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Abstract
Micro computed tomography (µCT) allows the noninvasive visualization and 3D reconstruction of internal structures of objects with high resolution. However, the current commercial µCT system relatively rotates the source-detector or objects to collect projections, referred as RCT in this paper, and has difficulties in imaging large objects with high resolutions because fabrication of large-area, inexpensive flat-panel detectors remains a challenge. In this paper, we proposed a source translation based CT (STCT) for imaging large objects with high resolution to get rid of the limitation of the detector size, where the field of view is primarily determined by the source translation distance. To compensate for the deficiency of incomplete data in STCT, we introduced multi-scanning STCT (mSTCT), from which the projections theoretically meet the conditions required for accurate reconstructions. Theoretical and numerical studies showed that mSTCT has the ability to accurately image large objects without any visible artifacts. Numerical simulations also indicated that mSTCT has a potential capability to precisely image the region of interest (ROI) inside objects, which remains a challenge in RCT due to truncated projections. In addition, an experimental platform for mSTCT has been established, from which the 2D and 3D reconstructed results demonstrated its feasibility for µCT applications. Moreover, STCT also has a great potential for security inspection and product screening by using two perpendicular STCTs, with advantages of low-cost equipment and high-speed examination.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Micro computed tomography (µCT) is a powerful noninvasive technique for investigating the inner structure of objects with high spatial resolutions, which is widely used for material science [1], biological investigations [2], and industrial inspections [3]. According to the variety of X-ray sources, µCT falls into two broad categories: synchrotron radiation-based µCT (SRµCT) and tube-based µCT (TBµCT) [4]. Using highly brilliant X-rays, X-ray optics, and Zernike, SRµCT can image objects at resolutions as high as a few nanometers [5]. Based on the principle of geometrical magnification, the commercially available TBµCT that employs nanofocus tube and flat-panel detector (FPD) can perform CT scans at resolutions around 100 nanometer, which is the highest resolution that has been reported until now [4]. Compared with SRµCT, TBµCT possesses several advantages, including lower costs, larger field of view (FOV), and a more flexible energy range. In this paper, we mainly discuss TBµCT, and the µCT mentioned later is regarded as TBµCT.
Conventional µCT system acquires projections by relatively rotating the source-detector or objects, which is referred as RCT in this paper. RCT requires that objects must fit within the X-ray beam formed by the source and detector to keep far away from truncation artifacts. However, RCT has difficulties in imaging large objects with high resolutions because of the limitations of the detector size. Fabrication of large-area, inexpensive photodetectors still remains a challenge [6]. Thus far, with the commercially available FPD, RCT can image an object as big as 0.03-1.0 mm with resolutions up to 0.1-3.0 µm. Nevertheless, the option of trimming a specimen to such a small size is undesirable.
Over the years, multiple techniques have been proposed for imaging large objects, including the second-generation scanning mode [7], displaced detector scanning mode [8,9], multiple translation and rotation of objects scanning mode [10,11]. However, these typical methods also have some limitations, such as inefficiency, difficulty to implement, and complex reconstruction methods. Some studies have been carried out to improve the reconstruction algorithm for these scanning modes [9].
Recently, linear scan trajectory-based CT, referred as LCT, has been continuously investigated for security inspections and clinical examinations, with many advantages, such as low-cost equipment, high-speed examination, and simple mechanical design [12–14]. For example, Liu developed an ultra-low-cost parallel-translation CT (PTCT) for clinical applications [15]. During the data acquisition, the X-ray source and detector array are oppositely translated in parallel, while the examined patient is fixed between them. To offer fast scans for security inspections with relatively low system cost, Zhang designed a symmetric-geometry CT (SGCT) where the distributed-source array and detector are linear distributed in a symmetric design [16,17]. During the scanning, the X-ray source points sequentially fire in an ultra-fast manner on one side, while the detector on the opposite side continuously collects data. As is known, projections from a single finite straight-line trajectory are incomplete for accurate reconstruction. Fortunately, it is feasible to compensate for this deficiency by using a multisegment straight-line trajectory [18–21].
Over the past decade, the inverse-geometry CT (IGCT) has been actively investigated to enlarge volumetric coverage in single rotation but free from cone-beam artifacts [11–13]. In IGCT, a large distributed source array and a relatively small area detector array are installed on the rotating gantry. A vital property of IGCT is that the FOV in the transverse direction is primarily determined by the source array size, as the same way that the FOV is determined by the detector size in a conventional CT system [20].
In this paper, inspired by LCT and IGCT, we used the nanofocus tube X-ray source and FPD, and proposed a source translation-based CT (STCT) method for imaging the object with a relatively large size at high resolution. During the data acquisition, the X-ray source is equidistantly translated along a straight-line trajectory, which is parallel to the fixed FPD, while the object is placed closer to the source. Because the object and detector keep fixed during the scanning and the object locates relatively close to the source, STCT can image the object with resolution as high as possible. Moreover, the whole object can be illuminated by translating the X-ray source in STCT, although each source position only illuminates a fraction of the object. In other words, the FOV of STCT is primarily determined by the translation distance of the source, as similar as that in IGCT. Therefore, STCT has the ability to image a large object by controlling the translation distance, which is too easy to implement in practice. However, there is still a problem that complete projections cannot be obtained from a single STCT. In this paper, we developed a multi-scanning STCT (mSTCT) to compensate for this deficiency.
Reconstruction is a crucial step for CT imaging, which maps projections to images. FBP is a typical reconstruction method in RCT, which requires that the object is totally within the FOV which is the region where each point is irradiated over 180°. The modified FBP also can be used for LCT reconstruction when projections are nontruncated [22]. Whereas, the projections collocated from IGCT are truncated because each source only illuminates a fraction of the object, and this problem also appears in STCT, resulting in artifacts in images when we directly use FBP for reconstruction [21]. The methods developed for IGCT are to rebin the truncated projections into nontruncated projections; however, the rebinning step will introduce some blurring in images [19]. Recently, compressed sensing is widely used for CT reconstructions [23–25], and provides a feasible approach to solve the truncated projections [26], and noise projections [27,28].
A theoretically accurate reconstruction is important for µCT applications. In LCT, accurate reconstructions can be achieved from a multi-segment line trajectory or an infinite straight-line trajectory, when projections are nontruncated [29,30]. Although STCT is similar to LCT in terms of geometry, whether it can theoretically achieve accurate reconstructions remains to be studied due to truncated projections. In this paper, we use the total variation (TV) method to reconstruct images from STCT and mSTCT. Besides, we theoretically analyze and experimentally validate whether mSTCT can achieve accurate imaging.
This paper is organized as follows. Section 2 describes the imaging model of STCT and mSTCT, analyzes the theory of mSTCT, and presents the reconstruction method. In Section 3, the numerical simulations and practical experiments are presented. We discuss some properties and issues of STCT and mSTCT in Section 4. Section 5 concludes this paper.
2. Method
2.1 Imaging model of STCT
The proposed STCT employs the nano-focus tube X-ray source and FPD for imaging. During the data acquisition, the FPD is fixed, the detected object locates relatively close to the source, while the source translates along a straight line that is parallel to the middle row of the fixed FPD. In STCT, the X-ray beam diverging from each source position can only illuminate a fraction of the object. However, the entire object can be covered by translating the source, as illustrated in Fig. 1(b).
To precisely describe the geometry of STCT, a fixed o-xyz coordinate system locates in the center of the object, with each axis direction depicted in Fig. 1(a). For simplification, we mainly investigate the geometry of STCT in the x-y plane because the difference of STCT mainly focuses on the transverse direction (x-y plane). We denote the length of the detector as 2d, and the translated distance of the source as 2s. h and l respectively represent the distance from the object to the detector and source trajectory. Then the position of the detector during the scanning can be expressed as
And each source position is
2.2 Projections acquired using STCT
Assuming that the density function of the object has circular support with R in radius, we have
We express the position of each detector unit as
Then the projection data, which is the line integral of the density function along the line oriented from ${\vec S_\lambda }$ to ${\vec D_u}$, can be formulated as
The one-dimensional function $p({{{\vec S}_\lambda, \cdot)}}$ is called a fan-beam projection. Because each projection only views a fraction of the object, $p({{{\vec S}_\lambda, \cdot)}}$ is truncated.
The projection data can also be expressed as the line integral of the density function along the line oriented at an angle $\varphi$ from x-axis and a signed distance r from the origin. So we have
Marking the parallel projections $\bar p({\varphi ,r} )$ in the Cartesian coordinate in terms of $({\varphi ,r} )$, the distribution of projections collected by STCT in the Radon space can be obtained, as shown in Fig. 2(b). This figure can help us to investigate whether complete projections can be collected by STCT for reconstruction. The complete projections for an accurate reconstruction of $f({x,y} )$ is that:
Fig. 2. (a) illustrates X-rays in the x-y plane, (b) is the distribution of projections in Radon space.
The projection values $\begin{array}{cc} {\bar p({\varphi ,r} ),}&{{\kern 1pt} \varphi \in [{0,\pi } ]} \end{array}$ are available for all $|r |\le R$.
We denote the angular coverage $\Delta \varphi$ as a function of r. According to Eq. (10) and the geometry of STCT, we have
As illustrated in Fig. 2(a), R1 is the distance from the origin to the line connecting the left-most source position with the right-most detector. The formula of angular coverage $\Delta \varphi (r )$ for ${R_1} < |r |\le {R_2}$ is very complex. However, from the distribution of projections in Radon space as displayed in Fig. 2(b), it is easily observed that the angular coverage decreased with $|r |$ increased. As presented in Fig. 2(a), R2 is the distance from the origin to the line connecting the left-most source position with the left-most detector, which can be calculated as
Based on Eq. (10) and the geometry of STCT, we can summarize the properties of STCT as following:
- 1) ${R_1}$ and ${R_2}$ are system parameters, which are dictated by $s,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} d,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} h,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l$. When ${\kern 1pt} d,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} h,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l$ are fixed, ${R_1}$ and ${R_2}$ can be enlarged by increasing the translated distance of the source.
- 2) For $|r |< {R_1}$, the angular coverage is primarily limited by the detector size, and it decreases with the object being closer to the source when the detector size is determined. For ${R_1} < |r |\le {R_2}$, the angular coverage is related to $s,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} d,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} h,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l$ and r. It decreases with $|r |$ increased when $s,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} d,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} h,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l$ are fixed.
2.3 mSTCT scanning mode
However, with a limited detector size, complete angular coverage is unable to be obtained from just a single STCT. Multisegment scanning is useful to provide complete projections for reconstruction. To guarantee that the projection value $\begin{array}{cc} {\bar p({\varphi ,r} ),}&{{\kern 1pt} \varphi \in [{0,\pi } ]} \end{array}$ are available for all $|r |\le {R_1}$, we developed a mSTCT scanning mode, as displayed in Fig. 3(b).
Fig. 3. (a) is one of the mSTCT segments. $\theta$ is the translated angle. (b) is the geometry of mSTCT in the x-y plane, with translated angles being ${\theta _1} ={-} \Delta \theta ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\theta _2} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\theta _3} = \Delta \theta$, respectively. $\Delta \theta$ is equal to $2\arctan ({{d \mathord{\left/ {\vphantom {d h}} \right.} h}} )$, which is the angular coverage from one STCT. (c) is the distribution of projections that are collocated from mSTCT in Radon space.
We denote the translated angle for each STCT as $\theta$, which is the angle between y-axis and the line perpendicular to the source trajectory, as shown in Fig. 3(a). $\Delta \theta$ represents the interval angle between two adjacent STCTs, which is set as
And the number of STCTs is determined by
where ${\rm{ceil}}(x )$ outputs the nearest integer that is greater than or equal to x. $\alpha$ is an angle as shown in Fig. 3(a), which can be calculated asDetail description can be found in Supplement 1. The presented mSTCT is similar to the multisegment scanning in [15,29,31], with a different interval angle which mainly depends on different scanning modes.
According to the parameters of mSTCT, the collected projections distributed in the Radon space are depicted in Fig. 3(c). From Fig. 3, we can conclude two vital properties of mSTCT as following:
- 1) For $|r |\le {R_1}$, the projection value $\begin{array}{cc} {\bar p({\varphi ,r} ),}&{{\kern 1pt} \varphi \in [{0,\pi } ]} \end{array}$ are available.
- 2) For ${R_1} < |r |\le {R_2}$, there are angles $\varphi \in [{0,\pi } ]$ for which the projection values $\bar p({\varphi ,r} )$ are not available. More precisely, $\bar p({\varphi ,r} )$ is known only for projection angles in some interval of length strictly less than $\pi$.
Therefore, in practice, mSTCT can accurately image objects with a radius smaller than R1. It also can be employed for imaging objects with a radius between R1 and R2, whereas the question that needs to be solved is whether the region of interest (ROI) with R1 in radius can be accurately reconstructed.
2.4 Theory of mSTCT
Firstly, we investigate whether mSTCT can accurately image objects with a radius smaller than R1. Because the filtering step requires all elements of projections, the traditional fan-beam algorithms cannot be directly used for reconstruction due to truncated projections. A general solution is rebinning the truncated fan-beam projection into a parallel projection based on Eqs. (9) and (10), and then using parallel-beam algorithms for reconstruction. As we know, accurate reconstruction can be guaranteed when the parallel projections are complete. As discussed above, mSTCT can collect complete parallel projections when $|r |\le {R_1}$, therefore, it has the ability to accurately image objects with a radius smaller than R1.
There is another way to prove that mSTCT can theoretically accurate image objects with a radius smaller than R1. Works in [29] have proved that accurate reconstruction can be achieved from multi-segment linear source trajectory when projections are nontruncated. Denote the fan-beam projection $p({{{\cdot, \vec D}_u )}}$ as a set of measured lines diverging from a detector element. Based on the line-integral model, the projection $p({{{\vec S}_\lambda, \cdot )}}$ diverging from the source can be reformed into the projection $p({{{\cdot, \vec D}_u )}}$ without introducing any error, as illustrated in Fig. 4. Because the projection $p({{{\cdot, \vec D}_u )}}$ is nontruncated in the case of $R \le {R_1}$, replacing the source trajectory ${\vec S_\lambda }$ with the detector element trajectory ${\vec D_u}$, we can draw the similar conclusion that accurate reconstruction can be guaranteed from mSTCT.
Fig. 4. (a) shows the fan-beam projection $p({{{\vec S}_\lambda, \cdot )}}$ diverging from the source, (b) depicts the fan-beam projection $p({{{\cdot, \vec D}_u )}}$ diverging from the detector unit, (c) displays the case of $R > {R_1}$.
Secondly, we study the case where the radius of detected objects is between R1 and R2. We define the following two regions:
- Region A = {$({x,y} )\in {\rm{support}}{\kern 1pt} {\kern 1pt} f$: all lines through $({x,y} )$ are measured},
- Region B = {$({x,y} )\in {\rm{support}}{\kern 1pt} {\kern 1pt} f$: $({x,y} )\notin$ region A}.
As displayed in Fig. 4(c), region A is a circular area centered at the origin with a radius of R1, and region B is a ring with a radius between R1 and R. By jointly looking at Fig. 3(c), because any point inside region B lacks full angular coverage, stable reconstruction within region B is impossible [32]. Thus the only candidates for practical quantitative ROI reconstruction lie within region A. However, the unicity and stability in region A are relatively difficult to establish due to region A completely inside the object. Some studies [33,34] illustrated a similar situation where region A is internal to the object support, and yet stable reconstruction is possible, which may be useful to analyze the case in this paper.
In this section, although these theoretical analyses do not provide direct analytic reconstruction algorithms, they are important for understanding the ability of mSTCT. Our purpose is to make sure whether projections provided by mSTCT meet the conditions required for accurate reconstructions.
2.5 Reconstruction method
To obtain a satisfying result from complete or incomplete projections collected from STCT and mSTCT, we utilize an iterative model for reconstructions. The iterative model can be expressed as a linear matrix equation
where ${\bf{f}}$ is the unknown reconstructed image, ${\bf{p}}$ is the log-normalized projections, ${\bf{A}}$ is the system matrix. This iterative model can combine with compressed sensing to improve the image quality in the case of incomplete data. TV is one of the popular methods used for incomplete data reconstruction and have form as follows,3. Experiments
To validate the ability of STCT and mSTCT, the simulated experiments were implemented, a prototype system was built for STCT and mSTCT. The reconstruction processes are written in MATLAB, accelerated by GPU. The experiments were tested on a computer with an Inter(R) Core i5-6400 CPU @ 2.70 GHz and an NVIDIA GT730. Reconstructing images in a 1536×1536 matrix from projections with a size of 1536×900 will cost 2.6s in each iteration. Because the results after 1000 iterations are visually indistinguishable, the SIRT-TV was stopped after 1000 iterations to balance the efficiency and accuracy.
3.1 STCT
Begin by evaluating the ability of STCT, and parameters listed in Table 1 were simulated. In STCT, the angular coverage is mainly restricted by the detector size, and it increases with the detector size increased. While the angular coverage is also affected by the distance between object and detector, it increases with the distance decreased. To guarantee at least 90° angular coverage, the simulated detector size is set as relatively large, and the magnification ratio (ratio of distance between source trajectory and detector to distance between object center and source trajectory) is relatively small. This configuration is feasible for security CT in practice.
Table 1. Geometry Parameters for STCT Simulation
A 2D Shepp–Logan head phantom with rectangular compact support of 200 mm×200 mm was used for simulation. Figure 5 presents the reconstructed result, which suffers from severe streaks and aliasing artifacts due to incomplete angular coverage. However, the structure in the data-know angles is not too bad.
Fig. 5. The result of STCT. (a) is the Shepp-Logan head phantom with rectangular compact support of 200 mm×200 mm (i.e. R=100 mm). (b) is the reconstructed image, which has 512×512 pixels with each pixel being 0.39 mm×0.39 mm. The display window is [0, 0.6].
3.2 mSTCT
To investigate the ability of mSTCT, the simulated experiments with parameters listed in Table 2 were implemented. Because mSTCT is anticipated to image large objects under high resolutions, the detector size is set as relatively small due to the small size of detector units, and the magnification ratio should be as large as possible. As discussed above, the scanning parameters determine two system parameters R1 and R2. mSTCT has ability to accurately image objects with a radius smaller than R1. However, its ability for imaging objects with a radius between R1 and R2 is unknown. Therefore, in addition to verifying the case of objects with radius R1, we also numerically evaluate the quality of results when mSTCT detects objects with a radius between R1 and R2. For comparison, the same detector size and magnification ratio are used for RCT and PTCT, with FOV being 3.4 mm in radius.
Table 2. Geometry Parameters for mSTCT Simulation
Firstly, a 7.6 mm radius (i.e. R = R1) wheel phantom was utilized for simulation. Figure 6 displays the reconstructed results. The mSTCT can faithfully recover the whole phantom without visible artifacts. However, RCT and PTCT can only image the ROI with 3.4 mm in radius.
Fig. 6. The first row is the results of the 7.6 mm radius wheel phantom (i.e. R = R1), where images have 512×512 pixels with each pixel being 0.029 mm×0.029 mm. The second row is the results of the 9.5 mm radius abdomen phantom (i.e. R1<R < R2), where images have 512×512 pixels with each pixel being 0.037 mm×0.037 mm. From left to right, images are phantoms and the results from RCT, PTCT, and mSTCT, respectively. The display window is [0, 0.8].
Secondly, an abdomen phantom with a radius of 9.5 mm (i.e. R1<R < R2) was scanned with the same parameters. Figure 6 presents the results from different scanning modes. Figure 7 plots the central horizontal profiles of the ROI which is centered at the origin with 3.4 mm in radius. As shown in Fig. 6, RCT and PTCT can only restore structures inside the ROI since projections outside the ROI are completely missed. Whereas, mSTCT is able to recover the whole phantom, because some projections outside region A can be acquired by mSTCT although they are incomplete. As displayed in Fig. 7, since the ROI in the abdomen is more complex than the piecewise constant, we can see that the results from RCT and PTCT deviate from true values. However, the result from mSTCT is more faithful than that of RCT and PTCT. If we neglect the smooth caused by TV regularization, it is reasonable to admit that mSTCT can exactly image an ROI within objects.
Fig. 7. The central horizontal profiles of the ROI in images which are shown in the second row in Fig. 6.
To further quantitatively evaluate the accuracy of results, the root means square error (RMSE) and structure similarity index (SSIM) are calculated on results, which are listed in Table 3. The RMSE quantifies the difference between results and the phantom. The SSIM measures the structural similarities between two images, with a big value meaning a better result. It can be seen that the results from mSTCT are better than those from RCT and PTCT in terms of RMSE and SSIM.
Table 3. Quantitative Metrics of ROIs Marked with Dotted Circle in Fig. 6.
3.3 Realistic experiments
To validate the feasibility of mSTCT in practice, an experimental platform was established, which is shown in Fig. 8. This experimental platform consists of a micro-focus cone-beam X-ray source (Hamamatsu), a rotation platform (Newport), an FPD (iRay), two translation platforms (Newport), a control system (Newport), and a data acquisition system. The X-ray source and detector are respectively mounted on two parallel translation platforms and can be translated under the system control. The detected object is placed on the rotation platform. This experimental platform can implement mSTCT, RCT, and PTCT without changing configurations.
The distances from the object to the source and the detector are 55 mm and 204 mm, respectively. The detector has 1536×1536 units, each of which has the size of 0.085 mm×0.085 mm, so the detector size is 130.56 mm×130.56 mm. The translation distance of the source is set as 104 mm, and then the resultant system parameters R1 and R2 are about 24 mm and 54 mm, respectively. Five translations with different angles were carried out in mSTCT. The spatial resolution can reach up to 0.018 mm×0.018 mm. For comparison, RCT and PTCT are implemented under the same conditions. In PTCT, 9 translations with different angles were implemented and 100 views were equidistance collected per translation.
Firstly, a 24 mm radius artificial phantom consisted of some spades, peanuts, and red beans has been scanned under different scanning modes. Secondly, this phantom was extended to a 44 mm radius by filling it with some beans, and it has been scanned under mSTCT without changing parameters.
The middle column of raw measure data in each view is extracted for 2D reconstructions. Figure 9 compares the results in the transverse direction from different scanning modes. It shows that RCT and PTCT can only recover a fraction of the object, whereas mSTCT has the ability to image whole objects. Also, we can see that the result from RCT suffers from artifacts due to truncated projections, which is more severe in PTCT. Since projections are complete when the radius of the object is 24 mm, mSTCT can obtain images without significant artifacts, as shown in Fig. 9(c). In the case of 44 mm radius, some structures in region B are blurred by artifacts due to missed projections, whereas those in region A are far away from the influence of artifacts, as displayed in Fig. 9(d).
Fig. 9. (a)(b)(c) are the results of the 24 mm radius phantom from RCT, PTCT, and mSTCT, respectively, with image size being 1536×1536 pixels. (d) is the result of the 44 mm radius phantom from mSTCT, with image size being 2816×2816 pixels. (e) is the 3D rendered result of the 24 mm radius phantom from mSTCT, with volume size being 512×512×310 pixels. The resolutions are 0.018 mm×0.018 mm except for (e). The display window is [0, 1.5].
In this paper, a 3D SIRT reconstruction method was developed for mSTCT. A 3D image was reconstructed for the 24 mm radius phantom, with a size of 512×512×310 pixels. Figure 9(e) shows the 3D rendered image. It shows that mSTCT can beautifully image a larger object than RCT and PTCT do.
4. Discussions
In this paper, STCT/mSTCT were developed to deal with the problems that the FOV of the RCT in the transversal direction is limited and trimming samples to an appropriate size matching the FOV is difficult. The specifics of STCT are that objects locate relatively close to the source and the FOV is primarily determined by the source translation, which is possible to image objects as large as possible with high resolutions.
Since STCT is unable to offer complete projections for the accurate reconstruction which is critical for quantitative inspection, mSTCT is developed with sacrificing efficiency, which is acceptable in µCT applications. In mSTCT, a decided geometry determines two vital radii R1, R2. The circular area centered at the origin with a radius being R1 has complete projections. However, the projections for the area with a radius between R1 and R2 are incomplete. As discussed above, for an object with radius R, theoretically accurate reconstruction can be achieved when R is smaller than R1. In the case of ${R_1} < R \le {R_2}$, we mainly concern region A whether can be accurately reconstructed, which is difficult to prove from a theoretical aspect. Whereas, the numerical experiments demonstrated that this ROI is able to be faithfully recovered by the SIRT-TV method if we neglect the error caused by smoothness.
4.1 Potential application of STCT
Even though STCT is incapable of providing complete projections, it also has the potential for qualitative detections which may be suitable for security inspections and product screening. Furthermore, a dual-source STCT (dSTCT) is introduced to collect more projections, where the first STCT is perpendicular to the second one, as presented in Fig. 10(a).
Fig. 10. (a) is the geometry of dSTCT. (b) is the distribution of projections from dSTCT. (c) is the result from dSTCT.
To validate the feasibility of dSTCT, a simulation was implemented, with parameters listed in Table 1. Figure 10(b) shows the corresponding data distributions in Radon space. The Shepp–Logan phantom with a radius of 100 mm was also used for computing projections. Figure 10(c) displays the reconstruction by using SIRT-TV. Although dSTCT still has an incomplete data problem when the detector size is not large enough, it can obtain an acceptable image with slight artifacts. Since qualitative inspections are more significant in security inspections, therefore, we recommend that the radius of objects could be slightly larger than R1 to improve the efficiency, and can develop dedicated algorithms to improve the image quality from incomplete data set.
4.2 Architectures for mSTCT and dSTCT
Figure 11(a) illustrates a feasible architecture for mSTCT. It consists of a cone-beam x-ray source, a turntable, an FPD, three translational platforms. The x-ray source is mounted on a translation platform, which allows the source to move parallel to the FPD. The FPD and turntable mount on two translation platforms to mechanically adjust the distances from the object to the detector and source, respectively. The detected object is placed on the turntable, which allows rotations of the object after each translation to realize mSTCT. In practice, this architecture can be easily implemented by mounting an additional translation platform in the traditional µCT system.
Figure 11(b) depicts a schematic illustration for dSTCT, which can be used for security inspection and product screening. In this dSTCT system, the two STCTs are orthogonally arranged, and can simultaneously work. Objects move into the imaging region by conveyor, whose projections can be successively collected by two STCTs. This configuration is feasible to apply in the assembly line in terms of low system cost and simple architecture.
4.3 Difference among SGCT, PTCT and STCT
The proposed STCT is similar to SGCT [17] in terms of geometry, as illustrated in Fig. 12. However, there are substantial differences between them as following:
- 1) The object is placed relatively close to the detector in SGCT, while it locates as close to the source as possible in STCT. Hence, STCT is able to image an object with a higher magnification ratio than SGCT does.
- 2) The projections are nontruncated in SGCT due to the object located within the X-ray beam, whereas they are all truncated in STCT because the X-ray beam diverging from each source position can only illuminate a fraction of the object.
- 3) The FOV is mainly limited by the detector size in SGCT, but that is primarily determined by the translation distance of the source in STCT.
PTCT is a technique that collects projections by oppositely translating the X-ray source and detector, as shown in Fig. 12(b). The FOV of PTCT is the intersectional area of all X-ray beams. The projections outside the FOV are completely missed when the detected object extends over the detector boundaries, resulting in truncation artifacts in reconstructions. Compared with PTCT, STCT differs in that it can provide a bigger FOV than PTCT when they are at the same magnification ratio, as presented in Fig. 12(b) and (c). Actually, both SGCT and PTCT are somewhat similar to tomosynthesis [36], while STCT is different from that. Since all of them can not provide projections over 180° from one segment, multiple segments are helpful to acquire sufficient data for reconstructions.
It is also worth noting that STCT is a supplement rather than a substitute for traditional methods. This paper presented the preliminary exploration of STCT. There are still a few problems that we will solve in our further works. This paper employed SIRT-TV to reconstruct images from complete as well as incomplete projections. In our future investigations, we will derive FBP and BPF methods for complete data reconstructions. In addition, some theoretical proofs are still underway. At the same time, some methods [37,38] based on deep learning will be used for incomplete projections reconstruction.
5. Conclusion
In this paper, we proposed an STCT method for imaging large objects with high resolution. To compensate for the deficiency that projections from the STCT with finite detector size are incomplete, mSTCT was developed. We also demonstrated that projections collected from mSTCT theoretically meet the conditions required for accurate reconstructions, when detected objects locate within the FOV where each point has complete angular coverage. Even though objects are larger than the FOV, mSTCT also has the potential capability to accurately image the ROI inside objects. At the same time, we introduced dSTCT to alleviate the burden of incomplete problems in STCT. Both numerical simulations and realistic experiments validated that mSTCT can faithfully image objects within the FOV and has satisfied image qualities when objects outside the FOV, also shown that STCT and dSTCT have ability to qualitatively inspect objects. As a supplement to traditional methods, the proposed STCT-based method can not only be used for imaging large objects with high resolutions in µCT applications, but also for security inspection and product screening.
Funding
National Instrument Program of China (2013YQ030629); National Key Scientific Instrument and Equipment Development Projects of China (11827809).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
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