1. Introduction

Micro-computed tomography (Micro-CT) is a crucial technology and piece of equipment that utilizes radiation penetrability and reconstruction algorithms to achieve high-resolution, non-destructive imaging and detection of the interior of objects. It plays an indispensable role in various fields, including material characterization [1], biological research [2], medical research [3], and industrial inspections [4]. In practical applications, it is inevitable to require high-resolution imaging of large and long objects [5–8], such as lithium-ion batteries [9], plant stem tissue [10], and critical workpieces in aerospace [11]. However, achieving both high spatial resolution and a large field of view (FOV) is often challenging in traditional CT [4,12]. The size of the detector typically determines the FOV, and manufacturing large-sized detectors is difficult and expensive [7].

To enlarge the FOV, several efficient scanning methods have been proposed, including detector-offset scanning [13–16], traverse-continuous-rotate scanning [17], rotation-translation-translation multi-scan mode [18], rotation-translation multi-scan mode [19], elliptical trajectory [20], complementary circular scanning [21], and rotated detector [22]. Recently, straight-line scanning methods have gained significant attention [4,12,23–26], especially with the development of linearly distributed source arrays based on carbon nanotubes [23–25]. To extend the FOV, single and multiple source translation computed tomography (STCT and mSTCT) imaging geometries, along with their corresponding reconstruction algorithms, were previously proposed [4,12]. In STCT, the detector remains fixed, the measured object is positioned close to the X-ray source to achieve high geometric magnification, and the source is translated parallel to the detector for scanning.

Due to the limited angle, STCT can only reconstruct partial portions. Therefore, mSTCT is designed to acquire complete projections in order to reconstruct the image inside the FOV. Moreover, the tube X-ray source typically has a highly consistent and wide beam angle [4,24]. As a result, STCT theoretically has a long source translation distance to enlarge the FOV. However, the imaging geometry involved in extending the FOV inevitably leads to truncated projections from each perspective, and mSTCT is no exception [12]. To avoid reconstruction errors caused by truncation in STCT, an iterative algorithm combined with compressed sensing was initially developed [4]. Considering its low reconstruction efficiency, the virtual projection-based filtering backprojection (V-FBP) algorithm for mSTCT was proposed. The virtual projection was designed to circumvent truncated projections and the set of measured X-ray divergences for each detector element [12]. Due to the ramp filtering of V-FBP along the source, its cutoff frequency is affected by the source sampling points (or projection numbers). Consequently, high-quality images can only be reconstructed with a sufficiently large number of projections. Furthermore, ramp filtering is a global operator. In the extended FOV, it is necessary to have sufficiently wide redundant data and smooth weighting functions. Otherwise, the error at the truncation point will significantly impact the entire image. To further address truncated errors and improve scanning efficiency, our research group previously proposed a novel cone-beam approximate backprojection filtration (BPF) algorithm for mSTCT reconstruction [26]. Although BPF for mSTCT can reconstruct high-quality images under large geometric magnification and an extended FOV, it reduces the axial height of the cone-beam artifact-free volume [26]. This is consistent with the aforementioned scanning methods for enlarging the FOV, which cannot solve the so-called “long object” problem. One solution is to use segmented mSTCT cone-beam scanning along the measured object axis, but it affects the detection speed and axial consistency of the image [27]. Additionally, the additional image registration and mosaic operations introduce some errors.

In reality, the above-mentioned scanning methods are mainly extending the FOV in the lateral direction, whereas the FOV of the standard helical CT (HCT) is a cylinder, so it has the ability to image long objects [28]. HCT has gradually become a primary focus of CT research and development, as one scan can not only obtain more reconstruction layers but also high detection efficiency and good axial consistency [27–30]. Though HCT can solve the problem of long objects, it is actually equivalent to the circular trajectory in the top view and cannot extend the FOV laterally. The existing scanning methods for extending the lateral FOV of HCT mainly include deflection-based helical CT and dual helical CT. The deflection-based helix requires that the area covered by the ray beam at each projection angle exceeds half of the cross-sectional area of the scanning area to ensure numerical stability. The lateral FOV of the deflection-based helix is determined by the size of the detector. Dual helical CT, including the dual-source-based dual helix [31] and single-source-based dual helix [5,7,8]. The dual-source-based dual helix [31] requires that the central layer of the ray beam at each helix angle covers the entire cross-sectional area to be measured, with the aim of improving scanning speed. Each helix of the single-source-based dual helix illuminates a portion of the measured object at each viewing angle. By translating the measured object in two opposite directions along the detector direction and completing two helical scans, the cross-sectional area of the measured object can be fully covered by the two ray beams, with the aim of extending the lateral FOV [5–7]. Compared to the deflection-based helix, the double helix increases the projection data, but its data volume better meets Tuy’s data sufficiency condition for exact reconstruction [7,8].

In this paper, our inspiration comes from the transformation from the circular trajectory into the helical trajectory to satisfy Tuy’s data sufficiency condition. For this reason, we investigate the mSTCT imaging geometry and discover that the full-scan mSTCT (F-mSTCT) [32] can be represented as a non-standard helix. Therefore, we propose a novel segmented helical CT (SHCT) to enlarge the helical FOV, which consists of multiple slant source translation (SST) trajectories, to address the challenge of imaging large and long objects with high resolution. To ensure efficient reconstruction and avoid lateral projection truncation, based on our previously proposed BPF algorithm for mSTCT [26], we generalize it to SHCT and propose a generalized BPF reconstruction algorithm, which we refer to as G-BPF. The inspiration for constructing G-BPF for SHCT is derived from the general Feldkamp-Davis-Kress (G-FDK) algorithm for HCT proposed by Wang et al. [33]. Furthermore, we design various experiments to demonstrate the performance of the proposed SHCT approaches.

This paper is structured as follows: Section 2 introduces the F-mSTCT imaging geometry as a reference theory. In Section 3, we describe the proposed SHCT imaging geometry and an approximate cone-beam analytic algorithm. The numerical experiments are performed to investigate the performance and properties of the proposed SHCT and its reconstruction algorithm in Section 4. Finally, we discuss some issues with SHCT and differences with other CT scanning methods in Section 5. Section 6 concludes this work.

2. Preliminary

In STCT, the flat panel detector is fixed, and the X-ray source focus is translated parallel to the detector [4]. Due to the incomplete projections of one STCT [4], it is necessary to introduce mSTCT to acquire a complete projection data set within the FOV from different angles. In fact, to achieve complete projection data collection, mSTCT (half-scan) [12] and F-mSTCT [32] can be used, and the reconstructed results with F-mSTCT (full-scan) are more stable under the large FOVs [32]. Figure 1 illustrates the different view geometries and sinogram of F-mSTCT, the fixed coordinate origin is set at the axis of the rotary table, where R denotes the distance from the source trajectory to the fixed coordinate origin, $dod$ represents the detector-to-origin distance, $\lambda$ is the local coordinate of X-ray source focal spot on the translation trajectory, u and v are Cartesian coordinates on the flat panel detector.

 

Fig. 1. Geometry model and sinogram of F-mSTCT with six segments STCT: (a) 3D imaging model of F-mSTCT; (b) sinogram of F-mSTCT; (c) front view and illustration of (a); and (d) top view and illustration of (a).

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Some properties of mSTCT include [4]: 1) the geometric magnification can be set as large as possible by placing the measured object close to the source; and 2) the large geometric magnification can cause truncated projection, i.e., the beam only illuminates a part of the measured object, but the complete projections can be obtained by adjusting the source translation distance; 3) the FOV of mSTCT is defined as the circular area centered at the origin o with the radius of ${R_1}$ where the measured projection data set is complete, as shown in Fig. 1(b), and the formula of ${R_1}$ is

Here, the detailed derivation of ${R_1}$ can be found in the paper [4]. ${\lambda _m}$ is the half length of the source translation trajectory, ${u_m}$ is the half length of the flat panel detector, and ${v_m}$ is the half height of the flat panel detector.

In mSTCT, to achieve a large magnification by the object close to the source (i.e., the magnification is $r = (R + dod)/R$), with the source-to-detector distance $sdd$ being invariant (i.e., $sdd = R + dod$), when $R$ is decreased, $dod$ is increased. In this case, the intersection height ${Z_{rec}}$ of the reconstructed cylindrical region and the X-ray beam becomes narrower, as shown in Fig. 1(c), and the formula of ${Z_{rec}}$ according to this geometric relationship is

Here, the decrease of R inevitably leads to decrease of ${Z_{rec}}$. The FOV is enlarged, but the reconstructed axial region becomes narrow, which means that only a narrow, artifact-free cylindrical region can be reconstructed.

3. Methods

3.1 Imaging geometry of SHCT

3.1.1 Definition of SHCT trajectory

SHCT comprises multiple SST trajectories with different translation angles and z-coordinates. As illustrated in Fig. 2(a), the measured object is positioned in close proximity to the source to achieve high magnification. The detector is fixed and maintained parallel to the SST trajectory, with its midpoint aligned to the SST trajectory. When the source focus is quickly translated along the SST trajectory, the detector can acquire cone-beam projections from different views. Figures 2(b) and (c) illustrate the different views of the SST imaging geometry. The fixed coordinate system is a cartesian coordinate including x, y, and z, and its origin is located on the axis of the rotary table and at the axial height midpoint of the measured object. $\theta$ is the clockwise angle from the positive x-axis to the SST direction. $\gamma$ is the angle of elevation of the SST trajectory.

 

Fig. 2. Imaging model of SST: (a) 3D imaging model of SST; (b) and (c) are the planar geometries under different views of (a).

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To implement SHCT, the SST trajectory and the detector is fixed, and the turntable is rotated at the angle interval $\Delta \theta $ and translated at the spacing $\mathrm{\Delta }h$ along the z-axis. It is equivalent to the measured object is stationary, while the SST trajectory and detector rotate and translate around it. Figure 3 shows the imaging geometry of SHCT, and we can formulate each X-ray source focus trajectory as

Here, ${\lambda _i} \in [{ - {\lambda_m},\textrm{ }{\lambda_m}} ]$, i is the index of source sample point, $i = 1,\textrm{ }2,\textrm{ }\ldots ,\textrm{ }N$, so ${\lambda _i} = 2{\lambda _m}/(N - 1) \cdot (i - 1) - {\lambda _m}$ and N is the number of source sample points per SST ($N$ is generally set to an odd to align the midpoint correspondence with the origin of the detector). $j = 1,\textrm{ }2,\textrm{ }\ldots ,\textrm{ }J$, and J is the number of elements in the row direction of the detector, ${u_j} \in [{ - {u_m},\textrm{ }{u_m}} ]$. ${\theta _n}$ is the clockwise angle from the positive x-axis to the n-th SST direction, ${\theta _n} = ({n - 1} )\Delta \theta $, n represents the index of SST in SHCT, $n = 1,\textrm{ }2,\textrm{ }\ldots ,\textrm{ }{N_r} \cdot T$, T is the number of SST trajectories in a lap scan, and ${N_r}$ denotes the number of scan cycles of SHCT, i.e., ${N_r} \cdot T$ is total number of SST. $\Delta \theta $ is determined by the equation

 

Fig. 3. Imaging models of SHCT: (a)–(c) Imaging model and its different view when SHCT consists of six SST trajectories with scanning a lap; and (d)–(f) Imaging model and its different view when SHCT consists of 10 SST trajectories with scanning a lap.

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In Eq. (3), $z(i,n)$ is a function of the z-axis that represents X-ray source in the n-th SST, it can be expressed as

Here, the axial spacing of each source sample point in one SST trajectory is ${h_s} = \mathrm{\Delta }h\mathrm{^{\prime}}/(N - 1)$. L is the edge length of a closed polygon in the top view of SHCT, $L = \sqrt {2{R^2}(1 - \textrm{cos}\Delta \theta )} $. The total axial height of the SHCT trajectory is H, i.e.,

In Eq. (4), ${z_v}(k,n)$ denotes the z-coordinates of the detector elements in the column direction, i.e.,

3.1.2 Properties of SHCT

In Figs. 3(b) and (e), it is not difficult to find that the top-view imaging geometries of SHCT are equivalent to the F-mSTCT geometries (see Fig. 1). Therefore, according to Eq. (1) of the FOV radius ${R_1}$ of the F-mSTCT, we can get the enlarged lateral FOV radius $R_1^\mathrm{^{\prime}}$ of the SHCT with complete projection data as follows:

Here, it can be found that there is only a need to change the original ${\lambda _m}$ to ${\lambda _m}\textrm{cos}\gamma $ to achieve equivalence. Therefore, compared with HCT, SHCT not only combines the characteristics of mSTCT but also has additional properties. The summary is as follows:

Property 1-2. SHCT inherits the first two properties of mSTCT (see Section 2.1).

Property 3. In SHCT, by adjusting the length of the SST trajectory $2{\lambda _m}$ and the angle $\gamma $ of elevation (see Eq. (13)), the size of the lateral FOV can be changed easily. The longer the SST trajectory (provided that the detector can receive the large offset beam without exceeding the physical limit) and the smaller the elevation, the larger the lateral FOV.

Property 4. In SHCT, by adjusting the pitch h, the number ${N_r}$ of scan cycles of SHCT, and the SST trajectory length $2{\lambda _m}$, the axial height H of SHCT can be adjusted (see Eqs. (9)–(11)) to meet the data sufficient condition for imaging the long object.

3.1.3 Cone-beam projection

Let $f({\vec{r}} )$ be the 3D volumetric function, where $\vec{r} = (x,y,z)$ denotes a point in this function space. We consider that $f({\vec{r}} )$ is smooth and may be nonzero everywhere inside the cylinder region $\Omega $, whose definition is

Mathematically, the cone beam projections measured for source sample point ${\vec{S}_{{\theta _n}}}({\lambda _i})$ can be formulated by the local coordinate ${\lambda _i}$ of X-ray source on the n-th SST trajectory as follows:

When the cone-beam projection in the detector is denoted by $p({u,v,{\lambda_i}(n)} )$, where u and v are Cartesian coordinates in the detector, the orthogonal projection of the middle point of the SST trajectory onto the detector is chosen as the origin, and we can also get

We define the unit vectors ${\vec{e}_u}(n)$, ${\vec{e}_v}(n)$, and ${\vec{e}_w}(n)$ as related to the fixed coordinate system as

 

Fig. 4. Illustration of scanning and forward projection in SHCT.

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The detector coordinates of the line that connects $\vec{r}$ to ${\vec{S}_{{\theta _n}}}({\lambda _i})$ can be written as:

3.1.4 Differences between SHCT and other helical scans

Figures 5(a) and (b) illustrate the schematics of the deflection-based helix and dual helix CT, respectively. They are improved scanning methods to extend the lateral FOV based on HCT. In the industrial CT, the deflection-based helix often deviates the measured object along the detector direction by a small distance and rotates, while the source detector shifts along the rotation axis or the measured object shifts to achieve this. It usually requires that the area covered by the beam at each angle exceed more than half of the cross-sectional area of the scanning area to ensure numerical stability. Moreover, there is an increase in missed projections when the pitch is large. In dual helix, each helix illuminates a portion of the measured object at each view. However, by translating the measured object in two opposite directions along the detector row and completing two spiral scans, we can obtain a complete projection data set that fully covers the cross-section of the measured object [5–8].

 

Fig. 5. Schematic illustrations of different imaging geometries: (a) deflection-based helix CT; (b) dual-helix CT; and (c) broken-line helix CT.

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Overall, the proposed SHCT is geometrically similar to broken-line helix [33], as shown in Fig. 5(c). Nevertheless, there are significant differences between them, as follows:

Note that SHCT is only a supplement to traditional scanning methods, not a substitute. Additionally, with the emergence and rapid development of linearly distributed sources based on carbon nanotubes [3,23–25], they can be introduced to improve efficiency and precision.

3.2 Approximate cone-beam analytical algorithm

As is well known, the G-FDK algorithm was first proposed by Wang et al. from the raw FDK algorithm of the circular trajectory and has been the most practical method for solving HCT imaging so far [30,33–35]. Inspired by G-FDK and based on our previously developed BPF for mSTCT [26,32], we propose a G-BPF algorithm for SHCT approximate reconstruction. More specifically, referring to the previous derivative backprojection (DBP) formula for STCT [26], the DBP formula of the n-th SST (the detail is described in Supplement 1) is written as follows:

For the volumetric DBP image of the n-th SST, to achieve the transformation from ${b_{{\theta _n}}}(x,y,z)$ to the function ${f_{{\theta _n}}}(x,y,z)$, we need to perform a one-dimensional finite Hilbert inverse along parallel to the direction of the SST top-view on its z-axial layer-by-layer image, which is equivalent to the finite Hilbert inverse in the cone-beam mSTCT reconstruction [26]. Finally, adding reconstructed results of all SST scans, i.e., $f(x,y,z) = \mathop \sum \nolimits_{n = 1}^{{N_r} \cdot T} {f_{{\theta _n}}}(x,y,z)$, we get a volumetric image. However, there is a crossing between adjacent SST trajectories, not only does truncation projection occur, but there is also redundancy in the projection data. Therefore, we introduce the smooth redundant weighting function designed for F-mSTCT in the paper [32] into Eq. (22).

4. Experiments

To verify the effectiveness and performance of the proposed SHCT imaging geometry and the derived approximate analytic algorithm, we designed some numerical experiments. The reconstruction is programmed using the Astra 1.9 tool and accelerated using the GPU [36]. The experiments were conducted on a computer with an Intel Core i7-8550U CPU @ 1.80 GHz and an NVIDIA GeForce MX250. The 3D Shepp-Logan phantom was chosen as the measured object, and its projections were acquired by the line integrals along the directions of the ray.

4.1 Reconstructions with F-mSTCT, HCT and SHCT

The purpose of this experiment was to compare the reconstruction performance of HCT in G-FDK, SHCT and F-mSTCT in BPF. The setup of HCT is set to the same with partial parameters (including R, $dod$, p, ${N_r}$, and the parameters of the detector) listed in Table 1.

Table 1. Geometry parameters for SHCT experiments

View Table

In each helix turn of HCT, 1080 projections were obtained, and 251 projections were acquired per STCT and SST. With the parameter setups in Table 1, the trajectory of the SHCT and F-mSTCT scanning a lap can be determined, and the number of SST trajectories is six (see Figs. 3(a) and (b)). The Shepp-Logan phantom size was about 16.76 mm × 16.76 mm × 14 mm, and it was constructed on a 256 × 256 × 256 grid. Noted that F-mSTCT had a slightly larger FOV (see Eq. (1)) than that of SHCT (see Eq. (13)) with the same length of source translation trajectory, as a result of the SST of SHCT having an elevation angle $\gamma$.

As shown in Figs. 6(a) and (b), the reconstruction results indicate that F-mSTCT can only reconstruct large lateral objects completely, but it cannot scan and image truncated axially long objects. In Figs. 6(c)-(d) and (e)-(f), the reconstruction results show HCT and SHCT can all scan and image an axially truncated long object. SHCT can image the large object, but HCT shows serious truncated artifacts in the lateral directions as a result of its small FOV.

 

Fig. 6. Reconstruction results via BPF for F-mSTCT, G-FDK for HCT, and G-BPF for SHCT, respectively: (a)-(b), (c)-(d) and (e)-(f) are the scanning trajectories and central reconstructed slices of F-mSTCT, HCT and SHCT, respectively. A display window is [0, 1] for all.

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4.2 Reconstructions with different pitches

We perform quantitative experiments to investigate the performance of G-BPF for SHCT with different pitches. Based on Table 1, the pitch p of the SHCT trajectory was set at 1 mm, 1.5 mm, 3 mm, and 6 mm, respectively, which corresponds to the number ${N_r}$ of scan cycles of 18, 12, 6, and 3 so that the scanning trajectory would cover the phantom, and the corresponding reconstruction results are displayed in Fig. 7. The common concept of the normalized pitch $p^{\prime}$ (the definition is $p^{\prime} = p/V$, where $V$ represents the height of the detector along the z-direction at the origin [35]) was introduced, thus $p^{\prime}$ was about 0.134, 0.202, 0.403, and 0.807, respectively.

 

Fig. 7. Reconstruction results via G-BPF for SHCT with different pitches: (a)-(d) central slices when the normalized pitches are 0.807, 0.403, 0.202, and 0.134, respectively; (e)-(g) central profiles along the x, y, and z-axis of (a)-(d), respectively. A display window is [0, 1] for all.

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We apply some numerical metrics, including root mean square error (RMSE), peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM), to quantitatively evaluate the reconstruction quality. The smaller the RMSE and SSIM, the higher the PSNR, indicating higher reconstruction quality. Some axial slices show some artifacts similar to the trajectory shape of SHCT, which may be due to the approximate use of the 2D redundant weighting function to handle the redundancy of SHCT cone-beam projections [33–35]. These reconstruction results with different normalized pitches indicate the image quality increases as the normalized pitch $p^{\prime}$ decreases, which is consistent with the trend of approximate cone-beam reconstruction in HCT [33–35].

4.3 Reconstructions with different geometric magnification

We change the geometric magnification into about 13.67 by adjusting the source-to-origin distances to 15 mm and the detector-to-origin distances to 190 mm. The parameters about the detector, the pitch p, and the number of scan cycles ${N_r}$ were the same as those listed in Table 1. The length of SST trajectory $2{\lambda _m}$ was set to 24 mm. With these setups, we can know that scanning a lap requires 10 SST trajectories (see Eq. (7) of calculating T and Figs. 3(d)-(f)), an interval translation angle $\Delta \theta$ of 36°, and an angle of elevation per SST $\gamma$ of 1.8535°, the total height of SHCT trajectory H of 18.4762 mm, the radius of lateral FOV $R_1^{\prime}$ of 5.9521 mm.

To image the phantom completely, the measured Shepp-Logan phantom size was set to about 11.9 × 11.9 × 14 mm³. The reconstruction results are displayed in Fig. 8. These results basically illustrate that changing the geometric magnification allows SHCT and its BPF algorithm to be adaptive to scan and image. The RMSE, PSNR, and SSIM of the reconstruction volumes are 0.0443, 28.5962, and 0.9330, respectively. Although the slices along the axis display some artifacts due to approximate reconstruction, the profiles and numerical metrics can be accepted.

 

Fig. 8. Reconstruction results via G-BPF for SHCT with different geometric magnification and number of SST trajectories with scanning a lap: (a) the trajectory of SHCT with 10 SST trajectories when scanning a lap; (b) the three central reconstructed slices; and (c)-(e) the central profiles along the x, y, and z-axis of the reconstructed volume. A display window is [0, 1].

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4.4 Reconstructions with real data

To validate the availability of SHCT in practice, an experimental platform was constructed, as shown in Fig. 9(a). This platform mainly consists of a micro-focus X-ray source, a rotary and horizontal propulsion device, a flat panel detector, a lifting platform. The X-ray source is installed on the lifting platform, and the measured object is a circular steel pipe about 1.5 m long and 16 mm in diameter, which is filled with materials composed of aluminum powder agglomerates and tungsten particles. The geometrical parameters in Table 1 was used to scan the measured object at the ways of HCT and SHCT, respectively. The number of scan cycles ${N_r}$ is 16, and the reconstruction matrix is 256 × 256 × 600.

 

Fig. 9. The experimental platform and reconstruction results with real data: (a) the experimental platform; (b) a segment reconstruction volume, x-y slice and x-z slice, respectively, via HCT scanning and G-FDK algorithm; and (c) reconstruction results via SHCT and its G-BPF algorithm. A gray window is [0, 65535] (the bit-width is 16 bits).

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This platform can implement HCT and SHCT without changing configurations. To achieve SHCT, each SST scan can be formed by two types of composed motions, i.e., the X-ray source is raised or lowered vertically, and the measured object is continuously translated by the horizontal propulsion mechanism. The measured object is then rotated at an angle interval after one SST scan, and the next SST scan is performed. Figure 9(b) shows three orthogonal central slices of the part-reconstructed volume via HCT scanning and the G-FDK algorithm. As the comparison, Fig. 9(c) also displays the relevant reconstruction results via SHCT scanning and its G-BPF algorithm. We can see that SHCT can extend the FOV in the lateral direction to image a large and long object, while HCT exhibit severe cupping artifacts due to a small FOV. Due to the occurrence of metal artifacts, the slices may not be high-contrast, but we can still observe the internal fill material from the reconstruction results of SHCT.

5. Discussion

In this paper, we develop the SHCT to scan and image large and long objects with high resolution. SHCT can provide a larger lateral FOV than HCT at the same geometric magnification. In other words, SHCT has a larger geometric magnification or higher spatial resolution at the same lateral FOV as HCT. HCT can image the long object, but the FOV is limited in the lateral direction. Although mSTCT can conveniently enlarge the FOV in the lateral direction by controlling the source translation distance, it can only reconstruct a narrow volume as a result of the object being placed closer to the source. Briefly, SHCT inherits some of the characteristics of HCT and mSTCT simultaneously. It is possible to image a large lateral object with high resolution by adjusting the length of SST and its elevation $\gamma$. Besides, it can also image a long axial object with high resolution by setting a small pitch p and a large number of scan cycles ${N_r}$.

In the cylindrical FOV of SHCT, there is complete projection data set in the lateral area with a radius of $R_1^{\prime}$, while the projection slightly larger than the lateral area with a radius of $R_1^{\prime}$ is incomplete. When the radius of the measured object is less than $R_1^{\prime}$, theoretically exact reconstruction can be achieved. However, when the lateral radius of the measured object is slightly greater than $R_1^{\prime}$, theoretically it is not possible to use analytic reconstruction to obtain high-quality complete images, but iterative algorithms and deep learning can be used. In the axial direction of the cylindrical FOV, there is complete projection data set in the area within the height of H. When the height of the measured object is less than H, an exact volume reconstruction theoretically may be achieved.

In fact, we only perform some preliminary studies for SHCT reconstruction. Considering efficiency, we only give the approximate cone-beam reconstruction algorithm, i.e., G-BPF. Although G-BPF can reconstruct high-quality volumes with a small pitch, this kind of approximate reconstruction is practical, and its efficiency makes it popular in applications. Considering the complexity and efficiency, we only adopt the redundant weighting function in the 2D situation. However, the redundancy of SHCT is complex in the 3D layers, so it will exhibit some visible artifacts in the case of large pitches. In the future, we will try our best to address these problems. We will derive Katsevich-type [37] and BPF-type [38] exact reconstruction algorithms for SHCT with the complete projection data set. Furthermore, we will apply deep learning methods to achieve high-resolution reconstruction of SHCT with large pitches, robust 3D redundancy weighting, and reconstruction of areas larger than the radius $R_1^{\prime}$ with incomplete projection data.

Besides the developed SHCT equipment in this work, we can also design other architectures to implement the SHCT scanning method for different applications, as shown in Fig. 10. To high-resolution inspect the large workpiece placed on the rotary table, Fig. 10(a) set a lifting platform under the rotary table and install the X-ray source on the slanted linear module. Figure 10(b) displays the schematic of a vertical SHCT system used for imaging chip arrays. In this SHCT system, the slant translation scanning of the X-ray source is horizontal, the detector is fixed directly above, and the batch of chips to be tested are horizontally placed on carbon fiber boards and can be controlled by the mechanisms for rotation and translation. To achieve the SHCT scanning method, these architectures avoid a strict requirement for the source-detector system to highly synchronize with the measured object to form composite motion. This can be achieved by independent, separated motions, making control simpler and easier to achieve, such as first translating the source along the slant line, then rotating the measured object at an angle interval, and finally raising or lowering it for a certain distance. Additionally, Fig. 10(c) depicts the conceptual diagram of SHCT for medical imaging by introducing the linearly distributed source array (multi-focus).

 

Fig. 10. Different schematics of SHCT implementation: (a) the SHCT system to image the large workpiece with a certain height; (b) the small-sized vertical SHCT equipment to image chip arrays; and (c) the conceptual diagram of SHCT for medical imaging.

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6. Conclusion

In this paper, we propose a SHCT method to easily extend the FOV in both lateral and axial directions to meet Tuy's data sufficiency condition for high-resolution imaging of large and long objects. The top-view of SHCT is equivalent to the previously developed F-mSTCT imaging geometry, so its characteristics are basically consistent with those of F-mSTCT and own axial characteristics similar to HCT. To achieve effective analytical reconstruction of SHCT and avoid lateral truncation, we are inspired by the idea of G-FDK for HCT. We generalize the previously proposed BPF algorithm for mSTCT to the SHCT imaging geometry and further propose a G-BPF approximate reconstruction algorithm. Experimental results demonstrate that the G-BPF for SHCT can image a large and long object within the cylindrical FOV with high quality, especially in the case of a small pitch. As a supplement to traditional CT scanning and imaging methods, the proposed SHCT methods address the challenge of imaging large and long objects in micro-CT applications, and can be easily implemented in practical systems.