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Abstract
Linear Computed Laminography (LCL) is used to yield slice images of plate-like objects (PLO) for the advantage of short exposure time, high control precision and low cost. Shift and Add (SAA) is a widely used reconstruction algorithm for LCL. One limitation of SAA is that the reconstructed image of the in-focus layer (IFL) contains information from off-focus layers (OFL), resulting in inter-slice aliasing and blurring. In this paper, an Iterative Difference Deblurring (IDD) algorithm based on LCL is proposed to reduce the blur in reconstructed images. The core idea of the IDD algorithm is: contributions from OFL are subtracted from the projection data to remove the blur from the IFL. The corrected projections are then reconstructed using the SAA to remove the superimposed contributions of OFL from the IFL. An iterative approach is utilized to adjust a weighting factor applied during the subtraction stage. The results demonstrate that IDD algorithm can achieve PLO reconstruction in the LCL system under extremely sparse sampling conditions, and can effectively reduce the inter-slice aliasing and blurring.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Computed Tomography (CT) is a nondestructive testing technology which can reconstruct the internal structure of an object. Originally applied to medicine, it has now been applied to many fields, such as aerospace, precision machinery, geology and archaeology [1–4]. However, CT is not suitable for imaging plate-like objects (PLO) for the following reasons: the transmitted thickness of PLO changes dramatically under different rotation angles, which may lead to the penetration failure; PLO may go out of the imaging field during the rotation process, resulting in collision or loss of projection information. Computed Laminography (CL) is an alternative technology to CT in detecting and imaging large or planar objects. As shown in Fig. 1, the x-ray source and detector keep stationary and the sample is rotated in CT and CL. In the CT system, the incident beam is perpendicular to the rotation axis. In the CL system, the incident beam is tilted by a certain angle relative to the axis of rotation. During the CL scanning, the normal direction of the sample surface is almost parallel to the axis of rotation, resulting in similar global transmission [5]. Therefore, CL is more suitable for nondestructive testing of PLO.
Fig. 1. Schematic illustration of CT and CL. (a) Scanning structure of CT. (b) Scanning structure of CL in rotation mode.
Various motion modes have been applied in CL system, including linear, circular, elliptic, sinusoidal and helical, each having their advantages and disadvantages [6]. Linear scanning motion is widely used in CL imaging for the advantages of short exposure time, high control precision and relatively low cost. Shift and Add (SAA) [7] is a conventional reconstruction algorithm in the field of laminography, which can obtain any slice image by shifting and adding the projections. One limitation of SAA is that the reconstructed image of in-focus layer (IFL) contains information from off-focus layers (OFL), resulting in inter-slice aliasing and blurring. Various methods have been proposed for reducing or eliminating the blurry artifacts.
In 1969, Edholm and Quiding [8] first applied frequency filter technique to remove stripe blur in CL imaging. The authors made a photographic negative of the original reconstruction, blurred it in the tomographic direction, and added it to the original. This method was equivalent to a high-pass frequency filter. In subsequent work, many efforts were made to reduce blur by applying high-pass or band-pass filters [9–12]. Filtering technique reduces the blur nicely from OFL but also leads to slight blur in IFL, since the filtering technique is a process of deblurring all the layers including IFL and OFL. Selective plane removal method is proposed by Ghosh Roy [13], who used the knowledge of the blurring functions to solve exactly for the distortion generated by a handful of planes immediately adjacent to the plane of interest. Similar works were implemented by Kolitsi [14] and Sone [15]. Selective plane removal method is easy to be affected by noise amplification. ‘Extreme-value reconstruction’ methods [16–18] were to backproject only the projection with the smallest (or largest) value for the voxel in question, which were more suitable for high contrast regions of interest. Iterative technique is another way to reduce blur in IFL. Ruttimann [19] proposed ‘constrained iterative restoration’ in which deblurring was realized by convolution and difference [20–23]. Iterative technique is not affected by low-frequency noise amplification but requires complex and massive computation. Algebraic reconstruction technique (ART) [24–26] is a kind of iterative technology, which inevitably has the problem of massive computation. Commercially implemented ART are often GPU-accelerated. The ART is generally considered to achieve good results when prior information is available.
In this paper, we proposed the Iterative Difference Deblurring (IDD) algorithm to reduce the inter-slice aliasing and blurring in LCL imaging. IDD algorithm can realize PLO reconstruction under extremely sparse sampling conditions and requires less exposure time, which is difficult to achieve by other algorithms. IDD algorithm belongs to iterative technique and inherits the advantage of not being affected by noise amplification. In addition, IDD applies simple operations including SAA, difference, and dichotomy, which reduces the operation complexity and improves the reconstruction efficiency compared with traditional iterative techniques.
2. System and method
2.1 LCL scanning system
In a LCL scanning system, the trajectory which defines the relative motion among the x-ray source, detector and object is linear. A schematic diagram of a linear scanning system is shown as Fig. 2(a). The x-ray source and the detector are placed on both sides of the stationary PLO and move relative to each other with a parallel line track. In the process, the x-ray beam irradiates and penetrates through the sample, and then it is acquired by detector. This scanning geometry enables the x-ray penetrating the IFL to reach the same position on the detector. The geometric magnification of the internal layers of the PLO is adjusted by the distance among the x-ray source, detector and PLO. The two-dimensional (2D) projections obtained under digital radiography (DR) scanning mode is the superposition of multilayer structure along the direction of PLO thickness. Another linear scanning system is shown in Fig. 2(b) with a fixed x-ray source, where the PLO and detector move linearly in the same direction, equivalent to Fig. 2(a).
Fig. 2. Schematic diagram of linear scanning system. (a) x-ray source and detector move in opposite directions. (b) x-ray source and detector move in same directions.
The linear scanning system (EI Scan 2350) used in this paper is shown in Fig. 3, which is provided by Sanying Precision Instruments Co., Ltd.. The EI Scan 2350 includes a x-ray source (Hamamatsu 130S), a 5-axis drive control system, a displacement platform, and a flat panel detector (iRay 1012 m). The number of pixels of the detector is 3008×2496, and the size of each pixel is 0.1mm×0.1 mm. The scanning mode of the EI Scan 2350 corresponds to Fig. 2(b).
2.2 SAA reconstruction algorithm
The basic principle of SAA based on LCL is illustrated in Fig. 4. The projections of the IFL labeled as a red circle keep staying at the same position on the detector during scanning. For the IFL, the sharp reconstruction image can be obtained by simply adding the multiple projections. However, for any OFL, such as the plane 1 labeled as a blue triangle, the position of projection on detector changes constantly due to the influence of scanning angle and the distance between the IFL and plane 1. To obtain sharp reconstruction image of plane1, every projection needs to be moved specific offset first to achieve alignment as shown in Fig. 4, and then be added together. According to this method, reconstruction can be realized for any layer of the PLO.
As mentioned above, correctly calculating the specific offset for each projection is the key to the SAA algorithm. The geometric structure of linear scanning of SAA algorithm is illustrated in Fig. 5. The x-ray source and detector move relative to each other in parallel paths around a fixed fulcrum. The motion direction of the detector is defined as $x$-direction. The center of the detector trajectory is defined as the origin O. The direction that passes through $O$ and is perpendicular to the $x$-axis is defined as the $z$-direction. The x-ray source is at the height $z = D$ above trajectory of the detector. The initial position of the x-ray is at $x = {a_1}$ and the initial position of the detector is at $x = {b_1}$. The focal plane is at the height $z = {Z_f}$ above the trajectory of the detector.
When x-ray source is located at the initial position $x = {a_1}$, according to the triangle similarity relation:
$b_{1}$ can be expressed as: $M_{f} = D/(D - z_{f})$ is the geometric magnification multiple of the layer at the height $z = {Z_f}$ (focal plane).For any OFL, such as the layer at the position $z = Z$, the first coordinate of the projection is:
Similarly, $M_{z} = D/(D - z)$ is the geometric magnification multiple of the layer at the position $z = Z$.
For the ${k^{th}}$ projection, the x-ray source is at the position $x = {a_k}$ and the center of the detector is at the position $x = {b_k}$. The projection of the layer at the height of $z = Z$ is at the position $x = {x_k}(z)$. The offset between the projection and the center of detector is:
Finally, to realize the reconstruction of the layer at the position $z = Z$, the offset $shift_{k}(z)$ of the ${k^{th}}$ projection is
The ${k^{th}}$ projection is defined as $I_{k}(x,y)$, $k = 1,2,3\ldots N$. The sharp image of the layer at the position $z = Z$ is defined as $T_{z}(x^{\prime},y)$. $T_{z}(x^{\prime},y)$ can be obtained:
The distinct advantage of SAA algorithm is high speed and simplicity. However, an obvious shortcoming of SAA is that the structural information of OFL is blurred and superimposed on IFL resulting in inter-slice aliasing and blurring in the reconstructed images.
2.3 IDD algorithm
For SAA algorithm, the cause of blurry artifact is that the projections contain the superimposed information of multi-layer structure. To solve the question, we propose IDD algorithm. The core idea of IDD algorithm is: for any IFL reconstruction, the information of OFL is subtracted from the projections to obtain projections only containing the structure of IFL. SAA algorithm is then applied to obtain sharper image of IFL. Assuming that the number of samples is 9, that is, 9 projection images are obtained in each LCL scanning, and the specific process of IDD algorithm is described as follows:
For a $n$-layer PLO, nine projections ${I_1} - {I_9}$ are obtained by Digital Radiography (DR) scanning in LCL system. Each projection is the sum of the projection of each layer at its corresponding location. Now, to achieve the reconstruction of IFL, the information of $n - 1$ OFL should be subtracted from the corresponding position of original projections. The difference formula is as follows:
Taking the 3-layer phantom as an example to demonstrate the relationship among the original projection and the orthographic projection of each layer. For the phantom, the total number of layers is set to 256. Three shapes are set at the 118th, 127th, and 151st layer respectively. Figures 6(a), 6(b), and 6(c) are the orthographic projections of the three layers. Figure 6(g) is the sum of 6(d), 6(e), and 6(f) that have been moved to achieve alignment with 6(g). To obtain the projection of IFL, such as 6(d), the projections of OFL: 6(e), 6(f) should be subtracted from the original projection 6(g).
Fig. 6. Relationship among the original projection and orthographic projection of each layer. (a) Orthographic projection of 118th layer. (b) Orthographic projection of 127th layer. (c) Orthographic projection of 151st layer. (d) Aligned projection of 118th layer. (e) Aligned projection of 127th layer. (f) Aligned projection of 151st layer . (g) Original projection.
However, in the actual scanning experiment, the information of the orthographic projection of each layer is unknown, so the truth value $Tj$ in formula (7) is unknown and finally solved. The implementation process diagram of IDD algorithm is shown in Fig. 7. Here, each layer image ${T_1}^0 - {T_\textrm{n}}^0$ reconstructed by SAA is regarded as the initial “truth values”. At this time, all the images ${T_1}^0 - {T_\textrm{n}}^0$ are blurry images with information from OFL. To obtain the projections only containing IFL, the simple sum of $n - 1$ layers cannot be directly subtracted from the projections ${I_1} - {I_9}$. The processing method is to multiply the sum of $n - 1$ layers by a coefficient ${x_m}^1$ less than 1. The difference process is expressed in formula (8):
Similarly, taking the 3-layer phantom as an example, the relationship among the original projection and SAA reconstruction of each layer image is shown in Fig. 8. Due to the existence of blurry artifacts, the sum of the reconstructed images 8(d), 8(e), 8(f) is greater than original projection 8(g). Therefore, a coefficient less than 1 should be multiplied by the sum.
Fig. 8. Relationship among the original projection and SAA reconstruction of each layer image. (a) SAA reconstruction of 118th layer. (b) SAA reconstruction of 127th layer. (c) SAA reconstruction of 151st layer. (d) Aligned SAA reconstruction of 118th layer. (e) Aligned SAA reconstruction of 127th layer. (f) Aligned SAA reconstruction of 151st layer. (g) Original projection.
The coefficient is selected by dichotomy [27], and the threshold value of the coefficient is set at [0,1.1]. In the first difference process, the coefficient is set as 0.5. Then SAA reconstruction is performed again for the projections obtained after the first difference. The new layer images ${T_1}^1 - {T_\textrm{n}}^\textrm{1}$ are obtained:
We use the dichotomy judgment condition to select the coefficients: compare the new layer image ${T_m}^1$ with the former layer image ${T_m}^0$. The sum of pixel value is taken as the comparison index, as shown in formula (10). If formula (10) does not satisfy the relation $|{P({T_m}^1) - {x_m}^1 \times P({T_m}^0)} |\in [ - 1, + 1]$, the coefficient is reselected according to the dichotomy and substituted into formula (8). Until the relation $|{P({T_m}^1) - {x_m}^1 \times P({T_m}^0)} |\in [ - 1, + 1]$ is established, the coefficient selection can be completed.
where P represents the sum of the pixels value of the image.The above coefficient selection is based on the following principle: for the coefficients that have been selected by dichotomy, each projection ${I_1} - {I_9}$ can be approximately separated into the sum of ${T_1}^0 - {T_\textrm{n}}^\textrm{0}$ multiplied by the weighting coefficients ${x_m}^1$:
The layer images ${T_1}^1 - {T_\textrm{n}}^\textrm{1}$ have less blurry artifacts than ${T_1}^0 - {T_\textrm{n}}^\textrm{0}$. ${T_1}^1 - {T_\textrm{n}}^\textrm{1}$ are used as the updated truth values in formula (7) to repeat difference, coefficient selection and SAA reconstruction process. With the increase of the times of difference, the sharper layer images are obtained as the truth values, the superimposed information contained in ${T_j}$ is eliminated successively, and the corresponding weight coefficient increases. When the layer images are sharp enough, it means the projections can be completely separated into the sum of the projection of each layer image, which is equivalent to satisfying the relationship shown in Fig. 6. In this case, the weighting coefficient converges to 1. Therefore, the weighting coefficient ${x_m} \in [0.99,1.01]$ is taken as the condition for the iteration difference termination.
Based on the above difference, SAA reconstruction and coefficient selection process, the following step of IDD algorithm can be obtained:
- (3) Dichotomy Judgement. If the relationship $|{P({T_m}^1) - {x_m}^1 \times P({T_m}^0)} |\in [ - 1, + 1]$ is not satisfied, step (1) is returned. The coefficient is searched in the interval [0,1.1] by dichotomy until the relationship is satisfied. Then step (4) is performed.
- (4) Iteration Termination Condition Judgement. Judge whether the coefficients ${x_m} \in [0.99,1]$ is satisfied. When the coefficients converge to 1, it means that clear images have been obtained and the deblurring process has been completed. The output images are the final reconstruction results.
Taking the 3-layer phantom as an example, the flow chart of IDD reconstruction algorithm is shown in Fig. 9.
3. Result
3.1 Simulation result
In this section, simulation experiments are implemented to verify the effectiveness of IDD algorithm. The LCL simulation system as shown in Fig. 5 is designed and the simulation parameters is shown in Table 1.
Table 1. Simulation parameters of LCL system
A 3-layer, 5-layer, and 7-layer phantoms with different structures are designed. Considering that the internal structures of actual PLO have certain thickness, each structure layer is set as continuous three layers. The layer shapes are shown in Fig. 10 and the layer information is shown in Table 2. The 3-layer phantom contains the 3rd, 4th, and 6th layers, the 5-layer phantom contains the 2nd to 6th layer, and the 7-layer phantom contains the 1st to 7th layer in Fig. 10. The orthographic projections of each layer of three phantoms are shown in Fig. 11 and are regarded as truth values. Compared with the layer images shown in Fig. 10, the size and grayscale value of the layers of orthographic projections have been varied. The comparison of simulation results of SAA and IDD algorithm are shown in Fig. 12.
Fig. 11. Orthographic projections of each layer of the three phantoms. (a) Orthographic projections of each layer of the 3-layer phantom. (b) Orthographic projections of each layer of the 5-layer phantom. (c) Orthographic projections of each layer of the 7-layer phantom.
Fig. 12. Comparison of simulation results of SAA and IDD algorithm for the three phantoms. (a) Comparison of simulation results of the 3-layer phantom. (b) Comparison of simulation results of the 5-layer phantom. (c) Comparison of simulation results of the 3-layer phantom.
Table 2. Layer information of simulation phantoms
Compared with the images obtained by SAA algorithm, IDD algorithm can basically eliminate the blurry artifacts in the reconstructed images. To quantitatively evaluate the deblurring effect of IDD algorithm, the index of slice independence is introduced. For the reconstruction results obtained by LCL scanning, the degree of blur caused by information superposition from OFL is proportional to the distance between the IFL and OFL. The independence of slices is evaluated by the correlation coefficient of adjacent structures. The correlation coefficient r is defined as:
where $Cov(x,y)$ represents the covariance of the gray-scale images of adjacent slice x and y, and $S_x$($S_y$)represents the standard deviation of the gray-scale images of slice $x$($y$).The smaller the correlation coefficient is, the stronger the independence is, and the better the deblurring effect is. The mean of correlation coefficients of adjacent slices of each phantom is shown in Fig. 13. The red dotted line is the mean value of the correlation coefficients of SAA algorithm, and the blue solid line represents the mean value of the correlation coefficients of IDD algorithm. The calculation results show that the correlation coefficients of adjacent layers obtained by IDD algorithm is smaller than those obtained by SAA, indicating that IDD algorithm has a better deblurring effect. In addition, with the increase of the number of layers, the correlation coefficients of adjacent layers reconstructed by SAA and IDD become larger.
In addition to slice independence, three Image Quality Assessment (IQA) indexes including Root Mean Square Error (RMSE), Peak Signal to Noise Ratio(PSNR) and Structural Similarity(SSIM) are also used to compare the images reconstructed by SAA and IDD. The mean values of 15 layers (3-layer, 5-layer, and 7-layer) shown in Fig. 12 are calculated, and the results are shown in Table 3. Compared with SAA, the mean of RMSE in reconstructed images obtained by IDD algorithm is reduced by 80.65%, the mean of SSIM is increased by 39.13%, and the mean of PSNR is increased by 15.29dB. The result indicates IDD algorithm effectively reduces the inter-slice aliasing and blurring and yields an improvement of image quality.
Table 3. Mean of RMSE, PSNR, SSIM for total 15 layers
In order to further verify the superiority of IDD algorithm, we used ART and FBP algorithm respectively to reconstruct the 3-layer phantom with the same number of samples. The comparison of reconstruction results is shown in Fig. 14. The results show that the FBP algorithm has serious inter-slice aliasing and blurring and poor image contrast. The ART algorithm reduces inter-slice aliasing and blurring nicely, but the reconstruction details are not accurate enough and the reconstruction time is slightly longer. The IDD algorithm can not only reduce the aliasing and blur basically, but also recover the image details better, and the reconstruction efficiency is higher.
For above simulations, to gain better visual effect, we set high and similar grayscale value for above slices in all phantoms. Considering that the attenuation coefficients of the different layers of the actual PLO may vary greatly, we add another simulation of 3-layer phantom, and the gray values of the three layers are set to 0.9, 0.5, and 0.1 respectively. The simulation result is shown in Fig. 15. The result indicates that IDD algorithm is still suitable and better than SAA algorithm. At the same time, we can find that compared with the layer with a larger gray value, the reconstruction effect of the layer with a smaller gray value is slightly worse. It is inferred that the reason for this phenomenon is that the IDD algorithm needs to move in the difference stage to achieve registration. Due to the limitation that only integer pixel values can be moved, the registration degree cannot reach 100%, resulting in the remaining blur of other layers after difference. Therefore, the reconstruction quality of the lower gray value layer is slightly worse than the higher gray value layer. Considering the difference degree of measured material attenuation coefficient in the applicable field of IDD algorithm, in general, IDD algorithm is universal and its reconstruction effect is much better than SAA algorithm.
3.2 Experiment result
In order to demonstrate the reconstruction effect of the IDD algorithm, a 3-layer substrate consisting of 3-layer transparent plastic sheets with each layer interval of 30mm is designed, as shown in Fig. 16. Two experiments are implemented on this substrate. Experiment 1: A wrench, a scissors, a fork, and a key are placed on the 3-layer substrate; Experiment 2: Two bank card and a phone card are placed on the 3-layer substrate.
In the experiments, the substate is placed on the displacement platform of the LCL system as shown in Fig. 3. The detailed scanning parameters are listed in Table 4. After scanning, nine DR projections are acquired. SAA and IDD are used to reconstruct the two phantoms respectively and the results are shown in Fig. 17.
Fig. 17. Comparison of experiment results of SAA and IDD algorithm for the two 3-layer phantoms. (a) Results of Experiment 1 consisting of a wrench, a scissors, a fork, and a key. (b) Results of Experiment 2 consisting of two bank chips and a phone card.
Table 4. Scanning parameter for the LCL
The results show that IDD algorithm basically eliminates the residual blur in the reconstruction images visually. The correlation coefficients of two adjacent slices are calculated and showed in Fig. 18. Compared with SAA algorithm, the correlation coefficients of adjacent slices obtained by IDD algorithm is smaller. The effectiveness and practicability of IDD algorithm proposed in this paper can be proved. In addition, according to Fig. 18, for the two sets of experiments based on same inter-slice spacing and the same LCL system, the improvement degree of correlation coefficients after applying IDD are different. Therefore, it can be reasonably inferred that the sample with large size and simple internal structure can achieve more obvious layer separation effect after applying IDD algorithm compared with the sample with small size and complex internal structure.
Fig. 18. Correlation coefficients of adjacent slices for two 3-layer phantoms. (a) Correlation coefficients of Experiment 1. (b) Correlation coefficients of Experiment 2.
4. Conclusion and discussion
The IDD algorithm proposed in this paper can reconstruct internal slices of PLO using 2D projections obtained from the LCL system, thus avoiding the problems of high cost, low precision and long exposure time caused by other complex scanning modes. The IDD algorithm is fast, simple and not affected by noise amplification. In addition, this algorithm can be used in extremely sparse sampling, such as 9 times of sampling or even less, which is difficult to achieve by other algorithms. Simulation and experimental results show that the IDD algorithm can effectively reduce inter-slice aliasing and blurring and obtain high quality reconstruction images, so it is suitable for LCL imaging of PLO. However, for the linear motion mode, there is a problem that for the structures parallel to the relative motion direction ($x$-direction), although they are on the OFL, there is little blur effect and they are just elongated along the direction of motion. In the simulation part of this paper, the structures parallel to the motion direction are avoided in the phantom design process. In practical experiments, this problem can be overcome by adjusting the direction of the PLO with known information. In order to solve the shape limitation from the root, we adopt the CL scanning method of circle trajectory and apply IDD algorithm. Simulation results show that the algorithm is also suitable for circular trajectory scanning without layer structure restriction. In addition, the number of original projections and layer spacing are also two factors affecting the reconstruction results. We respectively carried out IDD algorithm reconstruction for five, nine and seventeen projections. As the number of projections increases, the blurry artifacts in the reconstructed image become less. To balance the short exposure time, computation time and good reconstruction quality, nine projections are used in this paper for simulation and experiment. By varying the spacing between the layers, we find that the larger the spacing, the better the separation effect of each layer. For continuous layers without interval, IDD reconstruction results contain more artifacts, but the image quality is still far better than SAA. Another consideration for IDD algorithms is the more layers are reconstructed, the more time-consuming the algorithm need. After analysis, the IDD algorithm is composed of several linear operation processes including SAA, difference and dichotomy, so the IDD algorithm is also linear. By comparing the reconstruction time of the phantoms with different layers, it is found that there is a linear relationship between the consumption time of IDD algorithm and the number of reconstructed layers. Therefore, although the consumption time of IDD algorithm increases with the increase of the number of layers, it is still controllable. Due to the use of dichotomy in the coefficient selection process, the execution efficiency of the algorithm is relatively low. In the future, we plan to optimize the conditions and methods of coefficient selection and solve the problem of constraints on structures parallel to the direction of motion in LCL. Our on-going work is applying IDD algorithm to the reconstruction of arbitrary oblique layers and even arbitrary irregular surfaces. We believe the IDD algorithm will be beneficial to many applications, such as process control [28], security inspection [29,30], human medical diagnosis [31].
Funding
National Key Research and Development Program of China (2017YFB1103900); National Natural Science Foundation of China (No.61771328).
Acknowledgments
Ruobing Zhao thanks to Sanying Precision Instruments Co., Ltd for providing us with the experimental platform.
Disclosures
The authors declare that they have no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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