A fast tomosynthesis method for printed circuit boards based on a multiple multi-resolution reconstruction algorithm

Abstract

Digital tomosynthesis (DTS) technology has attracted much attention in the field of nondestructive testing of printed circuit boards (PCB) due to its high resolution and suitability to thin slab objects. However, the traditional DTS iterative algorithm is computationally demanding, and its real-time processing of high-resolution and large volume reconstruction is infeasible. To address this issue, we in this study propose a multiple multi-resolution algorithm, including two multi-resolution strategies: volume domain multi-resolution and projection domain multi-resolution. The first multi-resolution scheme employs a LeNet-based classification network to divide the roughly reconstructed low-resolution volume into two sub-volumes namely, (1) the region of interest (ROI) with welding layers that necessitates high-resolution reconstruction, and (2) the remaining volume with unimportant information which can be reconstructed in low-resolution. When X-rays in adjacent projection angles pass through many identical voxels, information redundancy is prevalent between the adjacent image projections. Therefore, the second multi-resolution scheme divides the projections into non-overlapping subsets, using only one subset for each iteration. The proposed algorithm is evaluated using both the simulated and real image data. The results demonstrate that the proposed algorithm is approximately 6.5 times faster than the full-resolution DTS iterative reconstruction algorithm without compromising image reconstruction quality.

Keywords

Multi-resolution tomosynthesis LeNet PCB iterative image reconstruction

1 Introduction

The development of PCB integration leads to the increasing density of circuit wiring and solder joints. Nondestructive testing is usually required to ensure PCB reliability. Digital Radiography (DR), Computed Tomography (CT), and Digital Tomosynthesis (DTS) are some of the nondestructive testing techniques available that can aid in the detection of voids/porosity, flaws, cracks, and inclusions within PCBs [1, 2]. Due to the serious overlap of DR, the high cost and motion limitation of CT, DTS technology has been introduced into the nondestructive testing of PCB [3, 4]. DTS is a finite-angle tomography, which is more suitable for thin slab objects. Iterative algorithms are widely used method for DTS reconstruction, which possess better noise tolerance, can effectively handle sparse or unevenly distributed projection data, and can integrate prior information [5]. Studies have demonstrated that iterative reconstruction algorithms yield higher imaging quality for PCB than analytical algorithms [6, 7]. To further reduce artifacts and random noise caused by incomplete or truncated DTS projections, the total variation (TV) regularization model is often utilized to constrain the iterative algorithm [8].

Although the iterative algorithms are superior in terms of reconstructed images quality, the computational cost of the iterative process can be quite high. In the cone-beam DTS, the X-rays pass through the entire measured object at the same time, resulting in projections containing contributions from various slices of the object. To reconstruct a specific slice, a complete volume comparable to the measured object must be generated and saved in memory for simultaneous processing. With high resolution, the reconstructed volume can reach several or even dozens of gigabytes, which consumes a significant amount of memory and computation resources, severely slowing down reconstruction speed. In recent years, graphics computing units (GPU) with highly parallel computing capability have developed rapidly and have been widely used for analytical and iterative reconstruction [9–11] to speed up calculations. However, in scenarios that require real-time processing of large amounts of data, GPU acceleration may not be sufficient to completely solve the problem. Some studies have introduced multi-resolution methods to reduce memory requirements and further lower computational costs by down-sampling large volume to lower resolutions or divide them into smaller sub-volumes [12–14]. However, these studies do not focus on the ROI or only manually differentiate between high- and low-resolution sub-volumes, and they only use a single multi-resolution strategy.

To accelerate the DTS iterative reconstruction process, we propose a novel approach called the multiple multi-resolution reconstruction algorithm. This algorithm incorporates two strategies: volume domain multi-resolution (VDMR) and projection domain multi-resolution (PDMR). In the case of nondestructive testing of PCB, the objective is to detect any defects within the welding layer such as virtual welding, solder voids, and solder bridging. Hence, there is no need to reconstruct an entire high-resolution volume. VDMR automatically distinguishes the reconstructed volume and solely reconstructs ROI with welding layers in high resolution. The total amount of data of the reconstructed volume is reduced so that the dimension of the projection matrix is reduced. Since X-rays in adjacent projection angles pass through many identical voxels, there exists information redundancy between adjacent projections. Therefore, PDMR divides the projections into non-overlapping subsets, utilizing only one subset per iteration to reduce the amount of redundant information in each iteration. The combination of VDMR and PDMR further accelerates the reconstruction process. The experimental results demonstrate that our algorithm significantly increases the computational speed of iterative reconstruction without compromising image quality.

2 Multiple multi-resolution reconstruction

In the conventional iterative reconstruction process for cone-beam DTS, the object is typically discretized into a high-resolution volume, resulting in storage and calculation difficulties. Although GPU acceleration can speed up computation, it remains challenging to directly store the entire volume in GPU memory. Moreover, in scenarios with huge amount of data or high real-time requirements, GPU acceleration alone may not be sufficient. To overcome these challenges, we propose a novel multiple multi-resolution reconstruction algorithm to accelerate the reconstruction process, including volume domain multi-resolution (VDMR) and projection domain multi-resolution (PDMR). The first multi-resolution scheme employs a LeNet-based classification network that automatically identifies and extracts ROI requiring high-resolution reconstruction, namely the welding layers, from the low-resolution reconstructed volume. The remaining volume, which contains unimportant information, can be reconstructed in low resolution. When using this scheme, there is no need to reconstruct an entire high-resolution volume of the measured object, so the amount of reconstructed data and projection matrix dimension are reduced. The second multi-resolution scheme exploits the information redundancy between projections to partition them into non-overlapping subsets, utilizing only one subset per iteration to reduce the amount of redundant information in each iteration. The number of subsets is determined by the partition interval. By employing VDMR and PDMR, separately or in combination, we can achieve faster and more efficient DTS reconstruction. Figure 1 provides a schematic illustration of the VDMR and PDMR approaches.

Fig. 1

Multiple multi-resolution reconstruction algorithm including volume domain multi-resolution (VDMR) and projection domain multi-resolution (PDMR). VDMR first roughly reconstructs a low-resolution volume $f_{low}^{K - 1}$ with K iterations, then uses the LeNet-based network to classify $f_{low}^{K - 1}$ into f_ROI and f_rest. f_ROI will be interpolated into $f_{res}^{0}$ as the high-resolution initial guess. The final result $f_{res}^{K - 1}$ is reconstructed in high resolution using projections of ROI p_ROI. PDMR divide p_ROI into multiple non-overlapping projection subsets s₁, s₁, … , s_N and only on subset is used for each iteration.

2.1 Volume domain multi-resolution (VDMR)

To detect potential issues such as solder voids and solder bridging, our attention is primarily focused on the welding layers located within the PCB (referred to as ROI), while less emphasis is placed on the components and substrates. Therefore, it is necessary to utilize a higher resolution for the reconstruction of ROI to prevent missing important details, whereas the unimportant parts can be reconstructed with a relatively lower resolution. VDMR is performed on the reconstructed volume to automatically identify ROI and reconstruct it in high resolution. To perform VDMR reconstruction, the original projections p_raw is firstly used to reconstruct a rough low-resolution volume f_low via SART-TVM algorithm (the basic DTS iterative algorithm). Then the pre-trained LeNet-based network is utilized to classify f_low into two sub-volumes: f_ROI, which requires high-resolution reconstruction to avoid missing crucial details, and the remaining volume, f_rest, which can be reconstructed in a lower resolution. Next, projections p_ROI containing only contributions of f_ROI are calculated using p_raw and the re-projections of f_rest. Finally, taking the linear interpolation result of f_ROI as the initial guess, the final result f_res is obtained by reconstructing the ROI sub-volume in high resolution using p_ROI. Figure 2 shows a local magnification comparison between the initial low-resolution DTS imaging result and the final high-resolution DTS imaging result in VDMR. We can see that the details in low-resolution result are too fuzzy, which cannot meet the image quality requirements of defect detection. Therefore, it is quite necessary to continue to carry out high-resolution DTS imaging on ROI layers. The pseudo-code of VDMR is outlined in Algorithm 1.

Fig. 2

Local magnification comparison between (a) initial low-resolution DTS imaging result and (b) the final high-resolution DTS imaging result in VDMR, display window [0, 1.05].

Algorithm 1:VDMR
Input: Initial low-resolution whole volume $f_{low}^{0}$ , original projections p_raw, number of SART-TVM iterations K,
relaxation coefficient λ, size of the input image for LeNet-based network M∘
Output: High resolution reconstructed ROI sub-volume f_res∘
1 FORi = 1, 2, . . . , K
2 $f_{low}^{i} = SARTTVM (f_{low}^{i - 1}, p_{raw}, λ)$ with low resolution
3 END
4 Resize each slice of $f_{low}^{K - 1}$ to M × M
5 $f_{ROI}, f_{rest} \leftarrow Lenet (f_{low}^{K - 1})$
6 p_ROI = p_raw - reproj (f_rest), reproj is the forward projection operation.
7 $f_{res}^{0} \leftarrow interpolation (f_{ROI})$ , interpolation is the linear interpolation algorithm.
8 FORi = 1, 2, . . . , K
9 $f_{res}^{i} = SARTTVM (f_{res}^{i - 1}, p_{POI}, λ)$ with high resolution
10 END
11 RETURN $f_{res}^{K - 1}$

The high-resolution ROI sub-volume is initialized from the low-resolution ROI sub-volume, which already contains a lot of information of the object to be measured. The reconstruction process can be completed with fewer high-resolution reconstruction iterations, which greatly reduces the number of calculations and accelerates the speed of reconstruction. However, due to the inherent limitations of iterative image reconstruction, it is impossible to recover accurate true data. Consequently, the forward re-projection of f_rest is only an approximation of the true projection, leading to a minor amount of interference of f_rest into p_ROI. Although the interference will be introduced into the high-resolution ROI reconstruction process, it has a negligible effect on the overall image quality, and the details are entirely preserved. Further analysis of this phenomenon will be conducted in subsequent experiments.

2.1.1 LeNet-based classification network

In the second step of VDMR, f_ROI can be determined manually, but it is time-consuming and labor-intensive. To overcome this problem, a classification network can be used to classify the volume blocks quickly and accurately. Due to the high real-time requirement, LeNet [15] and AlexNet [21] with small network scale are candidates in this paper. The preliminary experimental results showed that AlexNet had obvious classification errors in the test, while LeNet performed well. Thus, LeNet is chosen for the classification task. The network structure is simply modified to adapt to the image data used in this paper, as shown in Fig. 3. The input image size is 1000×1000, followed by three convolutional layers, three max pooling layers and two fully connected layers, and finally output two classification results. The network uses ReLu activation function, cross-entropy loss function and Adam optimizer.

Fig. 3

LeNet-based classification network. The input image size is 1000×1000, followed by three convolutional layers, three max pooling layers and two fully connected layers, and finally output two classification results.

Figure 4 shows the process of classifying low-resolution volume f_low. The volume is sliced, and each slice is passed through the network for classification. Each slice is classified into class High (C_H) with welding layers and class Low (C_L) with unimportant information. According to the index of slice, the positions of the two classes in f_low is determined respectively, and the entire volume is divided into the ROI to be reconstructed in high resolution f_ROI (stacked C_H) and the remaining volume f_rest (stacked C_L).

Fig. 4

The ROI determination process of VDMR. The input of the network is the slices of f_low. Each slice is classified into class High (C_H) which containing welding layers and class Low (C_L) containing other unimportant information. The positions of the two classes in f_low is determined respectively according to the index of slice, and f_low is divided into the ROI to be reconstructed with high resolution f_ROI and the remaining volume f_rest.

2.2 Projection domain multi-resolution (PDMR)

In the conventional DTS, the high-resolution projections obtained by the flat panel detectors can have a large amount of data, which can make the reconstruction process computationally expensive and time-consuming. Moreover, as the X-rays at adjacent source angles pass through many identical voxels, there will be a lot of information redundancy between adjacent projections. As shown in Fig. 5, rays at two adjacent projection angles will pass through the same shadow part.

Fig. 5

Information redundancy between adjacent projections. X-rays at adjacent source locations will pass through many identical voxels, as shown in the shadow part.

Considering the information redundancy between adjacent projections, it is not necessary to use the complete projection set in each iteration. By dividing the projections into subsets, we can effectively reduce the amount of redundant information used in each iteration, which can greatly improve the computational efficiency of the reconstruction process. This is because neighboring projections tend to share a significant amount of information, and by using only a subset of the projections, we can still obtain a high-quality reconstruction without the need to process all projections. This is especially useful in cases where the number of projections is very large, which can cause a significant computational burden for the reconstruction algorithm. As shown Fig. 1, p_ROI is divided into multiple non-overlapping projection subsets s₁, s₂, . . . , s_N, N is the number of subsets, and the specific division process is as follows.

Algorithm 2: PDMR
Input: ROI projections p_ROI, Number of projection subsets to be divided N, total number of projections P∘
Output: projection subsets s₁, s₂, . . . , s_N∘
1 U = P/N, the number of projections in each subset
2 FORi = 0, 2, . . . , N - 1
3 $s_{i} = {p_{ROI}^{i}, p_{ROI}^{i + N}, p_{ROI}^{i + 2 * N}, . . ., p_{ROI}^{i + (U - 1) * N}}$ , $p_{ROI}^{u}$ represents u th projection data
4 END
5 RETURNs₁, s₂, . . . , s_N

The first iteration uses s₁ subset, the second iteration uses s₂ subset, and so on. If K is greater than N, then N subsets are recycled until the end of the iteration. PDMR can be used alone when reconstructing a whole full-resolution volume data (divide the original projection p_raw) or combined with VDMR as a multiple multi-resolution method.

3 The tomosynthesis iterative algorithm SART

To use the multiple multi-resolution approach, we need a basic tomosynthesis iterative algorithm. There are three most used iterative methods include ART [16], SIRT [17] and SART [18]. In this paper, we adopt the SART algorithm, which is relatively fast and does not require additional storage space. The iteration formula of SART is as follows: $f_{j}^{(k + 1)} = f_{j}^{(k)} + λ \cdot \frac{1}{\sum_{i \in P_{φ}} ω_{ij}} \sum_{i \in P_{φ}} \frac{p_{i} - \sum_{j = 1}^{J} ω_{ij} f_{j}^{(k)}}{\sum_{j = 1}^{J} ω_{ij}} \cdot ω_{ij}$ (1) where f is the volume to be reconstructed, ω_ij represents the weight contributed by the j^th voxel to the i^th ray, p_i is the projection value of the i^th ray, k is the number of iterations, λ is the relaxation coefficient, and P_φ is the set of rays under the projection angle φ. The solution of SART includes four steps: forward projection, calculate projection error, back projection and image update. In this paper, the improved Siddon algorithm [20] based on the linear model is adopted, and the length of the intersection between j^th voxel and i^th ray is used as ω_ij [19].

Since DTS is essentially an incomplete angle problem, the exact solution can be approximated by adding a TV regularization constraint in the iterative process. The TV model based on the assumption of “piecewise constant", which means that there exist many local smooth regions in the image, and the discrete gradient transformation (gradient image) is sparse. The research target of this paper, PCB, also satisfies this assumption. In this paper, the TV minimization model is integrated into the DTS reconstruction of PCB to constrain SART, to solve the following reconstruction model with TV regularization added: $\min | | f | |_{TV} s . t . Wf - P ⩽ ɛ, f ⩾ 0_$ (2)

In the above formula, f represents the image to be reconstructed, W represents the projection matrix containing the weights ω_ij, P represents the projection data, and ɛ represents a small error value. The proposed model provides a solution to the DTS reconstruction problem based on adaptive TV minimization (TVM). The SART algorithm incorporating the TV minimization model (SART-TVM) can further suppress the artifacts and random noise caused by incomplete or truncated projection data, thereby improving the image quality. SART-TVM is solved in two steps; in the first step, data consistency is guaranteed by SART, and in the second step, TV is minimized.

4 Experiments

4.1 Experimental data and conditions

Simulated data and real PCB data were used to demonstrate the feasibility of the proposed method. The configuration of the computer is Windows10, 64-bit, NVIDIA GeForce GTX 1070 Ti, lntel (R) Core (TM) i5-9500 CPU @ 3.00GHz×6, 16 GB memory. To simulate the interference of various factors in the real projection process, scattering and Poisson noise were artificially added to the simulated projections. The SART-TVM algorithm is implemented using GPU for parallel computing.

4.1.1 Simulated data

A simulated PCB data of size 28.8mm*28.8mm*10 mm was created, including components, substrate, leads and solder joints. Its typical slices and 3D stereograph are shown in Fig. 6. The mass attenuation coefficients used to simulate the polyenergetic projection were provided by the table of X-ray mass attenuation coefficients of the National Institute of Standards and Technology (NIST) [22]. The simulated projections were obtained under a circular scanning trajectory: the detector and the ray source move in a relative circular motion in a plane, respectively. Other relevant experimental parameters are shown in Table 1.

Fig. 6

Simulated data. (a) and (b) show the 65^th slice and the 82^th slice of the volume. (c) shows the 3D stereograph of the volume.

Table 1

Experimental parameters

Parameter item	Simulated data	Real data
Full-resolution volume/voxel	15361536100	20002000100
Full-resolution voxel size/mm	0.018750.018750.1	0.0190.0190.05
Ray source - object distance (FOD)/mm	77.51044	68.3689
Detector - object distance (DOD)/mm	282.85964	260.1541
Angle between ray source and normal/°	40	42
Resolution of detector/pixel	1536*1536	1536*1536
Detector pixel size/mm	0.085	0.085
Number of projections	32	64
Range of projection angle/°	360	360

4.1.2 Real data

Different types of real PCB were used in experiment, including single-layer PCB, double-layer PCB, multi-layer PCB and heat sink. A multi-layer PCB was selected as an example demonstration purposes. Figure 7 depicts its appearance. The projections acquisition process concentrated on a certain ROI area of this PCB, as indicated by the red box. Relevant experimental parameters can also be found in Table 1. It is difficult to obtain the reference image in the actual tomosynthesis reconstruction, so there is no quantitative evaluation in real data experiment.

Fig. 7

The front and back of multi-layer PCB.

4.1.3 Dataset and parameters of classification network

The dataset comprised of 24 sets of reconstructed PCB volume, including one set of simulated data and 23 sets of real data, with each set containing 60 to 100 images. All images were divided into two classes, C_H and C_L. The images containing welding elements were taken as the C_H, and the remaining images containing unimportant information were taken as the C_L. In Fig. 8, four images in C_H ((a)-(d)) and four in C_L ((e)-(h)) are selected respectively for display. The images were augmented by rotation, flipping, and scaling. Especially, due to the small number of C_H images, data enhancement such as changing contrast and brightness were added to C_H. The final ratio of C_H to C_L is about 3:4. We shuffled all the images and randomly selected 10% as the test set, the remaining is divided into training set and validation set by 9:1. The initial learning rate was set t0.0001 and adjusted using equal step size strategy.

Fig. 8

Typical images in the C_H and C_L data set. (a)-(d) belong to C_H which contain solder joint and package information. (e)-(h) belong to C_L which contain unimportant information.

4.1.4 Full-resolution volume as reference

Before the multiple multi-resolution experiments, a full-resolution (the same as high-resolution) volume was first reconstructed as a gold standard for comparison, and both full-resolution and multi-resolution experiments were implemented using GPU. In the SART-TVM algorithm, the relaxation coefficient was set to 0.5 and the number of iterations was set to 20. The full-resolution iterative reconstruction time of the simulate data was 51 s, the single-layer PCB was 15.5 s, the double-layer PCB was 31 s, the multi-layer PCB was 56.5 s, the heat sink was 127 s.

4.1.5 Quantitative evaluation of image quality

Structural similarity index measure (SSIM) [23] and contrast to noise ratio (CNR) [24] were used as quantitative evaluation of reconstructed image quality. SSIM represents the structural similarity between the reconstructed image and the reference image, and CNR represents the ratio of the difference between the signal and background intensity to the background noise intensity in the image. The larger the two values are, the better the image quality is. SSIM and CNR are calculated using the following formula: $SSIM = \frac{(2 μ_{x} μ_{y} + C_{1}) (2 σ_{xy} + C 2)}{(μ_{x}^{2} + μ_{y}^{2} + C_{1}) (σ_{x}^{2} + σ_{y}^{2} + C_{2})}$ (3) $CNR = \frac{| μ_{signal} - μ_{back} |}{σ_{back}} (5)$ (4) where μ_k denotes the mean of patch k, σ_k denotes the standard deviation of patch k, and σ_xy denotes the covariance of patch x and y. signal denotes the region containing more details, and back denotes a more uniform background region.

4.2 Results

4.2.1 LeNet-based classification network

The pre-constructed data set was used for network training, validation and prediction. The initial rate was set to 0.0001 for 100 epochs of training. When the training was completed, the training loss was 0.03942 and the validation accuracy was 98.59%. Figure 9 shows the change curve of the loss function and the validation accuracy. The accuracy of the test set is 98.73%, and the misclassified images are in the transition between C_H and C_L, and can be divided into two classes, which has no obvious effect on the reconstruction results. The network can automatically classify a 70-slice low-resolution volume into f_ROI and f_rest within 3 seconds. The thickness of f_ROI of simulated data is 48 slices. In other word, the thickness of volume to be reconstructed in high resolution of simulated data is reduced from 100 to 48.

Fig. 9

The loss function change curve of training set and validation set. The accuracy changes curve of validation set. After 30 iterations, both the loss function and the accuracy curve tend to be stable.

4.2.2 Simulated data

In VDMR, the size of high-resolution voxels is the same as that of full-resolution voxels, and the size of low-resolution voxels is set to be double that of the high-resolution voxels, that is, the number of low-resolution voxels is half of high-resolution voxels. In PDMR, projections are divided into four non-overlapping subsets. Table 2 shows the quantitative evaluation and reconstruction time of different experiments. It can be seen from the values in the table that multiple multi-resolution greatly reduces the reconstruction time and has little impact on the quality of the reconstructed image. The multiple multi-resolution based reconstruction algorithm is approximately 6.5 times faster than the full-resolution iterative reconstruction.

Table 2
Comparison of quantitative evaluation and reconstruction time between full-resolution and multiple multi-resolution

CNR SSIM reconstruction time/s

Full-resolution 60.2896 0.7997 51.1

PDMR 59.9288 0.7879 26.3

VDMR 58.3555 0.7466 18.5

PDMR+VDMR 58.0821 0.7832 7.85

	CNR	SSIM	reconstruction time/s
Full-resolution	60.2896	0.7997	51.1
PDMR	59.9288	0.7879	26.3
VDMR	58.3555	0.7466	18.5
PDMR+VDMR	58.0821	0.7832	7.85

Figure 10 shows the reconstructed image and its localized magnification of each algorithm. The results demonstrate that the overall quality of the tomosynthesis image reconstructed by multiple multi-resolution is well maintained and the intricate details are clearly visible.

Fig. 10

Reconstruction of 65th slice of simulate data and its local magnification of each algorithm, display window [0, 1.05]. The image quality of PSMR and VSMR has almost no degradation compared with the full resolution, and the details in the image are very clearly visible.

4.2.3 Real data

The experimental results in the previous section verify the effectiveness of the proposed method. In this section, several sets of real PCB data are used for experiments, and the multi-layer PCB is taken as an example for display. Table 3 shows the comparison of reconstruction time of different sets of PCB data in full-resolution, PDMR, VDMR and the combination of the latter two. The numerical results show that the reconstruction algorithm based on multiple multi-resolution increases the speed by about 6.5 times.

Table 3
Comparison of iterative reconstruction time of different methods for real PCB (units)

Single-layer PCB Double-layer PCB Multi-layer PCB Heat sink

Full-resolution 15.5 31 56.5 127

PDMR 10.5 19.1 33.6 76.2

VDMR 8.3 15.5 16.8 29.6

PDMR+VDMR 3.5 4.5 8.7 19.7

	Single-layer PCB	Double-layer PCB	Multi-layer PCB	Heat sink
Full-resolution	15.5	31	56.5	127
PDMR	10.5	19.1	33.6	76.2
VDMR	8.3	15.5	16.8	29.6
PDMR+VDMR	3.5	4.5	8.7	19.7

Figure 11 shows the reconstructed 45^th slice and its local magnification of each algorithm. Compared with the full-resolution reconstructed tomosynthesis image, the tomosynthesis image obtained by multi-resolution method has very small degradation, and the overall quality of the image is still high, especially the details are still clear, which does not affect the subsequent detection and judgment of solder joint defects.

Fig. 11

Reconstruction of 45th slice of multi-layer PCB and its local magnification of each algorithm, display window [0, 0.94]. The tomosynthesis images obtained by PSMR and VSMR method still maintain high quality, and the details such as the voids in solder joint can be detected clearly.

5 Discussion

In this paper, we proposed a fast tomosynthesis method for PCB based on multiple multi-resolution reconstruction algorithm, which can be used in the field of nondestructive testing. The experiment is established SART-TVM algorithm implemented on GPU, and the effect of multiple multi-resolution reconstruction method on the quality and speed of PCB tomosynthesis is evaluated. We focus on the welding layers of the PCB, which need to be reconstructed using high-resolution to prevent missing details, and the remaining parts containing unimportant information are reconstructed in a lower resolution. A LeNet-based classification network is pre-trained using the available PCB dataset to automatically classify the roughly reconstructed low-resolution volume. The network can almost accurately divide the low-resolution volume into ROI to be reconstructed in high resolution and the remaining low-resolution volume. The experimental results of multiple multi-resolution reconstruction algorithm demonstrate that the reconstruction speed is improved by about 6.5 times.

From the quantitative evaluation on the reconstructed images of simulated data and real data, the image quality obtained by the strategy of PDMR and VDMR is slightly degraded. Due to the inherent flaw of iterative reconstruction, it is impossible to recover the completely accurate true data. As a result, the forward projections of the remaining low-resolution volume are approximation of the real projections, leading to the ROI projections still contain a minor amount of interference components of the remaining low-resolution volume, which will be introduced into the high-resolution ROI reconstruction process. Nonetheless, these minor degradation effects have an insignificant impact on the overall image quality, and the details are entirely preserved, which does not affect the subsequent detection and judgment of solder joint defects. Therefore, the proposed tomosynthesis reconstruction algorithm based on multiple multi-resolution can substantially enhance efficiency while ensuring the quality of reconstructed images.

In summary, this paper focuses on improving imaging speed through the design of a multi-resolution strategy, without altering the reconstruction or optimization algorithms. If higher-quality reconstructed images are desired, further improvement of the optimization algorithm can be applied, such as some variational approaches [25, 26].

Footnotes

Acknowledgments

The authors would like to thank the Unicomp Technology Company Ltd. for their help on the data acquisition.

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