Geometric parameters sensitivity evaluation based on projection trajectories for X-ray cone-beam computed laminography

Abstract

BACKGROUND:

X-ray cone-beam computed laminography (CL) is widely used for large flat objects that computed tomography (CT) cannot investigate. The rotation angle of axis tilt makes geometric correction of CL system more complicated and has more uncertain factors. Therefore, it is necessary to evaluate sensitivity of the geometric parameters of CL system in advance.

OBJECTIVE:

This study aims to objectively and comprehensively evaluate sensitivity of CL geometric parameters based on the projection trajectory.

METHODS:

This study proposes the Minimum Deviation Unit (MDU) to evaluate sensitivity of CL geometric parameters. First, the projection trajectory formulas are derived according to the spatial relationship of CL system geometric parameters. Next, the MDU of the geometric parameters is obtained based on the projection trajectories and used as the evaluation index to measure the sensitivity of parameters. Then, the influence of the rotation angle of the axis tilt and magnification on the MDU of the parameters is analyzed.

RESULTS:

At low magnification, three susceptible parameters (η, u₀, v₀) with MDU less than 1 (° or mm) must be calibrated accurately to avoid geometric artifacts. The sensitivity of CL parameters increases as the magnification increases, and all parameters become highly sensitive when the magnification power is greater than 10.

CONCLUSION:

The results of this study have important guiding significance for the subsequent further parameter calibration.

Keywords

Cone-beam computed laminography X-ray laminography geometric parameters sensitivity evaluation parameters sensitivity

1 Introduction

With the advantages of high-resolution characterization of the object’s internal structure under non-contact and nondestructive conditions, X-ray Computed Tomography has been widely used in quality inspection [1, 2], medical aid diagnosis [3, 4], safety inspection [5, 6], cultural relic archaeology [7, 8]. CT scanning requires the projections of the objects at 360 angles. However, in practice, large flat objects (such as printed circuit boards) cannot rotate at a whole angle due to space limitations in the scanning process. Moreover, when the incident rays are parallel to the flat direction of the objects, it is difficult for them to penetrate due to the long attenuation path, thus limiting the imaging quality.

Computed Laminography can overcome this problem. The main difference between the scanner for CL and CT is that the rotation axis of the CT scanner is parallel to the detector. In contrast, the CL scanner has a tilt angle between the rotation axis and the detector. The tilt angle makes the rotation of the flat object no longer restricted by space and avoids the radiation along the flat direction of the object. Therefore, Computed Laminography has unique advantages in the nondestructive testation of large flat objects (such as paintings [10], printed circuit boards [11], fossils [12], and industrial plates [13]).

In X-ray CB imaging, calibration of geometric parameters is essential to avoid geometric artifacts in the reconstructed images. However, not all geometric parameters need to be calibrated with high accuracy or the same precision. Therefore, before geometric parameters calibration, it is necessary to evaluate the parameters objectively to clarify the calibration requirements. In 2011, Kingston et al. [9] evaluated the sensitivity of geometric parameters of CT based on the reconstructed theoretical model. This manuscript proposes optimal units (ou) to measure the sensitivity of parameter misalignments. For any of the parameters, the ou is defined as the perturbation of the parameter value that causes a maximum one-voxel deviation in the back-projected. To ensure a sharp tomogram, all parameter values must be determined to a precision of < 0.5 ou. Kumar et al. [10] simulated the influence of the misalignment of each part of a CT system on measurement. These analyses provide a basis for many geometric artifact correction methods [11 –14].

Even with different structures, cone-beam CL and cone-beam CT share the same imaging principle. Therefore, the geometric parameter calibration of CL can learn from CT. However, in the CL system, the system geometry is more complicated than in the CT system due to the tilt angle of the rotation axis. The additional parameter will affect the calibration requirements for geometric parameters. Therefore, to calibrate the geometric parameters of the CL system or to refer to the CT self-calibration method, it is necessary to comprehensively analyze and objectively evaluate the parameter correction requirements of the CL system. In 2021, Zhang et al. [15] evaluated the influence of geometric errors through simulation experiments. However, no theoretical analysis was performed in this study, and not all parameters were analyzed.

Currently, the evaluation methods of geometric parameter sensitivity are all based on the reconstructed image, which is complicated and affected by the accuracy and accuracy of the reconstruction algorithm. The projection offset determines the quality of the reconstructed image. On the other hand, the projection offset is directly related to the deviation of parameters, which can be analyzed by modeling. Therefore, analyzing the parameter sensitivity based on projection is feasible and reliable.

This study proposes a geometric parameter sensitivity evaluation method based on projection trajectory for the CL system. Firstly, the projection trajectory formulas are derived according to the spatial relationship of CL system geometric parameters. Then, the minimum deviation element (MDU) of each geometric parameter is obtained based on the projection trajectory and used as the evaluation index to measure the sensitivity of each parameter. Furthermore, the influence of the tilt angle of the rotation axis and magnification on the MDU of each parameter is analyzed.

2 CL projection

2.1 CL system

The structure of the X-ray cone-beam CL system is shown in Fig. 1. The system hardware mainly includes the X-ray source, the mechanical turntable, and the flat-panel detector. The X-rays from the X-ray source penetrate the object and reach the detector, which acquires two-dimensional projections of the object. During the scan, the object is rotated 360 degrees by the mechanical turntable, and the projection image of the object at each angle is obtained. Then, the reconstruction algorithm obtains the 3D image of the object from 360 2D projection images. The difference between the CL system and the CT system is that the rotation axis of the ideal CT system is parallel to the middle column of the detector. At the same time, there is a tilt angle between the rotation axis of the CL system and the detector.

Fig. 1

Structure of CL imaging system.

As shown in Fig. 1, To describe the geometry of the CL system, the world coordinate system O - XYZ is established with the detector center O as the origin according to the right-handed rule and the coordinate object system O_L - X_LY_LZ_L with the rotation axis center O_L as the origin. The rotation axis lies in the plane ZOY at an angle a to the Z axis.

The ideal geometric of the CL system is that the main beam of the ray source passes through the center of rotation and perpendicularly intersects the detector at the center point. However, in the actual system, the ray source, rotation axis, and detector components may have deviations in all directions. The analysis shows that seven parameters are sufficient to calibrate a CB scanner [16]. Considering that the deviation of the ray source and rotation axis can be converted to the error of the detector [17], in this work, these seven parameters are denoted θ, φ, η, u₀, v₀, D, R and are defined as follows. As shown in Fig. 1, the parameter θ is the tilt angle of the detector around the X-axis, φ is the deviation angle of the detector around the Y-axis, η is the in-plane rotation angle of the detector around the Z-axis. Let u and v be cartesian coordinates along vectors X and Z in the detector plane with (u₀, v₀) the projection coordinates of the main beam. The parameter D is the distance from the ray source to the detector, and R is the distance from the ray source to the rotation center. The values of parameters θ,φ, η and u₀, v₀ are 0 in the ideal system.

2.2 Ideal geometric projection

The object rotates 360 degrees about the rotation axis during the scan, and the rotation angle is denoted by w. The coordinate (x_lw, y_lw, z_lw) of the point (x_l, y_l, z_l) after rotation by w angle can be calculated by the matrix rotation formula: $[\begin{matrix} x_{lw} \\ y_{lw} \\ z_{lw} \end{matrix}] = [\begin{matrix} cos w & - sin w & 0 \\ sin w & cos w & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} x_{l} \\ y_{l} \\ z_{l} \end{matrix}]$ (1)

Points (x_lw, y_lw, z_lw) in the object coordinate system can be mapped to the world coordinate system as (x, y, z) by the rotation axis tilt matrix and the translation matrix: $[\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & cos a & - sin a \\ 0 & sin a & cos a \end{matrix}] [\begin{matrix} x_{lw} \\ y_{lw} \\ z_{lw} \end{matrix}] + [\begin{matrix} 0 \\ D - R \\ 0 \end{matrix}]$ (2)

The projection coordinates (u, v) of the point in the world coordinate system on the detector can be expressed as: $u = \frac{D}{D - y} x$ (3) $v = \frac{D}{D - y} z$ (4)

where D/D - y is the magnification. By substituting (x, y, z) into Equation (3)(4), one can get the projection trajectories u (w), v (w) of the point (x_l, y_l, z_l) on the detector in the ideal geometry: $u (w) = \frac{D \cdot x_{l} \cdot cos w - D \cdot y_{l} \cdot sin w}{R - x_{l} \cdot sin w \cdot cos a - y_{l} \cdot cos w \cdot sin a + z_{l} \cdot sin a}$ (5) $v (w) = \frac{D \cdot x_{l} \cdot sin w \cdot sin a + D \cdot y_{l} \cdot cos w \cdot sin a + D \cdot z_{l} \cdot cos a}{R - x_{l} \cdot sin w \cdot cos a - y_{l} \cdot cos w \cdot sin a + z_{l} \cdot sin a}$ (6)

2.3 Misalignment geometric projection

In the actual CL system geometry, the projection position will change when the seven geometric parameters have deviations. As shown in Fig. 2, S represents the ray source, D₀ represents the ideal detector, and D’ represents the detector with deviation. The point (x_l, y_l, z_l) on the object is mapped to the detector coordinate system as (x, y, z). The transverse angle between the beam passing through this point and the central ray is γ_x, and the longitudinal angle is γ_z. The projection coordinates on the detector D₀ are P (u, v) (Equations 5, 6), and on the detector D’ are P′ (u′, v′).

Fig. 2

The projection of the actual system geometry with parameters deviation.

As shown in Fig. 2, Δsou and Δuou₁ are in the SOX plane of the world coordinate system, while Δsov₁ and Δv₁ov₂ in the SOZ plane. ∠uou₁ is the detector deviation angle φ, and ∠v₁ov₂ is the detector tilt angle θ. For triangle ding172, according to the law of sine, we have: $\frac{| o u_{1} |}{sin (π / - 2 - γ_{x})} = \frac{| ou |}{sin (π / - 2 - φ + γ_{x})} = \frac{Δ_{1}}{sin φ}$ (7)

Since it is known that $| SO | = D, | ou | = u, sin (γ_{x}) = u / - \sqrt{D^{2} + u^{2}}, cos (γ_{x}) = D / - \sqrt{D^{2} + u^{2}}$ , Equation (7) leads to the following relations: $| o u_{1} | = \frac{u \cdot D}{D \cdot cos φ + v \cdot sin φ}$ (8) $Δ_{1} = u \cdot sin φ / - cos (φ - γ_{x})$ (9)

In the plane SuP, draw the line u₁P₁ through the point u₁ parallel to uP, known |up| = v, we can get: $\frac{| u_{1} p_{1} |}{| up |} = \frac{D / - cos γ_{x} - Δ_{1}}{D / - cos γ_{x}}$ (10) $| u_{1} p_{1} | = | o v_{1} | = v (1 - \frac{u \cdot sin φ}{D \cdot cos φ + u \cdot sin φ})$ (11)

For triangle ding173, according to the law of sine, we have: $\frac{| o v_{2} |}{sin (π / - 2 - γ_{z})} = \frac{| o v_{1} |}{sin (π / - 2 - θ - γ_{z})} = \frac{Δ_{2}}{sin θ}$ (12)

Have known that $sin γ_{2} = v / - \sqrt{D^{2} + v^{2}}, cos γ_{2} = D / - \sqrt{D^{2} + v^{2}}$ , Equation (12) leads to the following relations: $| o v_{2} | = | p^{'} u_{2} | = \frac{| o v_{1} | \cdot D}{D \cdot cos θ + | o v_{1} | \cdot sin θ} = \frac{v \cdot D}{D \cdot cos θ + v \cdot sin θ} (1 - \frac{u \cdot sin φ}{D \cdot cos φ + u \cdot sin φ})$ (13) $Δ_{2} = | o v_{1} | \cdot sin θ / - cos (θ - γ_{z})$ (14)

In the plane Sv₁P₁, draw the line v₂P′ parallel to v₁P₁, known |up| = v, one then obtains: $\frac{| v_{2} p^{'} |}{| v_{1} p_{1} |} = \frac{D / - cos γ_{z} - Δ_{2}}{D / - cos γ_{z}}$ (15) $| v_{2} p^{'} | = | v_{1} p_{1} | (1 - \frac{v \cdot sin θ}{D \cdot sin θ + v \cdot sin θ}) = \frac{u \cdot D}{D \cdot cos φ + v \cdot sin φ} (1 - \frac{v \cdot sin θ}{D \cdot cos θ + v \cdot sin θ})$ (16)

Draw the line p′u′ perpendicular to the horizontal axis of the detector, and p′v′ perpendicular to the vertical axis of the detector. ∠v₂ov′ is the same as ∠u₂ou′, which is the rotation angles η of the detector. Considering that the central projection coordinates of the detector are shifted to (u₀, v₀), the coordinates of the P′ on detector D′ are obtained as follows: $u^{'} = \frac{u \cdot D \cdot cos η}{D \cdot cos φ + v \cdot sin φ} (1 - \frac{v \cdot sin θ}{D \cdot cos θ + v \cdot sin θ}) - \frac{v \cdot D \cdot sin η}{D \cdot cos θ + v \cdot sin θ} (1 - \frac{u \cdot sin φ}{D \cdot cos φ + u \cdot sin φ}) - u_{0}$ (17) $v^{'} = \frac{u \cdot D \cdot sin η}{D \cdot cos φ + v \cdot sin φ} (1 - \frac{v \cdot sin θ}{D \cdot cos θ + v \cdot sin θ}) + \frac{v \cdot D \cdot cos η}{D \cdot cos θ + v \cdot sin θ} (1 - \frac{u \cdot sin φ}{D \cdot cos φ + u \cdot sin φ}) - v_{0}$ (18)

3 Evaluation Method

3.1 Projection trajectory

The precision and accuracy of the projections determine the quality of the reconstructed image. However, a single projection cannot reflect the imaging process, nor can it directly measure the influence of geometric parameters on imaging. To analyze the influence of geometric parameters on projection accurately and systematically, the projection trajectories will be used as the analysis basis. As shown in Fig. 3 (a), the projection trajectory can be represented by a sinogram when the object rotates around for a parallel beam X-ray scanner. According to the back projection algorithm, the CT value at any point of the object corresponds to the sum (or average) of the data on the sinogram [18]. The influence of the geometric misalignment of the system on the projection can be reflected in the sinogram, so the geometric parameters can be analyzed and corrected according to the sinusoidal diagram [19, 20].

Fig. 3

(a) projection sinogram of the parallel beam X-ray scanner. (b) Projection trajectories of cone-beam X-ray scanner in two directions.

In the scanner composed of the cone-beam X-ray source and flat-panel detector, a single trajectory cannot fully describe the projection on the two-dimensional detector when the object rotates, and the projection trajectories thus are decomposed into U-direction (transverse) trajectory and V-direction (longitudinal) trajectory, as shown in Fig. 3 (b). In the cone-beam CL system, the deviation of the geometric parameters of the system will lead to the corresponding shift of the transverse or longitudinal coordinates of the projection and will inevitably cause the change of the U-direction trajectory and V-direction trajectory. Therefore, the projection trajectories contain all the geometric information of the system and thus can be used to analyze the influence of geometric parameters on the projections.

3.2 Minimum deviation unit

In CL imaging, the projection position will shift from the ideal position when the geometric parameters deviate. The deviation of the different parameters will cause a different offset of the projection position, and the tilt angle of the rotation axis will also affect the offset. To analyze the sensitivity of each parameter at the same scale, the minimum deviation unit (MDU) is proposed as the evaluation index. The minimum deviation unit of a geometric parameter is the deviation value of a geometric parameter that causes the projection to shift to one-pixel size. The image artifact will be caused when the parameter deviation exceeds the MDU value. Considering it will lead to the reconstructed information error when the projection deviation is larger than the pixel size at any angle of the rotation axis, this method uses the maximum offset in the projection trajectory as the projection offset to determine the MDU of geometric parameters, as the Formula (19,20). The smaller the MDU of the parameter, the more sensitive the parameter. $p_{MDU} = m s . t . max (| u_{p} (w, m) - u (w) |, | v_{p} (w, m) - v (w) |) = s w \in (0 \sim 2 π)$ (19)

In Formula (19), the parameter p is any of the seven parameters (θ, φ, η, u₀, v₀, D, R). m is the deviation value of the parameter p. When p is the angle parameter (θ, φ, η), the unit of m is °. when p is the distance parameter (u₀, v₀, D, R), the unit of m is mm. When one parameter is analyzed, the other parameters have no deviation. u_p (w, m) and v_p (w, m) are respectively the u and v coordinates of the projection when the deviation of parameter p is m. u (w) and v (w) are the projection coordinates with no deviation of all parameters. The parameter w is the rotation angle of the rotation axis. The parameter s is the pixel size of the detector.

4 Simulation analysis

4.1 Influence of parameters on projection trajectories

The influence of all seven geometric parameters on projection trajectories of CBCL system is analyzed by MATLAB simulation. The parameter values of the system simulation are shown in Table 1.

Table 1
System simulation parameters

Model Coordinates (mm) Voxel size (mm) Pixel size (mm) Detector size (mm) D (mm) R (mm)

x y z

Dot 25 25 0.15 0.05 0.1 80*80 1000 500

Model	Coordinates (mm)	Voxel size (mm)	Pixel size (mm)	Detector size (mm)	D (mm)	R (mm)
Dot	25	25	0.15	0.05	0.1	80*80	1000	500

The changes of the projection trajectories when the deviation of the parameters increases gradually are obtained by simulation, as shown in Fig. 4. In the figure, red, orange, yellow, green, blue, indigo, and purple represent the projection trajectory, respectively when the parameters deviate 0, 5, 10, 15, 20, 25 and 30 units (angle or element dimension). As can be seen from Fig. 4, the influences of geometric parameters on projection trajectory are mainly manifested as lateral translation of trajectories, linear and nonlinear increase of amplitude or the combination of various changes. The V-direction projection trajectory changes greater when the detector tilt angle θ increases, mainly manifested as the overall up and downshift of the trajectory. However, the relationship between the shift amount and the parameter θ is nonlinear. The amplitude of the U-direction trajectory increases with the parameter φ increasing, and both U-direction and V-direction trajectories change significantly when η increases, where the nonlinear upward shift of trajectory mainly manifests U-direction change. At the same time, the V-direction is the change of trajectory amplitude. The deviation of u₀ leads to the overall translation of the U-direction trajectory with the V-direction trajectory unchanged, While the deviation v₀ of leads to the overall translation of the V-direction trajectory with the U-direction trace trajectory unchanged. The trajectory changes are not apparent when D and R change.

Fig. 4

The change of projection trajectories caused by parameters deviation.

4.2 MDU of parameters

The variation of the maximum projection offset with parameters is shown in Fig. 5. It shows that the three parameters η, μ₀, v₀ cause larger projection offsets for the same amount of parameter deviation.

Fig. 5

The maximum projection offset caused by the deviation of each parameter.

The MDU of each parameter is calculated according to Equation 5, as shown in Table 2. It can be seen that at the same scale, the sensitivity of the rotation angle η is much higher than that of the other two angle parameters (θ,φ), and the sensitivity of the projection center (u₀,v₀) is significantly higher than that of the other two translation parameters(D, R).

Table 2

MDU of each parameter

Parameter	θ (°)	φ (°)	η (°)	u₀ (mm)	v₀ (mm)	D (mm)	R (mm)
MDU	3.80	3.89	0.23	0.1	0.1	3.91	1.92

4.3 Effect of imaging parameters on parameter sensitivity

In CL imaging, different magnification and rotation axis tilt angles will be used according to the sample requirements. The change of these parameters will affect the projection trajectories and the MDU of the geometric parameters.

Figure 6 (a) depicts the variation of the MDU of each parameter when the tilt angle a of the rotation axis changes from 0 to 45. It can be seen that the effects of the tilt angle of the rotation axis on the parameters θ, D, R are apparent, showing as the MDU first increased and then decreased with a increasing. However, the change of the parameter a has a minor effect on the MDU of η and φ, and does no effect on u₀ and v₀. Although a has different effects on each parameter, general situation of each MDU is not changed, and the sensitivity of parameters η, u₀, v₀ is still much higher than the other four parameters.

Fig. 6

(a) Influence of the rotation axis tilt angle on MDU. (b) Influence of magnification on MDU.

For the CB scanner, the distance D from the ray source to the detector and the distance R from the object to the ray source can be adjusted to change the magnification. According to the simulation parameters in this manuscript, to facilitate magnification adjustment, let D remain unchanged. The magnification varies nonlinearly from 1.105 to 20 When R varies uniformly from 950 50 50 mm. The Fig. 6(b) shows the corresponding changes of each parameter’s MDU. It can be seen that the sensitivity of the four parameters(θ,φ, D and R) originally insensitive gradually increases with the increase of the magnification. The MDU of the seven parameters is similar and less than 1 (° or mm) when the magnification is greater than 10. However, imaging magnification is less than 10 for general imaging requirements.

According to the above analysis, the sensitivity of the four parameters (θ,φ,D and R) is lower for general imaging requirements. Moreover, three parameters (θ,φ and D) of them are relatively fixed after the installation of the CL system and are less affected by each scan. Therefore, the parameter deviation can be controlled within the MDU by high-precision measurement means during system installation and scan, which will not cause geometric artifacts. Thus, there is no need for additional means to calibrate the above parameters in subsequent scanning. The deviation of the three parameters (η, u₀ and v₀) with higher sensitivity is required to be in the order of 0.1° or 0.1 mm, which cannot be achieved by mechanical measurement. Therefore, high-precision calibration is needed for the three parameters to avoid artifacts in the reconstructed image. In addition, Fig. 6 shows that when a is greater than 5 degrees, the sensitivity of the parameters increases with the a increasing. Therefore, on the premise of satisfying the scanning field of view and resolution, the parameter a should be as small as possible during CL scanning to improve the robustness of the system to geometric errors.

5 Conclusion

This study proposes a method to evaluate the sensitivity of CL system geometric parameters by using MDU based on projection trajectories. Firstly, the projection trajectory expressions are derived according to the spatial relationship of CL system geometric parameters. Then, the MDU of each geometric parameter is obtained based on the projection trajectory and used as the evaluation index to measure the sensitivity of each parameter. Furthermore, the influence of rotation axis tilt angle and magnification on the minimum deviation element of each parameter is analyzed. The results show that the sensitivity of the four parameters (θ, φ,D,R) is lower, and the three parameters (η, u₀, v₀) are more sensitive in general CL imaging. Three susceptible parameters must be calibrated accurately to avoid geometric artifacts in the reconstructed image. Through the above analysis, we know the geometric parameter calibration requirements in the CL system, and the projection trajectory expressions derived in this study provide a basis for the calibration of geometric parameters, which have important guiding significance for the subsequent further research and the design of correction algorithm for CL system.

Code availability

MATLAB script used in this study and the data used to make the charts are published on github: https://github.com/siyu-tan/cl-parameters-evaluation.

Footnotes

Acknowledgments

This work was supported by the National Key Research and Development Program of China (Grant No. 2020YFC1522002) and the National Natural Science Foundation of China (Grant No. 62201618).

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Geometric parameters sensitivity evaluation based on projection trajectories for X-ray cone-beam computed laminography

Abstract

BACKGROUND:

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSION:

Keywords

1 Introduction

2 CL projection

2.1 CL system

3.1 Projection trajectory

4.1 Influence of parameters on projection trajectories

Table 1 System simulation parameters Model Coordinates (mm) Voxel size (mm) Pixel size (mm) Detector size (mm) D (mm) R (mm) x y z Dot 25 25 0.15 0.05 0.1 80*80 1000 500

Code availability

Footnotes

Acknowledgments

References

Table 1
System simulation parameters

Model Coordinates (mm) Voxel size (mm) Pixel size (mm) Detector size (mm) D (mm) R (mm)

x y z

Dot 25 25 0.15 0.05 0.1 80*80 1000 500