1. Introduction

In computed tomography (CT) systems, the imaging of large objects that exceed the imaging field of view (FOV) under conventional configurations is often required [1–4]. To enlarge the FOV of the CT system, over the decades, many scanning methods have been developed, such as the second-generation scanning mode [5], traverse-continuous-rotate scan mode [6], rotation-translation-translation multi-scan mode [7], detector offset [8,9], elliptical scanning trajectory [10], and complementary circular scanning [11]. Recently, by simply extending the required FOV to fit the size of the measured object, schemes for source-translation CT (STCT) and multiple source-translation CT (mSTCT) have been developed [1,2]. In mSTCT, the detector is fixed, and the object is scanned by a source translation at several different views to acquire sufficient projection data within the extended FOV to reconstruct. The mSTCT scanning method, like many previous methods of extending FOV, can cause projection truncation, and direct filtered backprojection (FBP) reconstruction will lead to truncation artifacts. Therefore, considering the characteristic of backprojection filtration (BPF)-type algorithms that can avoid the global propagation of truncation errors in the image domain, our research group has previously derived two BPF-type algorithms for mSTCT, including the BPF with derivate along the detector and source, respectively (i.e., called D-BPF and S-BPF) [3]. Furthermore, we noticed that D-BPF and S-BPF in full-scan mSTCT (i.e., F-mSTCT, where the segment number of source translation is twice that in mSTCT and covers 360°) can obtain more stable and better reconstruction results compared to the previous half-scan mSTCT mode, especially in the large extended FOV imaging [4]. Thus, here, two algorithms are renamed FD-BPF and FS-BPF to indicate the F-mSTCT reconstruction scenarios.

In the practical application of CT, testers inevitably meet large cross-sectional and long objects. With a given detector and geometric magnification, for a large and long object, mSTCT, like other scanning methods, can only extend the FOV to image large cross-sections rather than the axial [12]. As we all know, the helical CT effectively solved the so-called “long object” problem, but its lateral FOV is limited. To simultaneously extend both lateral and axial FOVs, some scanning methods based on helical scanning for enlarging lateral FOVs have been proposed, mainly including deflection-based helical CT [13,14], dual helical CT [15–17], and one-sided two-helical scans [18]. Previously, to extend a more simple and easier to engineer and implement solution, our research group proposed a novel segmented helical computed tomography (SHCT) scanning method [12]. The inspiration for SHCT derives from the evolution of circular CT into helical CT. SHCT is continuously composed of multiple slant source-translation (SST) trajectories spiraling up or down, which is actually an axial extension of F-mSTCT [4]. Therefore, SHCT has characteristics that are basically consistent with mSTCT or F-mSTCT in the lateral direction.

The reconstruction of the helical CT was mathematically more complex compared to the circular CT. In addition, it is more challenging to reconstruct volume from the cone-beam SHCT projection data due to the unique geometry of SHCT. Under the premise of acquiring competitive and accepted reconstructed results, approximate cone-beam reconstruction algorithms are the earliest developed and the preferred practical algorithms to date [19,20]. Since helical CT satisfies Tuy's data sufficiency condition, exact reconstructions have been relentlessly pursued by a large number of researchers, among whom the most representative works are PI-lines-based Katsevich and BPF-type algorithms. Over the decades, under the premise of acquiring competitive and accepted reconstructed results, approximate cone-beam reconstruction algorithms have remained popular in practical application [20]. Inspired by the idea of the general Feldkamp (G-FDK) reconstruction for helical CT proposed by Wang et al. [21], we generalized a BPF-type reconstruction scheme for the cone-beam mSTCT to SHCT. Similarly, since the basic algorithms of mSTCT include D-BPF and S-BPF, the G-BPF type algorithm can also be specifically GD-BPF or GS-BPF. Theoretically and practically, this approximate reconstruction scheme indicates that acceptable image quality can only be reconstructed at small pitches [19,21].

In the approximate reconstruction algorithms of helical CT, in addition to the G-FDK approach, another influential and representative method is reconstruction based on the single-slice rebinning algorithm (SSRB). Noo et al. [22] combined the single-slice rebinning (SSRB) method of PET (positron emission tomography) imaging and the short-scan weighting schemes for helical CT to complete a fast and simple FBP reconstruction. The main idea of the SSRB algorithm is rebinning the three-dimensional (3D) cone-beam ray of helical CT to the fan-beam ray of the virtual circular trajectory of some axially reconstructed plane and then using the two-dimensional (2D) fan-beam reconstruction algorithm to reconstruct. The SSRB-based method was also extended to other helical CT scans with an extended FOV, such as detector-biased helical scans (or half-cover helix) [13,14] and one-sided two-helical scans [18]. Marc Kachelrieb et al. [23,24] proposed an advanced single-slice rebinning (ASSR) method, which processes the large cone beam and pitch through tilted or abrupt structural planes, with success for the standard helical and even tilted helical CT. However, interpolation is required to reformat the reconstruction coordinates into Cartesian coordinates after reconstruction. Subsequently, by changing the point where the cone-beam ray intersects the reconstructed slice and the corresponding rebinning formula, an improved SSRB method was presented to weaken the artifacts for helical CT when the pitch was too large [25]. Notably, the SSRB method to some extent promotes the convenient use of deep learning for volume CT image reconstruction and processing, as it can obtain more slices for use as datasets while avoiding high computational power and a large number of volume image datasets [26–30].

In this paper, to further develop an efficient approximate algorithm for SHCT, we investigated and indicated that the SSRB formula of common helical CT cannot be satisfied with the special SHCT geometry, and then introduced and constructed an effective SSRB formula for SHCT. In implementation, we proposed some key techniques to address the challenges of searching for the nearest source sampling point incorrectly, the truncation of projection data after rebinning, and the unknown corresponding STCT scanning segment. Finally, we successfully achieved the conversion from SHCT cone-beam projection to F-mSTCT fan-beam projection and completed the layer-by-layer 2D reconstruction of the volumetric image.

This paper is structured as follows. In Section 2, we introduce the imaging mode of SHCT. Section 3 gives the SSRB formula for SHCT and proposes some key techniques. In Section 4, we conduct some numerical experiments to investigate the performance of the SSRB algorithm for SHCT and perform the real experiment. Section 5 discusses the SSRB for SHCT. Finally, we give the conclusion of this work.

2. Imaging mode of SHCT

According to Ref. [12], the SHCT imaging model consists of multiple slant source-translation trajectories, as described in Fig. 1. During each SST scan, the X-ray source is translated along a slant rail with an angle of $\gamma$ from the x-y plane and a length of ${L_s}$ while the flat panel detector (FPD) is fixed. This global coordinate system $o - xyz$ is located at the center of the object. After acquisition of one-segment projections in the n-th SST, the object can be rotated at an angle $\Delta \theta $ (it is equivalent to the combination of the source and detector rotating around the z-axis and the object keeping stationary) and then rises or falls a distance $\mathrm{\Delta }h$, continuing with the next SST scan until forming the SHCT trajectory completely covering the object.

 

Fig. 1. Imaging model of SHCT with an example of T = 6 and N_r= 3: (a) 3D geometry of SHCT, (b) 2D top-view of SHCT, and (c) 2D front-view of SHCT.

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Mathematically, we define the angle between the t-th SST trajectory and the positive x-axis as ${\theta _t}$ (i.e., ${\theta _t} = \Delta \theta (t - 1)$), the distance from the source to the rotation center (z-axis) as R, and the trajectory formula of the X-ray source focal spot can be written in a discrete form as

Here, $\mathrm{\Delta }h$ actually represents the height of one SST that removes the intersecting parts at both ends, $\mathrm{\Delta }h = h/T$. T is the required number of SST trajectories in a lap covering 360°, it can be calculated by $T = ceil(2\mathrm{\pi }/\Delta \theta )$ because the top-view of SHCT is equivalent to F-mSTCT [4,12], and $ceil({\cdot} )$ is the upward rounding function. ${N_r}$ denotes the number of these laps in SHCT. The determination of $\Delta \theta $ can be found in the paper [1]. $H$ denotes the total z-axial height of the SHCT trajectory. As illustrated in Fig. 1, when setting a variable h as a special “pitch” of SHCT, we can get the relationship as

In SHCT, the lateral FOV radius $R_1^\mathrm{^{\prime}}$ [12] with complete projection data set can be determined as

Here, d is the distance from the detector to the rotation center (z-axis), ${U_m}$ is the length of detector and ${V_m}$ is its height, and the local horizontal and vertical coordinates of the element of the FPD are $u$ and $v$, respectively.

3. SSRB algorithm for SHCT

3.1 Mathematic model of SSRB for SHCT

Referring to the basic idea of SSRB in helical CT [22,31], the rebinning of our SSRB algorithm for SHCT is to convert the cone-beam projection data set of SHCT to a stack of the (virtual) fan-beam projection data set of F-mSTCT along the z-axial reconstructed slices.

First, we define a variable $\zeta$ about the index of the source sampling points of the total SHCT scan. According to the periodicity of the SHCT geometry, we can further determine the position of the source sampling point via the value $\zeta$ for all $\zeta \in [1,\textrm{ }{N_r} \cdot T \cdot N]$, such as the n-th source sampling point in the t-th SST in Eq. (1),

In addition, the position of the source sampling point can also be formulated as the $\chi$-th source sampling point in the $c$-th lap, where one lap includes $T \cdot N$ source sampling points,

Then, as illustrated in Fig. 2, the rebinning formula can be mathematically written as:

 

Fig. 2. Schematic of SSRB in one SST trajectory of SHCT: (a) – (c) Axonometric, top, and front views of a cone-beam ray being rebinned into a virtual fan-beam ray in SHCT, respectively.

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In fact, the rebinned fan-beam projection data set of F-mSTCT needs to be reconstructed by dividing STCT, thus it can be further converted according to Eqs. (6 )–(8),

In Eq. (9), the rebinning process from a cone-beam ray to a fan-beam ray only involves the v-coordinate of the FPD. In the standard SSRB, each (virtual) fan-beam projection value is estimated from a single oblique cone-beam ray-sum where passes through the middle point $\alpha$ of the intersection of the chord within the FOV and the source focal spot nearest to the reconstructed ${z_m}$ slice [22]. However, in SHCT, we conclude that the rebinned F-mSTCT projection data set by adopting this middle point $\alpha$ is incomplete and erroneous (the loca of $\alpha$ may only be suitable for circular or arc-shape scanning geometries), and its details can be found in the Supplement 1. Due to the special linear scanning structure of SHCT, we replace the standard SSRB with an improved SSRB of the helical CT [25], i.e., selecting the point $\alpha ^{\prime}$ locating in the u'-axis of the central virtual detector, as described in Fig. 2. Under this condition, the obtained F-mSTCT projection data set is complete. Therefore, as derived in the Supplement 1, the equation of the v-coordinate is

3.2 Challenges and techniques in SSRB for SHCT

To obtain the complete fan-beam projection data set of F-mSTCT for a given ${z_m}$ slice, i.e., including T segment STCT projections or $N \cdot T$ source sampling points, the rebinning operation of SHCT needs $N \cdot T$ source sampling points (or $N \cdot T$ cone-beam projections) that are centered on the ${z_m}$ slice before and after (see Figs. 3(a) and (b)). More specifically, we use the source point coordinate $\vec{S}({n,\textrm{ }t} )$ as the center, ${\vec{S}_1}^{\prime}$ and ${\vec{S}_2}^{\prime}$ as the lower and upper limits to index the cone-beam projection data of SHCT for rebinning, where ${\vec{S}_l} = \vec{S}({n,\textrm{ }t - T/2} )$ and ${\vec{S}_u} = \vec{S}({n - 1,\textrm{ }t + T/2} )$. Thus, the maximum of $\Delta z$ is equal to $0.5(h + {L_s}\sin \gamma - \Delta h)$, where $({L_s}\sin \gamma - \Delta h)$ is the up-down redundant height of one SST trajectory.

 

Fig. 3. Illustration of SSRB process from the SHCT projection data set to the F-mSTCT projection data set: (a) Schematic of the SHCT geometry, (b) The required projection geometry for the reconstructed ${z_m}$ slice in SHCT and illustration of challenge 1, (c) Rebinned F-mSTCT geometry, (d) Cone-beam projection data set of SHCT, (e) Rebinned F-mSTCT projection data set, and (f) Projection data set of F-mSTCT for the reconstructed ${z_m}$ slice.

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According to the above-derived Eqs. (9 –13), the SSRB algorithm for SHCT seems to be effective, as the obtained projection data shapes of each STCT are similar, as shown in Fig. 3. Nevertheless, to complete the final 2D analytical reconstruction of each z-slice, it still faces some practical challenges, as follows:

To address the above challenges, we propose following corresponding techniques in the SSRB process for SHCT.

Technique 1. Avoid jumping to another SST trajectory when searching for the source sampling point closest to the reconstructed ${z_m}$ slice.

More specifically, first construct a broken-line trajectory (each segment also includes N sampling points.) that is the same as the SHCT geometry without the cross parts between two SST trajectories as

Then, we subtract ${z_m}$ from the z-coordinates ${\vec{z}^\ast }(\zeta )$ of sampling points in the broken-line trajectory, take absolute values, and find the sampling point index ${\zeta ^\ast }$ with the absolute value being the smallest, further converting it to $({n^\ast },\textrm{ }{t^\ast })$ according to Eq. (6), i.e.,

We further the obtained ${t^\ast }$ as a limitation to search for the closest source point to the reconstructed ${z_m}$ slice and get an accurate nearest source point ${n_m}$ in the ${t^\ast }$-th SST (or the nearest source index ${\zeta _m}$ in the original SHCT trajectory),

Technique 2. Recording each ${n_m}$ and concatenating the projection data of the start and end STCT, then pinning the corresponding flag $\ell$ for each complete STCT projection data set with taking $\ell ^{\prime}$ as a start within the scope of a cycle: $\ell$  = 1, 2, …, $\ell ^{\prime}$ , …, T.

First, completing the splicing of the $\ell ^{\prime}$-th truncated STCT projection data, we define a splicing function ${f_{sp}}({\textbf x},{\textbf y})$ and a spliced $\ell ^{\prime}$-th STCT projection function $p_{z_{_m}^{}}^{{F_{{\theta _{\ell ^{\prime}}}}}}({\lambda _n}^{\prime},u)$,

3.3 2D analytical reconstruction for the rebinned F-mSTCT projection data set

As an illustrative reconstruction example, we use the previously proposed D-BPF algorithm for F-mSTCT (i.e., called FD-BPF) to achieve 2D reconstruction for the rebinned each STCT projection data set $p_{z_{_m}^{}}^{{F_\theta }}(\lambda ^{\prime},u)$ [4]. By reconstructing slice-by-slice along the z-axial direction, we can finally obtain the volume. D-BPF consists of the differentiated backprojection operator that differentiates along the detector (called D-DBP, where ‘D’ denotes differentiating along the detector, DBP is the differentiated backprojection) for the STCT projection data and the finite Hilbert inverse transform for each DBP image [3]. For a given reconstructed ${z_m}$ slice, the D-DBP formula of STCT is performed to obtain a D-DBP image,

After reconstructing $M$ slices along the z-axis, a volumetric image $f(x,y,z)$ is obtained. Note that we can also reconstruct volume using the mSTCT (half-scan, T will be reduced by half) projection data set, which is similar to Noo et al.'s work [22], i.e., applying the short-scan Parker-weighted reconstruction after rebinning the helical cone-beam projections into circular fan-beam projections. Based on the conclusion of our previous work, with the requirement of a large geometric magnification and FOV, more stable and good results can be obtained using the F-mSTCT mode than mSTCT [4].

4. Experiments

To demonstrate the effectiveness and performance of the proposed SSRB algorithm for SHCT, some experiments were designed and conducted, and the partial parameters are listed in Table 1. The reconstruction programs were developed based on the Astra toolkit, and the projection and backprojection operations were accelerated on the GPU [32]. The running platform was a computer with a 12th Gen Intel Core i9-12900KF @ 3.20 GHz and an NVIDIA GeForce RTX 3090 Ti. The reconstruction is on a grid of M ³ (M = 512) cubic voxels.

Table 1. Partial parameters for SHCT reconstruction experiments

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4.1 Effectiveness of SSRB for SHCT

A standard 3D Shepp-Logan head phantom was used to verify the effectiveness of SSRB for SHCT (including the designed techniques), which fits in a cylinder of radius 8.388 mm (equal to the radius $R_1^{\prime}$ of the extended FOV) and height 8.5 mm. In simulated scanning of SHCT, the pitch h and number of scan cycles ${N_r}$ are set to 2 mm and 6, respectively, so the angle $\gamma$ of elevation per SST and total height H of SHCT trajectory can be indirectly calculated as 1.3887° and 12.6361 mm, respectively.

In the stack of rebinned F-mSTCT projection data sets of multiple slices in Figs. 4(a) and (b), the direct searching method for the index of the source nearest to the reconstructed slice yields chaotic projections, whereas the rebinned projections are more regular via Technique 1. Next, for a rebinned F-mSTCT projection data set corresponding to a certain reconstructed slice, Fig. 4(c) depicts the basic process of 2D reconstruction carrying Technique 2. More specifically, it shows some schematic steps, including concatenating the truncated projection data set of the start and endpoints and sequentially extracting the projection data set of each STCT, then pinning the corresponding flag for each complete STCT projection data set. By processing the rebinned F-mSTCT (or mSTCT) projection data sets layer-by-layer, we can get multiple 2D reconstruction slices $f(x,y,{z_m})$ to stack along the z-axis ($m = 1,2,\ldots ,M$), forming the volume $f(x,y,z)$. The reconstructed central slices are shown in Figs. 4(d) and (e) using the corresponding rebinned projections in Figs. 4(a) and (b), respectively. The reconstructed results indicate the effectiveness of our designed SSRB for SHCT. Meanwhile, we can also notice that our designed Technique 1 can avoid the streak artifacts (see the positions indicated by the yellow arrows in the local enlarged images in Figs. 4(d) and (e)) that occurred in the results of the direct searching method.

 

Fig. 4. Illustration of the effectiveness of SSRB with techniques 1 and 2 in SHCT: (a) The rebinned result by directly searching the source point nearest to the reconstructed ${z_m}$ slice with $T$ = 6, (b) The rebinned result using technique 1, (c) The reconstruction process for each rebinned F-mSTCT projection data set with the technique 2, (d) Central vertical slices of the reconstructed image based on (a), and (d) Central vertical slices of the reconstructed image based on (b). Greyscale [0, 0.5] for (d) and (e).

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4.2 Reconstructions with different pitches

To investigate the performance of SSRB for SHCT and reconstruction using FD-BPF (i.e., called SSRB (FD-BPF)), except for the above 3D Shepp-Logan head phantom, a Defrise phantom was also introduced as the test object, which consists of seven discs of intensity 1.0 with a radius of 8.388 mm and a height of 8.5 mm. The pitch h was set to 1 mm and 2 mm, the corresponding ${N_r}$ being 12 and 6, respectively, and the simulated experiments were conducted. For a given height of the detector element, according to the formula of the normalized pitch $p = (R + d) \cdot h/(R \cdot \Delta v)$ [22,33], p was about equal to 68.86 and 137.72, respectively. Previously, we proposed a kind of generalized BPF (G-BPF) reconstruction approach for SHCT. The inspiration for the derivation of G-BPF is derived from the generalized Feldkamp cone-beam reconstruction algorithm from circular CT to helical CT. By analogy, the cone-beam reconstruction algorithm for F-mSTCT is extended to SHCT [12,21]. In this experiment, under the same scanned conditions, we used D-BPF of the G-BPF type algorithm (i.e., GD-BPF, also illustrated in the Introduction section) for reconstruction as a comparison.

Figures 5 and 6 illustrate the reconstruction results, obtained for the Shepp-Logan phantom and the Defrise phantom, respectively. Compared to the truth (Fig. 5(a)) and the reconstructed results of GD-BPF (Figs. 5(b) and (c)), we notice that the results of SSRB (FD-BPF) (Figs. 5(d) and (e)) do not exhibit artifacts like the SHCT trajectory on the vertical slices (analogy to helical-shaped artifacts in the approximate cone-beam reconstruction of helical CT), and the image quality is intuitively better. In Figs. 5(f)–(h), the profiles obtained based on SSRB (FD-BPF) are more closely fitting to the real profiles, even in the larger normalized pitch (see Figs. 5(i)–(k)), whereas GD-BPF shows large differences. Furthermore, we adopt the common metrics, including root mean square error (RMSE) and structural similarity (SSIM) (the lower RMSE and the higher SSIM, the greater quality), to quantitatively evaluate reconstruction quality, as listed in Figs. 5(b)–(e). With p = 68.86, the SSIM value of SSRB (FD-BPF) is superior to that of GD-BPF, but its RMSE value does not exceed that of GD-BPF. This may be due to the existence of some bright artifacts on the periphery of the reconstructed image of SSRB (FD-BPF), which have larger values compared to the artifacts in the results of GD-BPF, resulting in a larger global error. In addition, we can also notice that SSRB (FD-BPF) will distort some details, especially at a large pitch of p = 137.72 (see central x-y and y-z slices in Fig. 5(d)). Therefore, with p = 137.72, SSRB (FD-BPF) reconstruction intuitively has fewer artifacts than GD-BPF, but the distortion of some information results in metrics lower than GD-BPF. In the reconstruction results in Fig. 6, we also know that SSRB (FD-BPF) can achieve higher z-resolution and more uniform in-plane resolution.

 

Fig. 5. Central slices through the Shepp-Logan head phantom: (a) The slices in the first column are truth; (b) and (c) GD-BPF and SSRB (FD-BPF) for SHCT with the normalized pitch p equal to 68.86; (d) and (e) GD-BPF and SSRB (FD-BPF) for SHCT (normalized pitch p = 137.72). (f) – (h) are the profiles along the central x, y, and z-axes on (a), (b), and (c), respectively; (i) – (k) are also the central profiles on (a), (d), and (e), respectively. Greyscale [0, 0.5].

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Fig. 6. Central vertical slice and profiles through the Defrise phantom: (a) and (b) Central vertical slices of GD-BPF and SSRB (FD-BPF) with the normalized pitch equal to 68.86, respectively; (c) and (d) GD-BPF and SSRB (FD-BPF) (normalized pitch = 137.72). The Greyscale [-0.3, 1.3].

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In addition, in our programming of the basic D-BPF algorithm, the finite differential and Hilbert inverse operators were performed on the CPU, and the backprojection operation was run on the GPU. Under the conditions of normalized pitch p = 137.72 and ${N_r}$ = 6, we measured the total reconstruction times for GD-BPF and SSRB (FD-BPF) from the acquisition of the cone-beam projection data set to the final reconstructed volume. The reconstruction times of GD-BPF and SSRB (FD-BPF) are 319.10 s and 214.87 s, respectively. For p = 68.86 and ${N_r}$ = 12, the reconstruction times of GD-BPF and SSRB (FD-BPF) are 642.51 s and 217.87 s, respectively. When the pitch is reduced by one time (i.e. the number of projections is doubled), the processing times of SSRB (FD-BPF) do not seem to be significantly increased. However, GD-BPF not only requires significantly more time than SSRB (FD-BPF) under the same projection number conditions, but also doubles the reconstruction time as the projection number doubles. Instead of GD-BPF, SSRB for SHCT reconstruction can improve the efficiency by about 60% ((642.51 - 217.87)/ 642.51) under the conditions of p = 68.86 and ${N_r}$ = 12.

4.3 Reconstructions with real data

Implement the SHCT scanning method on the self-built CT platform as shown in Fig. 7(a), and the scanning parameters are the same as in Table 1. Due to the system already being equipped with a square panel detector, the actual rectangular detector size is achieved by extracting projection data from the middle rows. 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 scanning parameters of the actual system are shown in Table 1. Additionally, the reconstruction voxel sizes are 256 × 256 × 400, ${N_r}$ = 18, and $h$ = 1.5 mm. For the same real projection data, the 80^th y-z and x-z slices of the reconstructed volumes via two different algorithms are displayed in Figs. 7(b) and (c), respectively. The experimental results are basically consistent with the above simulated experiments. Subjectively, SSRB (FD-BPF) for SHCT can obtain better images than GD-BPF, especially for its almost invisible artifacts on the vertical slices.

 

Fig. 7. The experimental platform and reconstruction results with real data: (a) Experimental platform; (b) The 80^th y-z and x-z slices via GD-BPF reconstruction for SHCT; and (c) The 80^th y-z and x-z slices via SSRB (FD-BPF) reconstruction for SHCT.

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5. Discussion

This paper investigates a kind of SSRB algorithm for SHCT to enable highly effective imaging of large and long objects. Our purpose is to convert the 3D reconstruction of the cone-beam projection data set of the SHCT geometry to the 2D reconstruction task of F-mSTCT with low complexity. Although the SSRB for SHCT is an approximate approach, it is simple, practical, and highly effective. This work is an expansion rather than a substitution for the previously proposed G-BPF-type algorithms for SHCT.

In SSRB for SHCT, the pitch h is limited by the height ${V_m}$ of the detector. Combining with Eqs. (3) and (10), (12) and (13), and the maximum of $\Delta z$ being equal to $0.5(h + {L_s}\sin \gamma - \Delta h)$ is given (see section 3.2), for the given the detector size, the precise relationship depends on scanner parameters R, d, ${L_s}$, $\gamma$, and T as follows:

Here, when the number of the detector rows is 256, $h\textrm{ } \le$ 3.43 mm. With the detector rows being 128, the maximum allowable pitch can be set to 1.2 mm. This means that some detector rows are not effectively utilized when using a pitch $h\textrm{ }$ less than the maximum allowable pitch. In other words, when the pitch $h\textrm{ }$ is set to 1 mm, using only 128 detector rows is sufficient, as can be observed from the reconstructed image and metrics shown in Fig. 8(a). The above experiments have verified that a smaller pitch leads to better image quality in SSRB for SHCT, which is because a smaller pitch can improve the approximation in Eq. (9).

 

Fig. 8. The reconstruction results of SSRB (FD-BPF) with 128 rows of detectors and h = 1 (a); Central vertical slice and the horizontal and vertical profiles of adding Poisson noise to the simulated projections (b)–(d) and the 128^th horizontal slice with p is equal to 206.59 and 68.86, respectively (e)–(f). Greyscale [−0.3, 1.3] for the Shepp-Logan phantom and Greyscale [−0.3, 1.3] for the Defrise phantom.

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To illustrate that SSRB for SHCT is not overly sensitive to noise, we add Poisson noise (with a photon number of 5 × 10³) to the simulated projection data of the Defrise phantom and reconstruct it (the parameters are consistent with the above noise-free experiment with p = 68.86). The vertical slice of the reconstructed image is shown in Fig. 8(b), and the corresponding horizontal and vertical center profiles are shown in Figs. 8(c) and (d). The reconstructed cross-section of SSRB for SHCT can exhibit slight artifacts of axial distortion with p = 137.72 and m = 128^th, as shown in Fig. 8(e). It is similar to the artifacts in the approximate cone-beam reconstruction of helical CT when the pitch is too large, which also occur in the SSRB-based reconstruction of helical CT, and it gradually becomes almost invisible as the pitch decreases (see Fig. 8(f)) [22]. Regarding the other issues of SSRB for SHCT, we have provided illustrations in the Supplement 1. In the future, we will continue to research and improve it.

In addition, the other role of SSRB for SHCT has opened up a convenient path for us to apply deep learning technology to SHCT. Based on obtaining slices from the medical helical CT cone-beam projections through SSRB and 2D reconstruction, many deep learning medical image processing and reconstruction works are promoted [26]. This advantage not only solves the problem of insufficient volumetric samples by obtaining a larger and richer 2D dataset but also greatly reduces the pressure for deep learning training and processing.

6. Conclusion

In this paper, we propose a SSRB-type algorithm for SHCT to provide highly effective and competitive reconstruction for large and long objects. Our study demonstrated that the rebinning formula of SSRB, which traditionally serves standard helical CT, was not applicable to the special SHCT geometry. In the SSRB for SHCT, a rebinning model was introduced and mathematically constructed using the geometry of the point where the oblique cone beam rays pass through the reconstructed slice and intersect with the horizontal axis of the central virtual detector. Meanwhile, we also designed some techniques for challenges involved in SSRB for SHCT, and then successfully achieved layer-by-layer F-mSTCT reconstruction for the volumetric image. Although SSRB seems not to provide an improvement for the SHCT reconstruction on the metrics, it can intuitively obtain results with fewer artifacts, higher z-resolution, more uniform in-plane resolution, and higher reconstruction efficiency compared to the previously proposed G-BPF type algorithm. In addition, SSRB can also reduce the amount of cone-beam projection data required as fewer detector rows can be used. In fact, the proposed SSRB for SHCT still has certain limitations, such as helical artifacts at image edges and some distorted details. In the future, we will continue to improve SSRB, for example, with advanced SSRB methods and PI-line-based BPF algorithms. Our proposed SSRB could promote the application of deep learning techniques to SHCT.