Preamble

A rectangular cross-section FOV rotational computed laminography and its analytical reconstruction method Xiang Zou, Wu-Liang Shi, Mu-Ge Du, and Yu-Xiang Xing 1, 2, 1 Department of Engineering Physics, Tsinghua University, Beijing, 100084, China Key Laboratory of Particle and Radiation Imaging (Tsinghua University), Ministry of Education, Beijing, 100084, China Rotational computed laminography (CL) has broad application potential in three-dimensional imaging of plate-like objects, as it only needs x-ray to pass through the tested object in the thickness direction during the imaging process. In this study, a rectangular cross-section field of view rotational CL (RC-CL) was proposed to circuit board imaging. Compared with other rotational CL systems, its field of view is biggest and more suitable for rectangular circuit board. Meanwhile, because the imaging geometry of RC-CL is significantly different from cone beam CT, the Feldkamp-Davis-Kress (FDK) reconstruction algorithm cannot be used directly. On the other hand, transferring the projection data to fit into the CBCT geometry by two-dimensional interpolation will introduce interpolation error. Therefore, the FDK-type analytical reconstruction algorithm applicable to the RC-CL was derived. The effectiveness of the method was validated through numerical experiments and the influence of tilt angle on the reconstruction results was analyzed. Finally, the RC-CL was applied to the real defect detection research of circuit boards.

Keywords

Computed tomography (CT), Computed laminography (CL), Field of view, FDK, Analytical reconstruction

INTRODUCTION

Computed tomography (CT) technology is widely used in

the industrial field as a non-destructive testing technique[ 1 – 3

]. However, when imaging plate-like objects such as fos- sils, paintings, composite panels in the aerospace industry, printed circuit boards (PCB), the commonly used circular cone-beam CT (CBCT) is hard to obtain high-precision three- dimensional (3D) images due to the limitation of the imaging space and radiation source energy[ ]. At the same time, computed laminography (CL) technology only requires rays to pass through the object in the thickness direction, so it has great potential for the imaging of plate-like objects[ ]. In the early days, CL can only record the image on the focal plane of an object. With the development of computer, digital detector and CL reconstruction algorithms, CL can able to obtain 3D images of objects like CT[ According to the difference of scan trajectory, CL can be divided into translational CL[ ], rotational CL[ ], and swing CL[ ], among which rotational CL is widely used due to its strong adaptability, rich projection information and same resolution in the direction[ ]. In terms of the scan-

ning geometry, rotational CL is analogous to CBCT: the de- 23

tector and x-ray source rotate 360°around the rotation axis (i.e., axis) to collect projection information. But the angles between the central ray and the rotation axis (i.e., tilt angle in Fig. (a)) in both systems are different. In CBCT, the

central ray is perpendicular to the rotation axis ( α = 90 °), 28

while the tilt angle in rotational CL is less than 90°( α < 90 °), 29

as shown in Fig. (b). And it is this characteristic enables x- ray only to pass the plate-like object in the thickness direction

during the 360°scanning process[ 16 ]. 32

In both CBCT and CL, flat-panel detectors are widely used.

Supported by the National Key Research and Development Program of China (No. 2022YFF0607802) is the center of the object, is the source, is the center of the detector, are the vertical and horizontal axes of the detector local coordinate system, respectively. The ray passing through , and in sequence is called central ray.

is the angle between the projection line of central ray on the plane and the positive -axis direction.

However, the detector in CBCT is vertically set and facing the source during rotation, which can make full use of the detector[ ]. However, different from CT, there are vari- ous settings of flat-panel detector in rotational CL. As shown in Fig. (a), the first setting is that the detector is parallel to rotation axis, which is similar to CBCT. The second and third settings are currently commonly used. The detector is perpen- dicular to the central ray during rotation in the second setting, as shown in Fig. (b). In the third setting shown in Fig. the detector is perpendicular to rotation axis and with in-plane rotation so that its -axis always points to the rotation axis.

The fourth setting is our proposed one. Similar to the third setting, its detector is perpendicular to rotation axis. How- ever, the detector has only transitional motion and the orien- tation of detector remains unchanged during rotation. Differ-

ent settings of detector mean different scanning geometries, 49

which has direct impact on the image reconstruction[ Image reconstruction is an important part of CL imaging[ ]. The existing CL reconstruction methods can be divided into three categories, such as analytical method[ ], iter-

ative method[ 22 , 23 ], and deep learning method[ 2 , 24 ]. Al- 54

though some studies[ ] have shown that deep

learning method has excellent performance in computational 56

efficiency and accuracy, there are still many challenges in this

method (e.g., the lack of training data), and further optimiza- 58

tion is needed before it can be extensively accepted. Mean- while, analytical method and iterative method are widely used in practical application. Iterative method has good noise re- sistance and the ability to process incomplete projection data.

However, it needs large amount of calculating, so difficult to achieve real-time reconstruction. On the contrary, analyt- ical algorithms have less computational complexity and no

parameter needed, making them widely used in commercial fields. But analytical method specifically bonds to imaging geometry, and different geometries require different analyti- cal algorithms[ Different reconstruction methods have different applica- tion scenarios[ ]. Although there are worse artifacts in the reconstructed images of the analytical method compared with iterative methods, it is efficient and suitable for scenar- ios requiring efficiency, e.g., online detection of circuit board defects. In the analytical algorithm study of rotational CL, Yang et al. proposed a filtering backprojection reconstruc- tion formula suitable for rotational CL in 2010[ ]. However, this method only focuses on the backprojection process and does not discuss the filtering process. Sun et al. proposed a reconstruction algorithm based on projection transforma-

tion (PT-FDK)[ 15 ]. In this method, the CL scanning data and 81

parameters are converted into those of CT that conform the FDK conditions[ ] Then, the filtering backprojection oper- ation on the converted CL data is carried out. By this way, the CL projection data can be reconstructed through standard FDK algorithm. Compared with Yang’s job, this method con- verts projection data to standard geometry and adopt standard FDK then, thus with high applicability. However, this method requires a large amount of computation, and the interpolation error could be big to degrade image reconstruction.

In this study, for fast and high-precision imaging of circuit boards, we firstly proposed a rotational CL detector setting and compared its field of view (FOV) with other detector set- tings. Then, an FDK-type analytical reconstruction algorithm for our proposed detector setting was derived and verified by numerical experiments. Finally, the proposed rotational CL scheme was validated on a real system for PCB inspection.

FOV ANALYSIS WITH DIFFERENT DETECTOR SETTINGS As shown in Fig. , during the rotational CL imaging, the imaging range under a projection angle is the quadrangu- lar pyramid region formed by x-ray source

plane: (a) the first setting, (b) the second setting, (c) the third setting, (d) the fourth setting. The red circle/box is the corresponding FOV. the four vertices of detector. The intersection of the imaging ranges under all projection angles is the field of view (FOV) of CL imaging system. The projection information of voxel points within the FOV can be recorded by the detector at all projection angles. In CL imaging, in order to ensure the qual- ity of reconstruction, it is necessary that all voxels of interest are located within the FOV. Therefore, a larger FOV allows larger objects to be scanned.

Under a projection angle , let are the intersection points of the rays

the z = z 0 plane respectively, and the quadrilateral region 113

Correspondingly, the intersection of the quadrilateral region

CL on z = z 0 plane. Generally speaking, the FOV of CL 117

on z = z 0 plane varies with different coordinates z 0 . How- 118

ever, because the circuit board has small size in the thickness direction (i.e., the direction), it is mostly important to eval- uate the FOV of the CL system during imaging on the circuit

board by directly analyzing its FOV on the z = 0 plane. 122

According to the formula derivation (see Appendix for the detailed derivation process), as shown in Fig. , under the first three settings, the shape of the imaging region

CL on z = 0 plane does not change with the projection an- 126

gle, and just rigidly rotates around the origin during imag-

ing. Therefore, their FOV shapes on z = 0 plane are circles, 128

and the radius of these circles can be determined by finding

the minimum distance from origin O to the four sides (i.e., 130

) of quadrangle region rotate around the origin . Therefore, its FOV is the quad- rangle , which is a rectangle. , are

the distance in the i th ( i =1, 2, 3, 4) setting from origin O 137

to line , respectively. Because , (the expla- nation is given in appendix A), the circle radius of the first three settings can be expressed as:

R (1) = min � H (1) O − R 1 R 2 , H (1) O − R 4 R 1

R (2) = min � H (2) O − R 1 R 2 , H (2) O − R 4 R 1

R (3) = min � H (3) O − R 1 R 2 , H (3) O − R 4 R 1

voxel), (b) transverse plane ( voxel). In practical applications, the tilt angle of CL is less than 60°, i.e., . At this time, we can obtain:

H (1) O − R 1 R 2 = L u | SO | �

Therefore, R (1) < R (2) < R (3) . Further, the areas of circle 146

FOVs are:

S (1) = π � R (1) � 2 < S (2) = π � R (2) � 2 < S (3) = π � R (3) � 2

Meanwhile, because the FOV shape of the fourth setting is rectangular, and its area can be calculated:

S (4) = 2 L u | SO | | SD | × 2 L v | SO | | SD | = 4 | SO | 2

| SD | 2 L u L v > S (3) (4) 151

To sum up, when 0 ◦ < α ≤ 60 ◦ , S (1) < S (2) < S (3) < 152

. That is to say, under the same imaging conditions, the fourth setting has the largest FOV, followed by the third and second settings, and the worst is the first setting.

To more intuitively compare the FOVs, we compared the FOV of different detector settings using numerical test. In

( L 2 u sin 2 α +4 | SD | 2 ) < H (3) O − R 1 R 2 = L u | SO | | SD |

numerical test, four rotational CL systems with different de- tector settings as shown in Fig. were simulated using the ASTRA toolbox[ ]. These systems have the same imaging parameters except for the detector setting. tion, if the projections of a reconstruction point locate inside the detector at all projection angles, this point belongs to FOV.

The more the point, the larger the FOV. To present more intuitively, Fig. shows two mutually perpendicular sections and their area values of four FOVs.

= 0 168

voxel), and Fig. (b) is the transverse plane (i.e., cross-

section, z = 0 voxel). The volumes of FOVs are also given. 170

It can be seen that, in FOV distribution, the shapes of four FOVs are irregular in coronal planes. Meanwhile, as theo- retical formulas show, the cross-sections of the first three

( L 2 u sin2 α +4 | SD | 2 ) < H (2) O − R 1 R 2 = L u | SO | √

Source to origin distance (mm) Source to detector distance (mm) Size of detector bins (mm) Number of detector bins (pixel) Number of projections Angle are circular, while because the detector has the same size in two directions, the FOV of the fourth is a special rectangle: square. The volume of FOV in first setting is the smallest, fol- lowed by the third and the second, and the fourth is largest.

Although the volume of FOV in the second and fourth is simi- lar, the second is slenderer along the -direction, and not suit- able for imaging plate-like objects.

From the above analysis, it can be concluded that the pro- posed setting has the largest FOV under the same imaging parameters. Besides, its shape of cross-section is largest and rectangular, which is beneficial to plate-like objects using CL, for example circuit board. Majorities of these objects are rectangular. Finally, its detector is horizontal, which needs smaller installation space. Because this setting has a rect- angular FOV shape, it was named ‘rectangular cross-section FOV rotational CL (RC-CL)’ in this study.

Because the imaging geometry RC-CL is different from CBCT, the classical FDK algorithm cannot be directly used.

Although one can transfer the projection data of RC-CL to fit into the CBCT geometry by 2D interpolation so that FDK can applied to reconstruction similar to PT-FDK, there are two unfavorable factors need to be considered: 1) transferring projections of RC-CL to CBCT could require a much bigger virtual detector because RC-CL projections corresponding to big cone angle in CBCT. 2) 2D interpolation error in this sit-

uation will significantly reduce image quality of reconstruc- 201

tion. Therefore, an analytical reconstruction method specif- ically for RC-CL is necessary for efficient and good recon- struction.

The 3D schematic diagram of a RC-CL system is illus- trated in Fig. (a), where a global coordinate system is defined with being the rotation axis and the origin the intersection of axis and the center ray connecting the

source ( S ) and the center of detector ( D ).The Zenith angle α 209

is referred as the CL tilt angle. Plane is the plane where the detector locates and is the intersection of plane and rota- tion axis. is the projection of on plane . Fig. (b) is the 2D schematic diagram from top view on the plane . There is a native coordinate system in the detector. During ro- tation, the directions of axis remain parallel to the axes respectively. are locations of source/detector at two different projection angles, respectively.

Formulation of analytical reconstruction on a virtual 2D problem To derive the reconstruction formula for RC-CL, we follow the idea of FDK method and start from a 2D filtered back pro- jection (FBP) reconstruction on plane . Firstly, a rotational coordinate system is configured on the detector with points to the rotation axis during rotation to form a

2D virtual problem. For convenience, we define the angle be- 224

tween axis to be our projection angle . The relation between at projection angle

� u ′ = u cos β − v sin β v ′ = u sin β + v cos β (5) 227

As shown in Fig. , on the detector plane , if we regard point as the x-ray source in the 2D problem, projection

data along v ′ at a certain v ′ = v ′∗ gives a standard view of 230

a fan-beam CT. Hence, we can apply an FBP reconstruction algorithm in this situation:

, and representing the locations of detector center, isocenter and source respectively, and the projection of on detector.

f ( x, y ) = 1 2 � 2 π 0 ( | S ′ D | − v ′∗

where ( ) is the coordinates of a reconstruction point

is the point on v ′ axis where v ′ = v ′∗ . | S ′ R ′ | is the projected 235

distance of are the projection po- sitions of point on detector. are the dis- tance from source to the detector center and origin respectively. represents projection data under co- ordinate system , and is a ramp filter.

In Eq. , the FBP filtering is performed along axis. How- ever, the projection data in RC-CL is recorded along axes. Therefore, a reconstruction formula for RC-CL will be derived based on Eq.

Firstly, to simplify the derivation, let’s define: filtered

∆ = � u ′ max u ′ min | S ′ D | − v ′∗ √

It can be further expressed as: filtered Substituting Eq. into Eq. gives filtered filtered where is the projection data on physical detector grids, which is recorded on coordinate system are the corresponding coordinate in coordinate sys- tem of In Eq. , according to the scaling property of Dirac delta function, we can obtain

δ ( u sin β + v cos β − u ∗ sin β − v ∗ cos β ) = 1 | cos β | δ ( u sin β − u ∗ sin β − v ∗ cos β cos β + v ) . (10) 257

By substituting Eq. into Eq. , we obtain:

g filtered ( β,u ∗ ,v ∗ ) ( u, v ) =

, we obtain

v = u ∗ sin β + v ∗ cos β − u sin β cos β (12) 261

f ( x, y ) = 1 2 � 2 π 0 | S ′ D | − v ′∗

Extension to a 3D scenario In 3D situation, the dimension must be considered during the reconstruction. Same as the derivation of FDK algorithm for CBCT, when extending FBP algorithm from 2D to 3D in RC-CL, two items in FBP need to be modified.

The first one is the weighting factor prior to the filtering op- eration (recorded as ). According to Eq. , the expression in FBP is

η 1 = | S ′ D ′ | √

| S ′ D ′| 2 + u ′ 2 =

Physically speaking, in 2D case, stands for the cosine of fan angle (i.e., in Fig. ) of reconstruction point a 3D case, as shown in Fig. , fan angle of reconstruction point . Meanwhile, the influence of cone angle (i.e., in Fig. ) needs to be considered. Therefore, the expression of By substituting Eq. into Eq. , we obtain: filtered

According to Fourier transforming property: F { f ( at ) } = 264

, we obtain

h ( u ∗ − u cos β ) = (cos β ) 2 h ( u ∗ − u ) . (14) 266

By substituting Eq. into Eq. , we obtain:

η 1 = cos ∠ D s SP s × cos ∠ PSP s = cos ∠ O s SP s × cos ∠ PSP (18) 287

According to the cosine theorem, Eq. can be written as:

η 1 = cos ∠ O s SP s × cos ∠ PSP s = | SP s | 2 + | SO s | 2 −| O s P s | 2

2 ·| SP s |·| SO s | | SP s | | SP | = | SP s | 2 + | SO s | 2 −| O s P s | 2

global coordinate coordinate point , Eq. can be expressed as:

η 1 = | SD | sin α − v cos β − u sin β �

By substituting Eq. into Eq. , we obtain: filtered By combining Eq. and Eq. , we obtain an FBP-type reconstruction formula for RC-CL:

is located, is the point on axis with is the reconstruction point and its projection on detector is is the projection of point on line is the projection of -located plane.

η 1 = | SD | sin α − u ∗ sin β − v ∗ cos β �

The second one is the weighting factor for backprojection (recorded as ), according to Eq. , the expression of FBP is

η 2 = ( | S ′ D | − v ′∗

Physically speaking, is determined by the source-to- detector distance and the projected distance between the source and the reconstruction point on the central ray. Therefore, as shown in Fig. , its expression in 3D case

η 2 = ( | SQ | | SR 1 | ) 2 (23) 308

According to triangle similarity theorem, we can obtain:

| SQ | | SR 1 | = | QQ s | | R 1 R 2 | = | SD | · cos( α ) z + | SO | · cos( α ) (24) 310

where is the coordinate of reconstruction point Therefore, the Eq. can be written as:

η 2 = ( | SQ | | SR 1 | ) 2 = ( | SD | · cos( α ) z + | SO | · cos( α )) 2 (25) 313

Replacing the by Eq. and Eq. , the FDK- type reconstruction formula in RC-CL can be obtained. x, y, y The implementation steps for the proposed algorithm can be summarized into three steps:

can be obtained through 1D linear interpolation of the values of point and point on the grid around it. 1) Preweighting: Multiply the two-dimensional projection data by a weighting factor computed by:

factor = | cos β | | SD | sin α − u ∗ sin β − v ∗ cos β �

2) Filtration: In numerical implementation, to lower dis-

cretization error and avoid cos β = 0 at β = π 2 or β = 3 π 2 , 323

we divide into four parts, to perform filtration, as shown in Fig.

Specially, when (i.e., Eq. ) is adopted to re- place in Eq. . The formula derivations of Eq. are based on this situation. Accordingly, the value of ordinate can be manually specified, and the filtration is per- formed along the -axis.

Meanwhile, when

| cos β | . The u = u ∗ sin β + v ∗ cos β − v cos β sin β can be adopted to 333

replace in Eq. . For concise, we do not give the detailed formula derivation here but the reader can easily derive it ac- cording to Eq. . Now, the value of coordinate can be manually specified, and the filtration is performed along -axis.

During the filtration, although we can assign the coordi- nate when filtering along axis and the coordinate when filtering along axis as integer, the corresponding

coordinates need to calculated by v = u ∗ sin β + v ∗ cos β − u sin β cos β 342

and u = u ∗ sin β + v ∗ cos β − v cos β sin β , which are usually not an in- 343

teger. So, Interpolation is needed to get the projection values at these positions. However, different from the 2D interpola- tion in projection data transferring algorithm, we only need 1D interpolation and it is easy to be done as shown in Fig. 3 [FIGURE:3]) Weighted back projection:

The 3D back-projection weighted by is similar to other FDK type reconstructions.

EXPERIMENTAL ANALYSIS Simulation study To verify the proposed reconstruction method (referred to below as CL-FDK), we simulated a RC-CL system. In the simulation, the radiation source is regarded as point source.

Meanwhile,the ray- and voxel-driven models were chosen as forward- and back-projectors, respectively.

A PCB phan- tom as shown in Fig. was used. The phantom contains three copper circuit layers that are interconnected. The mass attenuation coefficients used references the table of x-ray mass attenuation coefficients of the National Institute of Stan- dards and Technology (NIST)[ ], and the range is [0.05, 0.46]. The detailed imaging parameters were: the title an- was 45°, the distances from source to origin and de- tector center were 45.79 mm and 194.58 mm, respectively.

The detector was simulated with a 768 768 array and a 0.17 mm pixel size, and 256 projection images were acquired. The reconstruction image grids are 300 with a 0.07 0.07 mm voxel size.

To quantitatively evaluate the quality of the reconstrued PCB images, three metrics were used to measure the similar- ity between the reconstructed image and the reference image: root mean square error (RMSE), mean structural similarity index (MSSIM) and peak signal to noise ratio (PSNR). The smaller the RMSE, the better the reconstruction quality. On the contrary, the larger the MSSIM and PSNR, the better the reconstruction quality.

For comparison, the reconstruction results by the PT-FDK method[ ], and the simultaneous iterative reconstruction

technique (SIRT) were also presented, with the SIRT iteration 381

count set to 200. The reconstructed results of three algorithms are shown in Fig. and Fig. . It can be seen that all three methods can reconstruct the main structural features in the phantom. The reconstructed 2D cross section images of slice

40 (i.e., z = 40 voxel) in Fig. 12 [FIGURE:12] clearly shows that artifacts 386

are unavoidable because of the incomplete data situation of CL scan. All reconstructed images are darker compared with original image. The difference between reference and recon- structed slice images shows that SIRT result is of relatively

least artifact and PT-FDK result is with most significant er- 391

ror. Horizontal profiles along the line and line in Fig. are plotted in Fig. , which confirmed that the difference be- tween the intensity of SIRT reconstruction and the phantom is the smallest, the error in CL-FDK result is smaller than PT-FDK. construction methods. It can be seen that the SIRT recon- struction method is the best, followed by CL-FDK, and the PT-FDK is the worst.

As a filtering backprojection algo- rithm, the PT-FDK algorithm has lower accuracy than CL- FDK mainly because PT-FDK requires out of plane interpo- lation of the projection image during the transformation pro- cess, and the additional interpolation operation not only in- creases the computational load but also brings interpolation

, (b) along line errors. Influence of tilt angle In CL imaging, title angle is an important parameter.

To study its influence on RC-CL, we experimented on set-

Comparison of evaluation indicators: (a) RMSE, (b) MSSIM, (c) PSNR. ting the tilt angle to 25°, 35°, 45°, 55°and 65°respectively, with other parameters kept. Fig. shows the reconstruction results at slice #30 and their difference from reference. As shown, CL-FDK can reconstruct the main internal features of the phantom at different tilt angles. However, the smaller the was, the severer the artifact occurred. The superimposed structure from other layers is less stronger when increasing the tilt angle.

spect to tilt angle. It can be seen that the RMSE decreases with the increase of tilt angle. Its value at 25°is 1.69 times larger than 65°. On the other hand, the MSSIM increases with the increase of tilt angle.

Real experimental study In this section, we did an experimental CL scan for a PCB sample. The experiment was carried out by a RC-CL system as shown in Fig. (a). The radiation source used in exper- iment is a microfocus X-ray source. Its main characteristics

include: X-ray tube voltage operational range 60 to 110 kvp, X-ray tube current operational range 10 to 800 A, and X- ray focal spot size (nominal value) is 4 m. In this study, the X-ray tube was set at 80 kvp and 20 A. The PCB sam- ple scanned is a computer motherboard with an L-shaped as shown in Fig. (b). Since the bottom of the PCB sample is not flat, it was placed on an aluminum base during imag- ing. In the experiment, the tilt angle is set to 45°and 512

projections uniformly distributed over 2 π are acquired. Each 436

projection is of 2048 2048 detector bins and each bin size is 0.14 mm . The distance between source and detector is 263.101 mm and the distance between source and origin is 28.681 mm.

Due to the large size of the PCB, we only select several representative areas in sample for imaging during the exper- iment. Fig. (a) show the reconstructed results of ball grid array (BGA) solder joints. Multiple bubble defects featured by black holes can be seen in the reconstructed images, for example, the ones pointed by the red arrows in the figure. leads (QFN) package solder joints. The square area in the image represents the QFN solder joints, and the irregular cir- cular area inside represents the internal bubble defects. Fig. 18 [FIGURE:18] (b) also shows the grey-value profile along the yellow line in three QFN, it can be seen that the change pattern of gray values is highly correlated with the location of defects. Ac- cording to these results, it can be concluded that the proposed CL-FDK algorithm can well reconstruct the main internal fea- tures of the tested objects and be applied in real system.

CONCLUSION

This work proposed a new rotational CL imaging system with horizontal and fixed-orientation detector, and the analyt- ical reconstruction algorithm suitable for it was derived. Re- search results shows that the proposed imaging system has the biggest FOV under the same condition. On the other hand, the proposed reconstruction algorithm has superior performance over the common-used projection re-sorting reconstruction algorithm. On the basis, the influence of tilt angle on recon- struction result was analyzed and larger tilt angle is suggested for better performance. Finally, the proposed imaging system and its reconstruction algorithm were validated on a system imaging circuit boards for defect detection.

Although the proposed method gives a new way for the plate-type object 3D imaging, due to the intrinsic shortcom- ing of rotational CL (i.e., lack projection information under same angles), the reconstruction image contains interlayer aliasing artifacts, which is difficult to eliminate through tradi-

tional methods. In this situation, deep learning methods can 475

be a good choice. we are to conduct further research on this topic.

Meanwhile, it should be pointed out that although motion artifact and scattering artifact are two common artifacts in CT/CL imaging, they are not the main error in this study. For motion artifact, the step-and-shoot mode is used to record the projection images and the motions of detector and source are well controlled in this study, so the motion artifact in the re- construction image is negligible. However, if using the con- tinuous mode to record projection image, for example in the online detection of circuit boards, the influence of motion ar- tifact cannot be ignored. On the other hand, for scattering artifact, because the research object of this study is circuit board, which high contrast ratio, the effect of scatter on the reconstruction is small and we are not conducted research on this question. But we also found that the CT value of air re- gion is not zeros during the CL image reconstruction, which indicate the existence of scattering artifact in CL imaging. We think if using CL for the low contrast ratio object, a detailed analysis for the scattering artifact is essential.

APPENDIX: CALCULATION OF FOV IN ROTATIONAL CL Taking the first setting as an example, its calculation pro- cess of the FOV is introduced in detail. During the imaging process, the coordinates of the x-ray source and the detector center of CL can be expressed as

S x = | SO | sin( α ) sin( β ) S y = −| SO | sin( α ) cos( β ) S z = −| SO | cos( β )

D x = −| OD | sin( α ) sin( β ) D y = | OD | sin( α ) cos( β ) D z = | OD | cos( β )

where is the tilt angle. is the projection angle. is the distance between , and is the distance between Meanwhile, the coordinates of four vertices , and of the detector can be represented as:

P 1 x = D x − 0 . 5 L u cos( β ) P 1 y = D y − 0 . 5 L u sin( β ) P 1 z = D z + 0 . 5 L v

P 2 x = D x − 0 . 5 L u cos( β ) P 2 y = D y − 0 . 5 L u sin( β ) P 2 z = D z − 0 . 5 L v

P 3 x = D x + 0 . 5 L u cos( β ) P 3 y = D y + 0 . 5 L u sin( β ) P 3 z = D z − 0 . 5 L v

P 4 x = D x + 0 . 5 L u cos( β ) P 4 y = D y + 0 . 5 L u sin( β ) P 4 z = D z + 0 . 5 L v

where are the length and width of detector, as shown in Fig. . Based on the coordinates of

and S , the coordinates of R 1 , R 2 , R 3 , and R 4 on z = 0 plane 511

can be calculated:

R 1 x = −| SO | (L u cos α cos β − L v sin α sin β ) L v +2 | SD | cos α R 1 y = −| SO | (L u cos α sin β +L v sin α cos β ) L v +2 | SD | cos α R 1 z = 0

R 2 x = | SO | (L u cos α cos β +L v sin α sin β ) L v − 2 | SD | cos α R 2 y = | SO | (L u cos α sin β − L v sin α cos β ) L v − 2 | SD | cos α R 2 z = 0

R 3 x = −| SO | (L u cos α cos β − L v sin α sin β ) L v − 2 | SD | cos α R 3 y = −| SO | (L u cos α sin β +L v sin α cos β ) L v − 2 | SD | cos α R 3 z = 0

R 4 x = | SO | (L u cos α cos β +L v sin α sin β ) L v +2 | SD | cos α R 4 y = | SO | (L u cos α sin β − L v sin α cos β ) L v +2 | SD | cos α R 4 z = 0

According to Eq. , we can obtain:

| OR 1 | = | SO | L v +2 | SD | cos α �

| OR 2 | = | SO | | L v − 2 | SD | cos α | �

| OR 3 | = | SO | | L v − 2 | SD | cos α | �

| OR 4 | = | SO | L v +2 | SD | cos α �

| R 1 R 2 | = 2 L v | SO | cos α √

4 | SD | 2 sin 2 α +L 2 u | L 2 u − 4 | SD | 2 cos 2 α | | R 2 R 3 | = 2 L u | SO | cos α | L v − 2 | SD | cos α |

| R 3 R 4 | = 2 L v | SO | cos α √

4 | SD | 2 sin 2 α + L 2 u | L 2 u − 4 | SD | 2 cos 2 α | | R 4 R 1 | = 2 L u | SO | cos α L v +2 | SD | cos α

From Eq. , it can be noted that the coordinates of , and are related to the projection angle , but the distance from each point to the origin is independent of the projection angle , and the distance between points is also independent of the projection angle . Therefore, the

imaging ranges on z = 0 plane under different projection an- 521

can be obtained by rigidly rotating the quadrilateral region around . Finding the intersection of the quadrilateral at all projection angles is equiva-

lent to finding the minimum inscribed circle of the quadrilat- 525

with the origin as the center of the circle,

that is, finding the minimum value of the distance from the 527

T. Wang, K. Nakamoto, H. Zhang, et al ., Reweighted

anisotropic total variation minimization for limited-angle CT 548

reconstruction. IEEE Trans. Nucl. Sci. , 2742–2760 (2017).

A. Zheng, H. Gao, L. Zhang, et al ., A dual-domain deep

learning-based reconstruction method for fully 3D sparse 552

data helical CT. Phys. Med. Biol. 245030 (2020). X. Ji, T. Zhang, B. Ji, et al ., Gray characteristics analysis of strain field of coal and rock bodies around boreholes dur- ing progressive damage based on digital Image. Rock Mech. origin to the four sides of the quadrilateral According to the calculation:

H (1) O − R 1 R 2 = L u | SO | sin α √

( L 2 u +4 | SD | 2 sin 2 α ) H (1) O − R 2 R 3 = L v | SO | sin α | L v − 2 | SD | cos α | H (1) O − R 3 R 4 = L u | SO | sin α √

( L 2 u +4 | SD | 2 sin 2 α ) H (1) O − R 4 R 1 = L v | SO | sin α L v +2 | SD | cos α

From Eq. , we can know

H (1) O − R 4 R 1 < H (1) O − R 2 R 3 . Because the value of H (1) O − R 4 R 1 532

is related to the tilt angle , detector size and , the radius of the inscribed circle can be expressed as

R (1) = min � H (1) O − R 1 R 2 , H (1) O − R 4 R 1

. Similarly, the distance formulas in the second setting are:

H (2) O − R 1 R 2 = L u | SO | √

( L 2 u sin 2 α +4 | SD | 2 ) H (2) O − R 2 R 3 = L v | SO | | L v sin α − 2 | SD | cos α | H (2) O − R 3 R 4 = L u | SO | √

( L 2 u sin 2 α +4 | SD | 2 ) H (2) O − R 4 R 1 = L v | SO | L v sin α +2 | SD | cos α

The radius of the inscribed circle can be expressed as

R (2) = min � H (2) O − R 1 R 2 , H (2) O − R 4 R 1

The distance formulas in the third setting are:

H (3) O − R 1 R 2 = L u | SO | | SD | H (3) O − R 2 R 3 = L v | SO | | SD | H (4) O − R 3 R 4 = L u | SO | | SD | H (3) O − R 4 R 1 = L v | SO | | SD |

The radius of the inscribed circle can be expressed as

R (3) = min � H (3) O − R 1 R 2 , H (3) O − R 4 R 1

. In the fourth setting, the distance formulas are the same as Eq. , and its FOV area is

S (4) = 2 L u | SO | | SD | × 2 L v | SO | | SD | = 4 L u L v | SO | 2

Rock Eng. , 5607–5620 (2023). 023-03351-x E.E. Ghandourah, S.H.A. Hamidi, K.A. Mohd Salleh, et al Evaluation of welding imperfections with X-ray computed laminography for NDT inspection of carbon steel plates. J.

Nondestruct. Eval. , 1–11 (2023). 00989-z L. Shi, C. Wei, T. Jia, et al ., An automatic measurement method of PCB stub based on rotational computed laminog- raphy imaging. IEEE Trans. Nucl. Sci. , 2201–2211 (2023).

Zhang, Shang, Improve- ments to conventional X-ray tube-based cone-beam com- puted tomography system. Nucl. Sci. Tech. , 43 (2018).

S.L. Fisher, D.J. Holmes, J.S. Jørgensen, et al ., Laminog- raphy in the lab: imaging planar objects using a conven- tional x-ray CT scanner. Meas. Sci. Technol. , 35401 (2019).

[8] Y. Li, M. Luo, X. Fei, et al ., Online learning for DBC 577

segmentation of new IGBT samples based on computed laminography imaging. Discov. Appl. Sci. , 145 (2024).

Y. Zhang, M. Yang, Y. Wu, et al ., A new CL recon- struction method under the displaced sample stage scan-

ning mode. IEEE Trans. Nucl. Sci. 68 , 2574–2586 (2021). 583

J. Fu, B.H. Jiang, B. Li, Large field of view computed laminog-

raphy with the asymmetric rotational scanning geometry. China 586

Technol. Sci. , 2261–2271 (2023). H. Deyhle, H. Towsyfyan, A. Biguri, et al ., Spatial resolu- tion of a laboratory based X-Ray cone-beam laminography

scanning system for various trajectories. NDT and E Int. 111 , 591

102222 (2020). Z.J. Tian, H.J. Yu, L.B. Wang, et al ., Orthogonal translation computed laminography. Acta Phys. Sin. , 2211002 (2020).

P. Ji, Y. Jiang, R. Zhao, et al ., Fusional laminography: A strategy for exact reconstruction on CL and CT informa- tion complementation. NDT and E Int. , 102991 (2024).

Y. Jiang, J. Zou, X. Hu, et al ., Study on optimal lamino- graphic tilt angle: a method for analyzing quantity information gained in projections. IEEE Access , 38164–38173 (2001).

L. Sun, G. Zhou, Z. Qin, et al ., A reconstruction method for cone-beam computed laminography based on projection transformation. Meas. Sci. Technol. , 45403 (2021). 10.1088/1361-6501/abc965 X. Zhao, W. Jiang, X. Zhang, et al ., Image reconstruc- tion based on nonlinear diffusion model for limited-angle computed tomography. Inverse Probl. , 45015 (2024). 10.1088/1361-6420/ad2695 X. Guo, L. Zhang, Y. Xing, Study on analytical noise propa- gation in convolutional neural network methods used in com- puted tomography imaging. Nucl. Sci. Tech. , 77 (2022). 10.1007/s41365-022-01057-3 G. Ma, X. Zhao, Y. Zhu, et al ., Projection-to-image transform frame: A lightweight block reconstruction network for com- puted tomography. Phys. Med. Biol. , 35010 (2022). 10.1088/1361-6560/ac4122 Y. Yang, L. Li, Z.Q. Chen, A review of geometric calibration for different 3-D X-ray imaging systems. Nucl. Sci. Tech. 76 (2016).

J.M. Que, D.Q. Cao, Y.L. Yan, et al ., Computed laminogra- phy and reconstruction algorithm. Chinese Phys. C , 777– 783 (2012).

Y. Zhao, J. Xu, H. Li, et al ., Edge information diffusion- based reconstruction for cone beam computed laminog- raphy. IEEE T. Image Process. 4663–4675 (2018).

J. Lu, Y. Liu, Y. Chen, et al ., Cone beam computed laminogra- phy based on adaptive-weighted dynamic-adjusted relative to- tal variation. Nucl. Instrum. Meth. , 168200 (2023).

T. Jia, S.Q. Liu, X.L. Shun, et al ., GPU implementation of an iterative reconstruction algorithm for computed Laminog- raphy. Chin. J. Stereol. and Image Anal. , 393–400 (2020).

Y. Shi, P. Ou, M. Zheng, et al ., Artifact noise suppression of particle-field computed tomography based on lightweight residual and enhanced convergence neural network. Acta Phys. , 1–13 (2024).

J. Di, J. Lin, L. Zhong, et al ., Review of sparse-view or limited-

angle CT reconstruction based on deep learning. Laser Opto- 643

electron. Prog. , 32–69 (2023). Chinese) Y.S. Hao, Z. Wu, Y.H. Pu, et al ., Research on inver- sion method for complex source-term distributions based on deep neural networks. Nucl. Sci. Tech. , 195 (2023).

H. Gu, X. Bi, D. Wang, et al ., Limited-angle CT im- age reconstruction based resnet and deconvolution network model. J. Sys. Sci. and Math. Scis. , 2349–2360 (2021).

Y. Sun, Y. Han, S. Tan, et al ., Geometric parameters sensitiv- ity evaluation based on projection trajectories for X-ray cone- beam computed laminography. J. X-Ray Sci. Technol. , 423– 434 (2023).

Y. Xing, L. Zhang, A free-geometry cone beam CT and its FDK-type reconstruction. J. X-Ray Sci. Technol. , 157–167 (2007).

H. Yang, K. Liang, K. Kang, et al ., Slice-wise reconstruction for low-dose cone-beam CT using a deep residual convolu- tional neural network. Nucl. Sci. Tech. , 53–61 (2019). 10.1007/s41365-019-0581-7 Y.M. Sun, Y. Han, S.Y. Tan, et al ., Geometric parameters sen- sitivity evaluation based on projection trajectories for X-ray cone-beam computed laminography. J X-Ray Sci. Technol. 423–434 (2023).

Q. Lin, M. Yang, L.F. He, et al ., Design of detachable com-

puted laminography scanning mechanism and neutron tomog- 670

raphy detection method for plate-like component. NDT and E , 102712 (2022).

M. Yang, G. Wang, Y. Liu, New reconstruction method for x- ray testing of multilayer printed circuit board. Opt. Eng. 56501 (2010).

L.A. Feldkamp, L.C. Davis, J.M. JKress, Practical cone-beam algorithm. Journal of the optical society of America A. , 612– 619 (1984).

W. van Aarle, W.J. Palenstijn, J. Cant, et al ., Fast and flexible X-ray tomography using the ASTRA toolbox. Opt. Express. , 25129–25147 (2016).

J.H. Hubbell, S.M. Seltzer, Tables of X-ray mass attenuation coefficients and mass energy-absorption coefficients 1 keV to

20 MeV for elements Z = 1 to 92 and 48 additional sub- 684

stances of dosimetric interest. Pramana, , 1–21 (1995). 10.1007/s12043-009-0033-8 H. Tang, T. Li, Y.B. Lin, et al ., A fast tomosynthesis method for printed circuit boards based on a multiple multi-resolution reconstruction algorithm. J. X-Ray Sci. Technol. , 965–979 (2023).