curves with high-order terms to calculate the contrasts by different methods. The schematic diagram of numerical validation processes is shown in Fig. 4. Given constant three parameters (amplitude, mean value and variance), the flat-field PSCs were generated as discrete Gaussian functions with 8 steps, whose Fourier series consisted of first-order to fourth-order components at least. This Gaussian-shaped PSCs can be obtained from the edge-illumination or analyzer-based imaging systems, and we also tried the cosine-shaped PSCs of GPCI systems, whose results were simpler since the PSCs consist of only zeroth-order and first-order components. The SAXS distributions are also approximated as Gaussian functions, and the required three parameters are generated from three transformed image channels (Y/U/V) of the classic RGB Lena image. YUV is a color encoding system, and encodes a color image taking human perception into account, thereby typically enabling transmission errors or compression artifacts to be more efficiently masked by the human perception than using a RGB-representation. Y stands for the luminance component (the brightness) and U and V are the chrominance (color) components, respectively for the blue projection and red projection. Then, according to the well-established convolution relationship in Eq. (6), we obtained the sample data by convolving the synthetic flat data with the SAXS data.

Figure 5 shows the given reference images from the Lena image, and the typical contrasts retrieved by FCA and TA-MMA, whose truncated maximum order is set as 4, i.e. $N_{\textrm {max}}=4$. Besides every channel Y/U/V, we also give the transformed RGB fusion images to show differences more clearly. From Fig. 5, it can be seen that our proposed TA-MMA can obtain almost the same contrasts with the references, which demonstrates its numerical accuracy. On the other hand, the DPC and DFC retrieved by FCA is a little different from those by TA-MMA and the references, and it indicates that the MMA can reflect all orders of components in the sample PSCs, while the FCA is only relative to the first-order component.

 

Fig. 4. The schematic diagrams of the validation processes.

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Fig. 5. The reference contrasts, and those contrasts retrieved by TA-MMA and FCA. The FCA images are calculated by only zeroth-order and first-order components, while the TA-MMA images consider additional second-order to fourth-order components according to Eqs. (27)–(29).

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Besides the numerical simulation, we carried out physical experiments on a Talbot-Lau interferometry at Tsinghua University. In the Talbot-Lau system, the utilized source was a conventional industrial x-ray source (Comet MXR-160HP/11), and the detector was a flat panel CMOS detector (Dexela 1512) with a pixel size of 75 $\mu m$. The tube voltage was set as 35 kV, and the tube current was 24 mA. We utilized the typical phase-stepping process with 8 steps for data acquisition, and the exposure time of every step is set as 800 ms. The grating periods of G0, G1 and G2 are 16.8 $\mu m$, 4.2 $\mu m$ and 2.4 $\mu m$, respectively, where G1 is a $\pi$ phase grating and the other two are absorption gratings. The imaging specimen was an euthanized rat, and the experiment was conducted with the approval of the Institutional Review Board of Tsinghua University.

The contrasts retrieved by FCA, DB-MMA and TA-MMA are shown in Fig. 6. As expected, one can see obvious phase-wrapping artifacts caused by the strong scattering of many tiny alveoli structures in the DPC image computed by FCA, while corresponding images retrieved by both MMA-type methods are more consistent, as they are indeed the combination of both DPC and DFC. Of course, in the cases of phase-wrapping situations where phase contrasts are truncated at $[-\pi , \pi ]$ range, a sine operation should be helpful as well. Moreover, the DFC images retrieved by all methods show the lower lung clearly, which is usually blocked by the heart and liver in the ATC images. In general, most detailed structures in contrasts by all three methods (FCA, DB-MMA and TA-MMA) can be observed, while DPC and DFC values retrieved by TA-MMA are higher than those by DB-MMA, especially for the DFC signals. These differences between TA-MMA and DB-MMA can be explained from two respects. On one hand, DB-MMA is often divergent and even obtains different results for different iteration times. Thus, it is almost impossible to obtain identical results for DB-MMA and TA-MMA in practice. On the other hand, although single high-order term in the phase-stepping curve is much smaller than the first order term as shown in Section 2.3, the summation from second to infinite order can be a considerable quantity. Thus, the abandonment of all high-order terms may cause some differences between the ideal moments and our truncated moments.

 

Fig. 6. Comparisons of the three modalities retrieved by FCA, DB-MMA and TA-MMA for the euthanized rat specimen. The red region is used as the ROI for calculating SNR and CNR, while the blue region is used as the background. Scale bar here is 6 mm.

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Considering the important meaning of DFC for the diagnosis of lung disease, we further conduct quantitative comparisons between relative contrasts retrieved by FCA, DB-MMA and TA-MMA. As shown in Fig. 6, we choose an ROI and a background to calculate their mean values and standard deviations as the signal and noise indexes. Then we calculate their signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR) as listed in Table 1. From Table 1, it can be found that the $\tilde {M}_2^{\textrm {TA}}$ retrieved by our proposed TA-MMA have much greater SNR than FCA or even DB-MMA thanks to both the larger signal value and smaller standard deviation. With the smallest noise in the background, DB-MMA have the best CNR performance, while the CNR of TA-MMA is still a little larger than that of FCA. On the other hand, as shown in Table 2, both FCA and TA-MMA are analytic algorithms, and can be therefore hundreds of times faster than the iterative DB-MMA in terms of the computational efficiency. The reason of DB-MMA’s superior background noise has been analyzed in [39], which is caused by the unimodal scattering ditribution assumption in Lucy-Richardson deconvolution, and its codomain of the $\tilde {M}_2$ is $[0.193; 1]$ rather than $[0, 4\pi ^2/12]$ of $D_{\textrm {FCA}}$. Moreover, to evaluate the potential hybrid representation of our proposed HFC, we also calculate the quantitative indexes in Table 1, where its same level of SNR with $\tilde {M}_2^{\textrm {TA}}$ supports our opinion that HFC can reflect both the DPC and DFC signals. Overall, our proposed TA-MMA is a good candidate of information retrieval algorithms for balancing the noise performance and computational efficiency.

Table 1. The noise performance comparisons of contrasts retrieved by FCA, DB-MMA and TA-MMA shown in Fig. 6. The ROI is the red region in Fig. 6, and the background is the blue region in Fig. 6. The unit for contrasts in first three rows are $10^{-12}$ rad$^2$, while it is $10^{-6}$ rad for the HFC contrast in last row.

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Table 2. The computational efficiency comparison of FCA, DB-MMA and TA-MMA. The DB-MMA method was implemented by 200 times Lucy-Richardson iterations (CPU-based parallel computing in Matlab 2020a software with an Intel i7-7700K CPU of 8 cores).

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From these results, it can be seen that the HFC image provides a compact fusion image of typical DPC and DFC. As we know, the DPC image is sensitive to light elements, while the DFC image is sensitive to ultra-small-angle scattering properties. Thus, in some cases, our proposed HFC image can reflect this sensitive information at the same time, such as the lung area of DFC and the tissues of DPC in Fig. 6. Furthermore, compared with original DPC and DFC, the HFC image can be of higher SNR in some cases, as shown in Table 1. However, the specific conditions where the HFC can have higher SNR have not been given, and we will conduct more analytic derivations and experimental studies in the future work. Besides, we will explore more potential applications of the HFC image, such as the diagnosis of breast cancer by combining the microcalcifications in DFC and the soft tissues in DPC.

4. Conclusion

In this paper, we presented a Fourier-based interpretation of the moments of small-angle x-ray scattering in grating-based x-ray phase-contrast imaging, and showed that it is further acted as an important bridge to connect the Fourier component analysis based on the wave optics and the multi-order moment analysis based on the particle scattering theory. Our proposed interpretation explains the high similarity between the contrasts retrieved by these two methods, i.e. containing similar information just with different forms, and also explains their differences. Moreover, a noise analysis was conducted to evaluate its performances. Also, the first-order moment computed by our proposed truncation method is actually the product of the original phase-contrast and the dark-field contrast retrieved by Fourier components analysis, providing a new hybrid-field contrast, which may have a potential to be directly used in practical applications.