A Novel Geometric Parameter Self-Calibration Method for Portable CBCT Systems

Abstract

In outdoor environments or environments with space restrictions, it is difficult to transport and use conventional computed tomography (CT) systems. Therefore, there is an urgent need for rapid reconstruction of portable cone-beam CT (CBCT) systems. However, owing to its portability and the characteristics of temporary construction environments, high precision spatial location is difficult to achieve with portable CBCT systems. To overcome these limitations, we propose an iterative self-calibration improvement method with a self-calculated initial value based on the projection relationship and image features. The CT value of an open field image was used as the weight value of the projection data in the subsequent experiments to reduce the nonlinear attenuation of the projection intensity. Subsequently, an initial value was obtained based on the invariance of the rotation axis. Finally, self-calibration was realized iteratively using the reconstructed image. This method overcomes the main problem of the rotation axis invariance calibration algorithm—high similarity between the adjacent positions of symmetrical homogeneous materials. The proposed method not only improves the precision of self-calibration based on the projection relationship, but also reduces the performance cost and solution time of the self-calibration algorithm based on the image features. Thus, it satisfies the precision requirements for self-calibration of portable CBCT systems.

1. Introduction

Owing to their portability and rapid imaging characteristics, portable cone-beam computed tomography (CBCT) systems are widely used in medical diagnosis, nondestructive testing [1,2], safety inspection, and cultural relic restoration. Technological constraints and diverse operating conditions, such as harsh outdoor environments or space restrictions, limit the use of conventional computed tomography (CT) systems and prevent rapid 3D reconstruction; therefore, there is an increasing demand for rapid reconstruction of portable CBCTs. However, owing to the mobile application of portable CBCTs [3,4], it is difficult to achieve high-precision spatial location. Moreover, the influence of the hardware equipment manufacturing process on the attenuation of the intensity distribution of the projection data increases the difficulty of obtaining the high-quality projection data required for portable CBCT 3D reconstruction, thereby reducing the reconstruction quality and interfering with the detection and diagnosis results.

To obtain high quality X-ray projection data and preprocess the projection data, the nonlinear decay of the projection data intensity distribution, geometric artifacts, and the tuning problem between the projection data must be solved. The relative rest between the ray source, sample, and detector in the CBCT system is crucial. To reduce the influence of the hardware and environment on the CT projection images and improve the CT image accuracy, empty field images without sample placement, which are usually collected during the CT experiment, can be used to deduct the background [5] from the projected data image. A deviation from the CT value of the empty field image indicates that the actual focus projection is not at the geometric optical center owing to hardware and environmental influences. Consequently, the CT value of the projection image does not decay linearly from the optical center when there’s no object on the turntable, resulting in the interference of the real projection data. Herein, the CT value of the empty field image is standardized as the weight value of the subsequent experimental projection data to reduce the influence of the focus eccentricity and the nonlinear decay of the projection intensity.

Considering the geometric artifacts of the projection data reconstruction owing to the deflection error of the detector during the CT scanning process, either the offline calibration method or the self-calibration method [6] are employed to solve the geometric parameters, depending on whether a calibration object is required. The offline calibration method solves the geometric parameters based on the correspondence between the coordinates of markers in standard modules and their projected coordinates. The offline calibration method has a high calculation accuracy and versatility and does not rely on the detected object. However, the disadvantages of this method are that the calibration accuracy depends on the machining accuracy of the standard phantom and additional rotational scanning of the phantom is required. Recently, CT imaging resolutions have entered the nanoscale and the CT imaging field is limited by several millimeters [7], which requires an accurately sized body mold, leading to severe challenges in the application of the offline calibration method. The geometric artifact self-calibration method was developed to overcome the problem of the high precision phantom. The self-calibration method [8] does not require extra calibration but uses the feature information of the detected object to establish the cost function or criterion related to the geometric parameters of the system; the geometric parameters are calibrated through iterative optimization to eliminate geometric artifacts. Considering the characteristics of a portable CBCT, calibration usually involves expensive and complex steps, which is inconsistent with the possible usage scenarios of portable CBCT systems. Therefore, self-calibration algorithms or semi-self-calibration algorithms are primarily considered for the calibration of portable CBCTs. Based on the principle of the self-calibration method, it can be divided into three categories: self-calibration based on projection relationship, self-calibration based on reconstructed image features, and self-calibration based on deep learning.

The self-calibration method based on the projection relationship uses some characteristics of the projection data as the criteria to construct the cost function of the system geometric parameters associated with the projection data characteristics; the geometric parameters of the system are calibrated using an optimization method to determine the minimum cost function. In 2018, Xiao et al. [9] proposed a geometric parameter self-calibration method based on the invariance of the rotational axis. This method solves two kinds of geometric parameters—imaging and system parameters—based on the invariance of the rotating axis in the flat plate detector projection position. It can achieve a calibration effect that is comparable to the two-sphere calibration mode, with a high calibration accuracy. However, in their algorithm, the imaging parameter was determined at the extreme point with a certain tilt angle. When calculating the extreme point, different feature projections can provide robust results owing to the double requirement of both the average gray value threshold and the similarity coefficient/RMSE. However, symmetrical homogeneous substances exhibit a high similarity between adjacent positions, leading to unsatisfactory results. The self-calibration method based on the projection relation has low computational complexity but a fast solution speed. However, self-calibration can only be realized for some geometric parameters, and all CT systems cannot be solved; therefore, the practical applicability of this method is low.

The self-calibration method based on reconstructed image features uses the feature information of the reconstructed image as the criterion to iteratively solve the geometric parameters through constant updates and realize geometric artifact calibration [10,11]. In 2013, Jun et al. [12] proposed a geometric artifact self-calibration method using high-frequency energy. The theoretical basis of this method is that a reconstructed image without geometric artifacts contains significant high frequency information, which can be lost. An optimization model is built using the high-frequency energy as the criterion and the geometric parameters of the system are solved iteratively using the simplex method. This method achieves a calibration effect that is similar to the image sharpness without increasing the Gaussian filtering operation, while reducing the computational complexity. To reduce the time required for iterative reconstruction, Tan et al. [13] proposed a geometric artifact self-calibration method based on the interval partition. This method first divides the initial search space into multiple small intervals and calculates the similarity of the reconstructed slice images using the endpoint values of two adjacent intervals. The search range is constantly reduced by the maximum similarity, and the final interval with the maximum similarity represents the location of the optimal geometric parameters. In 2016, Ouadah et al. [14] used 3D-2D image registration to calibrate the geometric parameters of a CBCT system. First, the 3D reference image of the CBCT was reconstructed using a known geometric structure and the geometric parameters were calibrated by aligning the 2D-3D image. The self-calibration method based on reconstructed image features uses the direct relationship between reconstructed images and mismatched geometric parameters and can solve most geometric parameters; therefore, it has extensive applications. However, the accuracy of the registration depends on the initialization accuracy of the first projection and the known geometric parameters of the CT system. Furthermore, it requires multiple reconstructions of the projection images, necessitating extremely high computational performance and a relatively high time cost.

The self-calibration method based on deep learning has powerful feature learning and recognition capabilities that can quickly and intelligently process large quantities of imaging data and improve the calibration efficiency [15,16]. In 2017, Yang et al. [17] calibrated the center of rotation parameters of a parallel-beam CT system using a convolutional neural network (CNN). Compared to the image registration calibration method, their method could calibrate the rotation center in a human-eye-like manner, with a higher calibration accuracy. In 2019, Xiao et al. [18] proposed a geometric artifact calibration algorithm based entirely on CNNs. In this method, the neural network is trained to learn end-to-end non-mapping relationships to reconstruct images with geometric artifacts. This method not only enables the removal of geometric artifacts in reconstructed images from sector beam and CBCT systems, but also provides a calibration level that is comparable to conventional two-sphere calibration bulk modes. In 2020, Zhu et al. [19] verified the accuracy and versatility of different network structures in geometric artifact image evaluation by studying the application of CNNs in image recognition and classification. In 2020, the Nguyen [20] attempted to establish a low-cost modular calibration method using a Lego block as the calibration object. A residual network was employed to learn the bead position in the projection data, minimize the experimental projection bead and the simulated projection interbead distance, derive the error parameters, and realize the calibration. Cho et al. [21] and Chetley et al. [22] proposed another model that used two spherical steel loops placed on an acrylic cylinder for calibration. Such low-cost, simple, and flexible self-calibration methods are well suited to the requirements of a portable CBCT system. However, self-calibration methods based on deep learning require vast amounts of sample data for training. Moreover, owing to insufficient generalization of the reconstructed objects, only similar objects can be successfully calibrated.

In this paper, considering the characteristics of a portable CBCT system and the lack of vast amounts of data on reconstructed objects, we propose a modified geometric parameter self-calibration method based on rotation axis invariance and the projection relationship with image features. This method provides a better initial value for 3D-2D image registration through rotation axis invariance. This 3D-2D image registration overcomes the problem of the high similarity between adjacent positions for symmetric uniform objects, which improves the accuracy of self-calibration based on the projection relationship. In the simulation experiment 1, we could obtain symmetric lung reconstruction using the method proposed in this paper where the average deviation in the xyz direction was [0.0956,0.1717,0.0084], respectively, while using the rotation axis invariance method it was [1.244,9.333,1.333], respectively. The average deviation decreased by [1.1484,9.1613,1.3246]. The deviation imp and our method reduce the performance cost and solution time of the image feature self-calibration algorithm. It took 29,815 s using our method in experiment 2 and direct 3D-2D registration without parameter initialization took 39,432 s with the rotation axis invariance method. Therefore, it achieves the robustness and accuracy required for the self-calibration of portable CBCTs.

2. Materials and Methods

2.1. Overview of the Method: Self-Calibration for CBCT Systems

The geometry of a CBCT system is a three-dimensional coordinate system that includes the source position, isocenter (rotation axis), and detector as shown in Figure 1. The description of the geometry is based on the international standard IEC 61217 [23], which was designed for cone-beam imagers in isocentric radiotherapy systems; the rotation axis was y. Nine parameters were used per projection to define the position of the source and detector relative to the fixed coordinate system: the source position $Ts={\{T{s}_x,T{s}_y,T{s}_z\}}^T$, detector position $Td={\{T{d}_x,T{d}_y,T{d}_z\}}^T$, and rotation angle of the detector $Rd={\{R{d}_x,R{d}_y,R{d}_z\}}^T$. Assuming that the position of the source is fixed relative to the detector, the geometric parameters changed from nine to six degrees of freedom, as did the corresponding error parameters, which were $\Delta Td={\{\Delta T{d}_x,\Delta T{d}_y,\Delta T{d}_z\}}^T$ and $\Delta Rd={\{\Delta R{d}_x,\Delta R{d}_y,\Delta R{d}_z\}}^T$. According to the literature [4], $\Delta R{d}_x$,$\Delta R{d}_y$, and $\Delta T{d}_z$ have a negligible influence on reconstruction results and can be ignored.

The proposed method primarily comprises four steps:

• Data standardization of the actual projection images: overcome the influence of the nonlinear decay of the projection intensity due to uneven light and ensure that the focus of the X-ray source is not in the geometric center during the experiment;
• Cross-correlation registration method based on the CT value: used to correct the displacement deviation of projection image caused by the relative motion between the ray source, sample and detector;
• Determining the initial deflection angle $\Delta R{d}_z^{Initial}$, the deflection displacement $\Delta T{d}_x^{Initial}$, and $\Delta T{d}_y^{Initial}$ of the rotation axis in the projection image pair (180° of the rotation angle difference between the image pair) based on the invariance of the rotation axis;
• Using the initial deflections $\Delta R{d}_z^{Initial}$, $\Delta T{d}_x^{Initial}$, and $\Delta T{d}_y^{Initial}$ as the input, reconstructing the 3D reference model using the Feldkamp–Davis–Kress (FDK) algorithm and performing forward projection to obtain the 2D reference image; the measure with the projection image is Structural Similarity (SSIM);
• The Covariance Matrix Adaptation Evolutionary Strategies (CMA-ES) algorithm is used to maximize the similarity, obtain the optimal deflection angle $\Delta {\widehat{Rd}}_z$ and deflection displacement, $\Delta {\widehat{Td}}_x$, $\Delta {\widehat{Td}}_y$, and realize the self-calibration of geometric parameters. The entire algorithm is shown in Figure 2.

2.2. Standardization of the Projection Data

To avoid the influence of the nonlinear attenuation of the projection intensity owing to the geometric light center error induced during the manufacturing process, an empty shot was captured before registering the actual sample to obtain an empty field image for standardized processing.

The CT projection images are represented by a matrix $\left\{f_{ij}\right\}$, wherein $f_{ij}$ represents the gray values at different locations $(i,j)$ of the image; j is the row direction and i is the column direction. $(i_c,j_c)$ indicates that the geometric center of the image is the center of the circle projection; if the radius $r(0<r<R)$ is a circle, then any pixel on the circle meets the criteria:

$${(i-i_c)}^2+{(j-j_c)}^2=r$$

(1)

$$\overline{f}\left(r\right)=\frac{1}{n}\sum _{{(i-i_c)}^2+{(j-j_c)}^2=r}{f}_{ij},$$

(2)

where $\overline{f}\left(r\right)$ represents the mean at the distance r which is from the geometric center of the image. R is the radius of the maximum effective range. Polynomial fitting is used to obtain the radial grayscale curve $f\left(r\right)=Fitting\left(\overline{f}\left(r\right)\right)(r\in [0,R])$ of the projection image. The normalized treatment yields:

$$f_{norm}\left(r\right)=\frac{f\left(r\right)}{f_{max}},0\le f_{norm}\le 1,$$

(3)

where $f_{max}$ is the maximum gray value of the projected image.

Theoretically, the empty field projection image should be a circular projection of the linear decay from the geometric center. However, the real empty field projection was obtained owing to uneven lighting and the X-ray source focus error induced by the manufacturing process. To obtain the unaffected real projection image from the sample projection, the sample projection must be divided by the standard empty field projection $f_{norm}$ to obtain the processed sample projection image $\left\{{\widehat{f}}_{ij}\right\}$, which excludes the effect of the nonlinear decay.

$${\widehat{f}}_{ij}=\frac{f_{ij}}{f_{norm}\left(r_{ij}\right)}.$$

(4)

Using the cross-correlation registration method based on the CT value, the projection image background of the relative movement between the ray source, sample, and detector is inconsistent. Setting the empty-field image $f^0$ as the original acquisition position, $f^n$ as the background image in the sample projection image to register an image (where n is the projection image sequence), the sample projection image relative to the empty-field image line offset u, and the column offset v, the cross-correlation function is:

$$c(u,v)=\frac{{\sum }_{i,j}(f_{ij}^n-\overline{f^n})(f_{ij}^0-\overline{f^0})}{\sqrt{{\sum }_{i,j}{\left(f_{ij}^n-\overline{f^n}\right)}^2}\sqrt{{\sum }_{i,j}{\left(f_{ij}^0-\overline{f^0}\right)}^2}}.$$

(5)

Formula (5) provides the corresponding background offset $\widehat{(u,v)}$ under the maximum correlation coefficient $c_{max}$; the sample projection image translation transformation is performed to realize the registration.

$$\widehat{(u,v)}=\underset{(u,v)}{argmax}c(u,v)$$

(6)

2.3. Geometrical Error Parameter Initialization

During a CT scan, the rotation axis maintains rotational movement without any translational motion, ensuring that the projection of the rotation axis is always along the same line on the detector. This is the invariance property of the rotation axis. Based on this property, the designed algorithm can calculate $\Delta R{d}_z^{Inital}$ through a set of mirrored projections, which involves two projection images with rotation axes that are 180° apart.

During a CT scan, the object is on a circular trajectory that is centered around the rotation axis, while the projections from each angle are recorded by the detector. When the projections are recorded with rotation axes that are 180° apart, they exhibit mirrored features in the projection images owing to their symmetry through the rotation axis. In the mirrored projection images, the CT values on the rotation axis are the same, whereas the other lines differ. Therefore, the rotation axis can be determined based on the similarity of the mirrored projections. The maximum similarity coefficient of the corresponding line is the rotation axis Y. The similarity coefficient S is defined as follows [9]:

$$S=\frac{{\sum }_{i=1}^{N_{num}}(L{A}_i-\overline{L{A}_i})(L{B}_i-\overline{L{B}_i})}{\sqrt{{\sum }_{i=1}^{N_{num}}{(L{A}_i-\overline{L{A}_i})}^2{(L{B}_i-\overline{L{B}_i})}^2}},i\in [1,N_{num}]$$

(7)

where $L{A}_i$ and $L{B}_i$ are the mirrored projection CT values of each pixel, and $\overline{L{A}_i}$ and $\overline{L{B}_i}$ are their mean values; $N_{num}$ is the detector pixels of the maximum length; and i is the range of the detector pixels. Subsequently, the projection of the rotation axis on the detector plane can be determined, and $\Delta R{d}_z^{Inital}$ and $\Delta T{d}_x^{Inital}$ can be obtained.

For mirrored projections, their average gray values of corresponding row are different except the mid-plane, which is inferred as follows:

$${\overline{L}}_{mid-plane}\left(R{d}_z\right)={\overline{L}}_{mid-plane}(R{d}_z+{180}^{\circ }),R{d}_z\in [0,{360}^{\circ }]$$

(8)

$${\overline{L}}_{other-plane}\left(R{d}_z\right)\ne {\overline{L}}_{other-plane}(R{d}_z+{180}^{\circ }),R{d}_z\in [0,{360}^{\circ }],$$

(9)

where the $\overline{L}$ is the projection’s average gray value of each plane. To calculate $\Delta T{d}_y^{Inital}$, we must obtain the mid-plane position on the detector and determine its intersection with the axis of rotation. A prerequisite to determining the mid-plane is the root mean square error (RMSE), which is defined as follows:

$$RMSE=\sqrt{\frac{{\sum }_{plane}(P_{plane,center+i}\left(R{d}_z\right)-P_{plane,center-i}\left(R{d}_z\right))}{N}},i\in [1,M_{num}],$$

(10)

where $M_{num}$ is the number of pixels on the detector vertical to the $R{d}_y$ axis and $P_{plane,center}$ is the CT value of each pixel on the plane line. $\Delta T{d}_y^{Inital}$ can be obtained by finding the minimum RMSE [4].

2.4. Optimization of the Geometric Error Parameters

Based on the invariance of the rotation axis, the initial geometric error parameters $\Delta R{d}_z^{Inital}$, $\Delta T{d}_x^{Inital}$, and $\Delta T{d}_y^{Inital}$ can be obtained. The FDK algorithm is used to obtain the 3D reference model, and the CBCT system with known geometry is used to forward project the 3D reference model to obtain a 2D reference image. Using SSIM to measure the similarity between the reference projection and the real projection image, the reference image $I_M$ and projection image $I_F$ are set.

$$SSIM(I_M,I_F)=\frac{(2{\mu }_M{\mu }_F+c_1)(2{\sigma }_{MF}+c_2)}{({\mu }_M^2+{\mu }_F^2+c_1)({\sigma }_M^2+{\sigma }_F^2+c_2)},$$

(11)

where ${\mu }_M$ and ${\mu }_F$ are the mean values of $I_M$ and $I_F$, respectively; ${\sigma }_M^2$ and ${\sigma }_F^2$ are the variances of $I_M$ and $I_F$, respectively; ${\sigma }_{MF}$ is the covariance of $I_M$ and $I_F$; and $c_1$ and $c_2$ are constants, usually 0.01 and 0.03.

The CMA-ES optimizer [24,25] is used to solve the transformation that maximizes SSIM:

$$\Delta {\widehat{Rd}}_z,\Delta {\widehat{Td}}_x,\Delta {\widehat{Td}}_y=\underset{\Delta R{d}_z,\Delta T{d}_x,\Delta T{d}_y}{argmax}\sum _{n=1}^{N}SSI{M}_n(I_M(\Delta R{d}_z,\Delta T{d}_x,\Delta T{d}_y),I_F).$$

(12)

Thus, we obtain the final calibration parameters $\Delta {\widehat{Rd}}_z$, $\Delta {\widehat{Td}}_x$, $\Delta {\widehat{Td}}_y$.

3. Results

3.1. Experimental Setup

The proposed geometrical hybrid calibration method for a CBCT system was applied to a set of real CT data of a human lung obtained from the cancer imaging archive (TCIA) [26] and a set of scanned physical object data acquired from a mobile CBCT system in our institute.

Considering the first dataset, the size of the real CT was $512\times 512\times 227$ voxels, the distance between the source and detector was 1536 mm, the distance between the source and rotation center was 1000 mm, the size of the detection pixel was 2 mm, and the size of each projection was $512\times 512$ voxels.

Considering the second experiment, the distance between the source and detector was 952 mm, the distance between the source and rotation center was 593 mm, the size of the detection pixel was 0.139 mm, the size of each projection was $1440\times 240$ voxels, and the number of projections was 180, with a uniformly spaced angle increment. The detector we used was Mars 1717V, X-ray is CANON D-045S. The detailed parameters were as follows and Figure 6:

• X-ray parameters:

  -

  Maximum X-ray tube voltage of 70;

  -

  Focus of 0.4 mm;

  -

  kV—the anode (or cathode) indirectly was 35 kV;

  -

  Minimum X-ray tube voltage of 50;

  -

  kV maximum X-ray tube current of 12;

  -

  mA maximum filament current of 3.0 A.

• Detector parameters:

  -

  Types of detectors was amorphous silicon;

  -

  Type of scintillator was CsI;

  -

  Effective Imaging area (inch) was $17\times 17$;

  -

  Pixel size (μm) of 139;

  -

  Spatial resolution (lp/mm) of 3.6;

  -

  AD converted bits (bit) of 16;

  -

  Dimensions (mm${}^3$) of $460\times 460\times 15$;

  -

  Weight (kg) of 4.6;

  -

  Power dissipation (W) was Max.20.

The calibration experiments were implemented in the Python 3.8.12 programming environment using the Reconstruction Toolkit (RTK) open-source software reconstruction and 3D Slicer visualization. The codes were implemented on a personal computer running the Windows 10 operating system with an Intel^® Core™ i9-10900K CPU @ 3.70 GHz and 32 GB memory. The elementwise computation of the transform and adjoint transform in Section 2.2 were accelerated by an NVIDIA^® GeForce RTX™ 3080 graphics processor.

3.2. Calibration of Downloaded CT Data

Using the CT data as the model, the error parameters were set and forward projection was performed. The calibration algorithm was used to calculate the simulation projection data, compare the error parameters with the set point, and compare the differences between the reconstructed and true model. The geometric parameters are shown in

Table 1. Geometric parameters.

Case	$\mathbf{\Delta }{\mathit{Td}}_{\mathit{x}}$/mm	$\mathbf{\Delta }{\mathit{Td}}_{\mathit{y}}$ /mm	$\mathbf{\Delta }{\mathit{Rd}}_{\mathit{z}}$/mm
1	0	0	0
2	0	0	1.5
3	0	0	3
4	0	8	0
5	12	0	0
6	12	8	3

. The original CT model and slice are shown in Figure 3.

The 3D reconstruction is carried out by using the rotation axis invari- ance method and the method proposed in this paper, and Figure 4 and Figure 5 are obtained. Through comparison, we can find that the method in this paper is more effective. See Section 4 for specific quantification

3.3. Calibration of Real CBCT Data

The experimental equipment shown in Figure 6 was used for the self-calibration of the CBCT geometric parameters; the equipment parameters were described in Section 3.1.

4. Discussion

As shown in

Table 2. Calculated parameters.

Initial Parameters			Iterative Parameters
$\mathsf{\Delta }{\mathit{Td}}_{\mathit{x}}^{\mathit{Initial}}$/mm	$\mathsf{\Delta }{\mathit{Td}}_{\mathit{y}}^{\mathit{Initial}}$/mm	$\mathsf{\Delta }{\mathit{Rd}}_{\mathit{z}}^{\mathit{Initial}}{/}^{\circ }$	$\Delta {\widehat{\mathit{Td}}}_{\mathit{x}}$/mm	$\mathsf{\Delta }{\widehat{\mathit{Td}}}_{\mathit{y}}$/mm	$\mathsf{\Delta }{\widehat{\mathit{Rd}}}_{\mathit{z}}{/}^{\circ }$
0.6667	8.0000	−0.4000	0.0010	−0.0007	0.0109
0.6667	8.0000	−0.7980	0.0435	−0.0219	1.4981
0.0000	4.0000	0.8000	0.0375	0.0162	2.9778
2.1686	−12.0000	−0.8000	0.0284	7.9534	−0.0010
13.4286	16.0000	0.1000	11.9954	−0.0477	0.0073
14.5342	8.0000	0.8000	12.4587	8.8970	3.0068

and

Table 3. Degree of deviation of three direction $x,y,z$: (a) the rotation axis invariance method deviation $|\Delta T{d}^{Initial}-\Delta Td|,|\Delta R{d}^{Initial}-\Delta Rd|$, (b) the proposed method deviation $|\Delta \widehat{Td}-\Delta Td|,|\Delta \widehat{Rd}-\Delta Rd|$.

Deviation	1	2	3	4	5	6	Average Deviation
(a) x/mm	0.6667	0.6667	0.0000	2.1686	1.4286	2.5342	1.244
(a) y/mm	8.0000	8.0000	4.0000	20.0000	16.0000	0.0000	9.333
(a) $z{/}^{\circ }$	0.4000	2.2980	2.2000	0.8000	0.1000	2.2000	1.333
(b) x/mm	0.0010	0.0435	0.0375	0.0284	0.0046	0.4587	0.0956
(b) y/mm	0.0007	0.0219	0.0162	0.0466	0.0477	0.897	0.1717
(b) $z{/}^{\circ }$	0.0109	0.0019	0.0222	0.0010	0.0073	0.0068	0.0084

, the average error accuracy of the six cases $|\Delta {\widehat{Td}}_x-\Delta T{d}_x|$, $|\Delta {\widehat{Td}}_y-\Delta T{d}_y|$, $|\Delta {\widehat{Rd}}_z-\Delta R{d}_z|$ and based on the proposed method, were 0.0956, 0.1717, and 0.0084, respectively. The same values based on the rotation axis invariance method were 0.0956, 0.1717, and 0.0670, respectively. The rotation axis invariance method resulted in a large deviation in $\Delta T{d}_y$ and small deviations in $\Delta T{d}_x$ and $\Delta R{d}_z$. Furthermore, the real geometry deviation using the proposed method was smaller than that using the rotation axis invariance method.

Direct 3D-2D registration without parameter initialization took 39,432 s with the rotation axis invariance method in the Real Experiment. The proposed method took 29,815 s, which represents an increase of $24.39$ in the computational speed. However, the slice accuracy of the test reconstruction model was significantly inferior to that of Simulation Experiment 1 through Figure 7 and Figure 8. Comparing the input projection data, the CT value of the projection graph in Simulation Experiment 1 had a single floating point accuracy of $[-1080252,74449.1]$. The CT value of the projection image obtained from the real detector had an integer accuracy of $[0,65535]$. This is primarily why the accuracy of the reconstruction model in Simulation Experiment 2 was not as good as that in Simulation Experiment 1.

5. Conclusions

Herein, we presented a simple yet effective method to construct and calibrate a portable CBCT system. Through simulation experiments, we demonstrated the effectiveness of the proposed method. The results indicated that the proposed method can quickly distinguish between the geometric parameters of the detector. Misalignment artifacts were suppressed in the CT reconstructed volumes after correcting the acquisition geometry of the portable CBCT system. Experiments with real object projections and CT datasets also demonstrated that the proposed method can be applied to practical X-ray CBCT systems. In conclusion, the proposed calibration method can be a valuable solution for calibrating the geometry of X-ray CBCT systems.

With this calibration method, we have initially achieved the rapid shooting and reconstruction work with a portable CBCT in an outdoor environment. It can monitor the failure of small equipment and predict whether a soldier’s ankle is broken. However, the current reconstruction work must meet the requirements of the Nyquist–Shannon sampling theorem. In future, we will attempt to apply this calibration procedure to the sparse projection reconstruction work of a portable CBCT to investigate rapid X-ray reconstruction in outdoor environments using minimal projection data and a high-accuracy detector.