1. Introduction

X-ray computed tomography (CT) technique has played an important role in clinical diagnoses for its nondestructive investigation of patients. However, since conventional x-ray imaging is based on the absorption properties of materials, it is difficult for current CT systems to distinguish low-density materials, such as soft tissues. Compared with traditional absorption-based x-ray imaging, x-ray phase-contrast imaging (PCI) technique has gained special attention in recent years for its superior soft-tissue imaging capability [1–4]. Among the various proposed methods for PCI, grating-based PCI technique has been extensively developed and successfully extended to use with a laboratory x-ray tube, which indicates its promise further in clinical imaging applications [5–7].

Compared with the traditional absorption-based x-ray CT, grating-based phase-contrast CT (GPC-CT) is much more effective in imaging low-density materials [8–10], but more complicated. During a general GPC-CT scan, phase-stepping (PS) method is used to capture projections with different grating transverse positions at each projection angle. Thus, the total radiation dose delivered to the scanned object is several times higher than that within a traditional CT scan. What’s worse, the stop-and-go motion in PS-based GPC-CT makes it much slower than the traditional CT scanning. These limitations restrict the standard GPC-CT system adopted in clinical imaging applications. To achieve a practical GPC-CT scanning for clinical diagnosis, fast data collection procedure and acceptable radiation dose are critical.

According to data collection procedures, the imaging method of GPC-CT can be classified into four categories: the PS method [11], Moiré analysis method [12, 13], interlaced PS method [14, 15] and reverse projection method [16]. Among them, the PS method is commonly used in experiments as the gold standard. Though the others can improve the shortcomings of the PS method in some respects, they are all imperfect, and sacrifice other performances as compensation. In this paper, we follow the imaging scheme of the interlaced PS method and propose a novel reconstruction method for it, named the inner-focusing (IF) reconstruction method. The proposed method can well address the limited field-of-view (FOV) problem of the interlaced GPC-CT without any additional requirements by having a more uniform noise texture [17]. Both numerical simulations and real experiments were carried out to investigate its effectiveness in achieving a fast and low-dose GPC-CT.

2. Materials and method

System geometry of GPC-CT with a synchrotron x-ray source is shown in Fig. 1(a) where parallel x-rays travel through the object and then analyzed by grating G1 and G2. The synchrotron-based GPC-CT can be easily translated to a laboratory system by inserting another grating, G0, placed in front of x-ray tube to ensure the incident x-rays are coherent enough. The imaging principle of grating-based PCI is well described by previous study [5]. As the real experiment in this paper was performed with a synchrotron radiation source, we mainly focused on the parallel-beam GPC-CT condition.

 

Fig. 1 (a) Schematic representation of GPC-CT with synchrotron x-rays. The data collection schemes with PS, and interlaced PS methods are illustrated in (b). The red dashed boxes indicate the data collection procedure at each projection angle.

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2.1 PS and interlaced PS

Data collection schemes of GPC-CT with PS and interlaced PS methods are plotted in Fig. 1(b). Each of them can be implemented with parallel-beam x-rays (using synchrotron radiation, Fig. 1(a)) or fan-beam x-rays (using laboratory x-ray tube). The PS method is the gold standard for data acquisition of GPC-CT which is based on Talbot effect [18]. Through the Talbot effect, the phase modulation generated by the phase grating G1 and the scanned object is transformed into intensity modulation downstream and analyzed by the absorption grating G2, which is placed at a distance from G1. When the PS method is adopted, G1 (or G2) is stepped along the transverse direction perpendicular to grating lines at each projection angle and a series of projections of the sample is recorded. The intensity signal in each detector cell (indexed by $t$) oscillates as a function of the grating transverse position $x$ and can be approximated by only the zeroth and the first Fourier components as

Instead of stepping grating at each projection angle in the standard GPC-CT, the interlaced PS method allows the grating movement and the sample rotation to occur at the same time. The interlaced method makes the GPC-CT compatible with a continuous rotation. Thus, the interlaced GPC-CT scanning can be completed within a much shorter time, having a comparable temporal resolution with existing CT systems [14]. Meanwhile, the radiation dose delivered to the sample during an interlaced GPC-CT scan is only a fraction of that with the standard PS method. Since the continuous sample rotation and grating stepping, the recorded data for reconstruction is incomplete. In the data analysis processing of interlaced GPC-CT, neighboring projections are employed to compose a stepping sequence. Thus, at each projection angle, there is a series of projections with different grating stepping positions for signal extraction, as with the data collected by standard GPC-CT. The following reconstruction is the same as standard GPC-CT [11]. As the sample is rotated along with the grating stepping, the intensity oscillation in each pixel is a function of both the grating stepping position and the projection angle. So, the interlaced PS method is an approximate method for image reconstruction. According to previous results, obvious artifacts will be introduced by this method, especially areas far from the rotation center [17, 19]. Thus, the effective FOV of GPC-CT employing the interlaced PS method is limited within the area around the rotation center.

2.1 IF reconstruction

According to the analysis above, the interlaced GPC-CT can be implemented in a continuous rotation mode, and the radiation doses delivered to the scanned object is only a fraction of that with the PS method. However, there are obvious artifacts in the reconstructed images as the signal extraction process at each projection angle is approximate. In this paper, we consider the interlaced GPC-CT method, and propose a novel reconstruction method to improve its performance, making it practical in clinical applications. To address the limited-FOV problem of the interlaced GPC-CT, an alternative reconstruction method is proposed, named the IF method.

Adopting the interlaced PS method, projections with neighboring viewing angles are used to compose a stepping sequence according to the rotation center of the sample as shown in Figs. 2(a) -2(b). There will be obvious artifacts around the structures which are far from the physical rotation center. While, in the IF reconstruction, several virtual rotation centers are established and corresponding stepping sequences are collected according to these virtual rotation centers. The final reconstructed image is composed by these local images around each virtual rotation center. To reconstruct a local area around $\text{f}(\overrightarrow{r})$ which is far from the physical rotation center of the sample (Fig. 2(c)-2(d)), projections are shifted along the horizontal direction making the point $\text{f}(\overrightarrow{r})$ as a virtual rotation center to compose stepping sequence. Let $\left\{P_j\right\}$ be the composed stepping sequence for calculating differential phase-contrast projection ${\phi }_i$ where$j\in \left\{i-4,i-3,\mathrm{...},i,\mathrm{...},i+4\right\}$. Here, we set 9 neighboring projections correspond to a full $2\pi$ stepping sequence as an example. $\left\{{\alpha }_i+(j-i)\Delta \alpha \right\}$ is the set of their corresponding projection angles, and $\Delta \alpha$ is the angle difference between neighboring projection. With conventional interlaced PS method, the intensity curve of stepping is represented by values at the fixed position $t$ on horizontal detector array with neighboring projection angles$\left\{P_j(t)\right\}$, as shown in Fig. 2(b). While in the proposed IF reconstruction, projections with projection angles are firstly shifted horizontally to focus at the virtual rotation center, such as $\text{f}(\overrightarrow{r})$. Then, the estimated intensity curve $\left\{P_j\left(t^{\prime }\right)\right\}$ is located at $t+\Delta t_{j-i}$ corresponding with slightly different projection angles, where $\Delta t_{j-i}$ is the position offset related with divergent angles$(j-i)\Delta \alpha$. Let $\gamma$ be the angle with x-axis as shown in Fig. 3 . The projection position of $\text{f}(\overrightarrow{r})$ on to the detector array is $t=\left|\overrightarrow{r}\right|\cdot \cos (\gamma -\alpha )$ and $t=R\frac{\left|\overrightarrow{r}\right|\cdot \cos (\gamma -\alpha )}{R+\left|\overrightarrow{r}\right|\cdot \sin (\gamma -\alpha )}$ respectively for parallel beam x-rays and fan beam x-rays. The position offsets for each condition are $\Delta t=\left|\overrightarrow{r}\right|\cdot (\cos (\gamma -\alpha )-\cos (\gamma -(\alpha +(j-i)\Delta \alpha )))$ and $\Delta t=R\left(\frac{\left|\overrightarrow{r}\right|\cdot \cos \left(\gamma -\alpha \right)}{R+\left|\overrightarrow{r}\right|\cdot \sin \left(\gamma -\alpha \right)}-\frac{\left|\overrightarrow{r}\right|\cdot \cos \left(\gamma -\left(\alpha +\left(j-i\right)\Delta \alpha \right)\right)}{R+\left|\overrightarrow{r}\right|\cdot \sin \left(\gamma -\left(\alpha +\left(j-i\right)\Delta \alpha \right)\right)}\right)$ correspondingly.

 

Fig. 2 (a-b) Data analysis of the interlaced PS method where neighboring projection images $\left\{P_i\right\}$ are utilized to retrieve differential phase-contrast signals. The red triangle indicates the physical rotation center. Black and grey lines in (a) are captured projection images with different projection angles and different grating positions. In the proposed IF method, to reconstruct a local area in object $\text{f}$ as marked by red dashed circle in (c), neighboring projections are firstly shifted along horizontal direction making $\text{f}(\overrightarrow{r})$ being a virtual rotation center (red triangle) where $\overrightarrow{r}$ is the position vector from the physical rotation center to the virtual rotation center. In (b) and (d), red dashed lines indicate projection positions of physical and virtual rotation centers and the black dashed line in (d) suggests the projection of physical rotation center of (c).

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Fig. 3 A geometrical illumination of imaging processes for parallel beam GPC-CT (a) and fan beam GPC-CT (b).

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At each projection angle, the differential phase-contrast projection corresponding to the virtual rotation center $\text{f}(\overrightarrow{r})$ is retrieved from $\left\{P_j\left(t^{\prime }\right)\right\}$. They are calculated by fitting cosine functions (as expressed by Eq. (1)) with the Levenberg–Marquardt algorithm [20] in Matlab software (MathWorks, Massachusetts, USA). By taking these differential phase-contrast projections around the sample, the refractive index of the local area in the sample can be reconstructed using a FBP algorithm [9]. This is the so-called inner-focusing process. To reconstruct a complete cross slice of the sample, multiple virtual rotation centers are set, and the final reconstructed complete image is composed with these sub-images around each virtual rotation center. With the IF reconstruction method, phase-contrast, absorption-contrast and scattering-contrast images can be reconstructed simultaneously, same as the interlaced PS method [14]. Since the refractive index is more suitable for representing low-density materials, in this paper, we show the reconstructed phase-contrast images in the following experiments.

3. Results and discussion

To evaluate the proposed IF reconstruction method for GPC-CT, a numerical phantom was designed with four kinds of rods embedded inside as shown in Fig. 4 . The diameter of the phantom was 4 mm. The setup of GPC-CT system is shown in Fig. 1(a) where the phase shift of grating G1 was set at $\pi /2$, and the periods of G1 and G2 were 2.4 μm. Projections of the sample were recorded by a detector with an effective pixel size of 5 μm. The visibility of the phase-stepping curve was set at 40%. In numerical experiments, a parallel-beam interlaced GPC-CT scan of the phantom was performed with x-ray energy of 25 keV. The number of projection angles was N_P with 180 degrees around the sample. According to the data collection scheme of interlaced GPC-CT plotted in Fig. 1(b), the stepping of grating G1 and the rotation of the sample were simultaneous. 9 steps of grating G1 movement covered a full period of grating G2. The sample was placed exactly on the center of the rotation stage.

 

Fig. 4 Numerical phantom used in simulations. (a) is its absorption coefficient map and (b) the distribution of refractive index.

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According to the reconstruction method of interlaced PS, the center area of the cross slice of the sample can be reconstructed with acceptable quality. Meanwhile, there will be obvious artifacts within regions far from the rotation center as shown in Fig. 5(a) . Increasing the angular sampling rate around the sample is an effective way to solve this problem. The reconstruction result with 1080 projections is shown in Fig. 5(b). Its image quality is much better than that with 360 projection angles. Linear profiles as marked in Fig. 5(a) are plotted in Fig. 5(c) where the base line is from the image with standard PS method and 1080 projection angles.

 

Fig. 5 Reconstructed phase-contrast images with the interlaced PS method (a–b) and IF reconstruction method (d–e). The display window is the same with Fig. 4(b). The number of projections (N_P) in (a), (d) is 360 and (b), (e) 1080. (c) and (f) Plots of the line profile indicated by the red line in (a) with the interlaced PS method and IF reconstruction method respectively. The red triangle in (d) marks the virtual rotation center (–200, 200) in IF reconstruction, and red circles show the reconstruction region. The images are displayed with the same window as in Fig. 4(b).

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Figures 5(d) and 5(e) show reconstructed sub-images with the IF reconstruction method with 360 and 1080 projection angles respectively. The virtual rotation center is marked by the red triangle in Fig. 5(d). Compared with reconstruction results using the interlaced PS method, the proposed IF method can effectively maintain structural boundaries within the sub-image as marked by red dashed circles in Figs. 5(d) and 5(e). The linear profiles demonstrate the edge preserve ability of the proposed IF reconstruction method (Fig. 5(f)). Consistent with the analysis above, regions far from the virtual rotation center are asymmetrically blurred, as indicated by red arrows in Figs. 5(d) and 5(e).

In the numerical experiment, virtual rotation centers are set in a two-dimensional plane with uniform spacing of 50 pixels, marked by red triangles as shown in Fig. 6(b) . The reconstructed region around each virtual rotation center is taken to compose a complete cross slice of the sample. The montaged image is shown in Fig. 6(b). The Image reconstructed with the interlaced PS method for the same projection numbers, 720, is shown in Fig. 6(a) correspondingly. Similar to previous results, there is boundary blurring around the structure far from rotation center as suggested by red arrow in Fig. 6(a). In contrast, these kinds of artifacts are not observed in the images reconstructed with the IF method. The residual error images of the reconstructed results with conventional interlaced PS method and the proposed IF method are shown in Figs. 6(c) and 6(d) displayed with a narrow window. It is obvious that the IF reconstruction method can effectively preserve the boundary of reconstructed structures and enlarge the effective FOV of the interlaced GPC-CT.

 

Fig. 6 Reconstructed phase-contrast images with interlaced PS method and IF reconstruction method are shown in (a) and (b) respectively. The two images are shown with the same display window as that in Fig. 4(b). The red arrow in (a) marks blurred boundary. The two-dimensional red-triangle array in (b) shows virtual rotational centers used in IF reconstruction method. The complete image is a montage of these reconstructed areas around virtual rotation points. (c) and (d) are their corresponding residual error images displayed with a narrow window.

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In our real experiment, a phantom consisting of polymethylmethacrylate (PMMA) balls was utilized to investigate the effectiveness of the proposed IF reconstruction method for interlaced GPC-CT. The phantom was scanned by a grating-based interferometer installed into the BL13W beam line at Shanghai Synchrotron Radiation Facility (SSRF) [8]. G1 was designed to be $\pi /2$, and its period is 2.396 µm. The period of G2 is 2.4 µm. The inner distance between G1 and G2 was 46.4 mm. X-ray tomography of the phantom was carried out at 20 keV, in which 720 projection angles were recorded over 180 degrees with an effective pixel size of 9 μm (Photonic Science, East Sussex, UK). The exposure time for each projection image was 10 ms. In each phase-stepping scan, the grating G1 moved along the transverse direction perpendicular to the grating line direction with 9 steps to cover one full period of the grating G2. To simulate the data collection schemes of GPC-CT with PS and interlaced PS methods, as illustrated in Fig. 1(b), we rearranged the complete CT data set into specific data sets according to their data collection protocols. In the interlaced GPC-CT, the sample rotation and the grating stepping occur at the same time, and 9 steps of grating G1 stepping cover one period of G2.

In the IF reconstruction, virtual rotation centers are configured with the same way as in the numerical simulation with 50-pixel uniform spacing grids in a two-dimensional plane. The reconstruction results with both the interlaced PS method and the proposed IF method are shown in Figs. 7(a) and 7(b). Taking the reconstruction result with standard PS method as the base image, the difference between the based image and the reconstructed images are shown in Figs. 7(c) and 7(d). Consistent with previous numerical analysis, there are obvious artifacts around the structures far from the physical rotation center with the conventional interlaced PS method (Fig. 7(c)). While adopting the IF reconstruction method, there are much more uniform noise textures in the reconstructed image as shown in Fig. 7(d). Though the background noise, near the boundary of FOV, is enhanced to some extent, the IF method can keep the shapes of reconstructed objects well. For radiologist diagnosis, the uniform noise distribution is important to make correct judgments. Predictably, under the same imaging condition, the imaging quality of GPC-CT with the proposed IF method is better than that with the conventional interlaced PS method.

 

Fig. 7 Reconstructed phase-contrast images using real experimental data. Images reconstructed with interlaced PS method and IF method are shown in (a) and (b) respectively. Bar: 1 mm. (c) and (d) are their corresponding residue errors, compared with the image reconstructed by standard PS method.

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4. Summary

Based on the data collection scheme of interlaced GPC-CT, we proposed a novel reconstruction method, namely the IF reconstruction method. This method can effectively enlarge the limited FOV of interlaced GPC-CT by preserving boundary in reconstructed images without any additional requirements. Combined with the proposed IF reconstruction method, GPC-CT scan can be implemented with fast speed and reasonable radiation dose with an acceptable imaging quality. Both numerical simulations and real experiments results have demonstrated that the IF reconstruction method is effective in archiving a low-dose and fast GPC-CT scan. We believe that the proposed IF reconstruction method will be one of critical factors promoting GPC-CT applied in clinical imaging applications.