Pole-piece position distance identification of cylindrical lithium-ion battery through x-ray testing technology

For x-ray imaging of cylindrical batteries, the paths that the x-ray penetrates vary dramatically. The intensities of the x-ray reaching the detector differ widely so that the gray levels of the DR image are distributed over a wide range. If the original DR images are adopted to recognize position distance defects directly without any pre-processing, the pole-pieces in the ROI will be very likely missed due to its low contrast. To avoid this situation, the homomorphic filtering algorithm based on the lighting reflection model is applied to enhance the DR images. According to this model, the given image $f(x,y)$ can be expressed as the combination of its illumination components $l(x,y)$ and reflectance components $r(x,y)$:

$$\begin{equation}f(x,y) = l(x,y)r(x,y)\end{equation} \tag{ 1 }$$

It is a remarkable fact that the illumination components $l(x,y)$ are mainly composed of low-frequency information, and the reflectance components $r(x,y)$ reflect the detail and boundary information of the object. The filtering operation aims to attenuate the contribution made by the low frequencies (illumination components) and amplify the contribution made by high frequencies (reflectance components) in the homomorphic filtering algorithm. Through this method, the percentage of the reflectance components $r(x,y)$ in the lighting reflection model will rise notably [18–20]. The flow chart of the homomorphic filtering algorithm is shown in figure 2.

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When the homomorphic filtering algorithm is used to process an image, the logarithmic operation in the original image $f(x,y)$ is carried out first to transform the multiplication operation to additive operation, as expressed by equation (2). Then the discrete Fourier transform (DFT) is operated on equation (2) as expressed by equation (3), where ${F_l}(u,v)$ and ${F_r}(u,v)$ are Fourier transforms of $\ln [l(x,y)]$ and $\ln [r(x,y)]$, respectively.

$$\begin{equation}z(x,y) = \ln f(x,y) = \ln l(x,y) + \ln r(x,y)\end{equation} \tag{ 2 }$$

$$\begin{equation}Z(u,v) = {F_l}(u,v) + {F_r}(u,v)\end{equation} \tag{ 3 }$$

Next, an appropriate frequency-domain filter $H(u,v)$ is designed to act on $Z(u,v)$ to decrease the percentage of illumination components $l(x,y)$ and enhance the percentage of reflectance components $r(x,y)$ as expressed by equation (4). The enhanced image $g(x,y)$ is finally obtained by inverse DFT and exponential operation as expressed by equation (5),

$$\begin{equation}S(u,v) = H(u,v)Z(u,v) = H(u,v){F_l}(u,v) + H(u,v){F_r}(u,v)\end{equation} \tag{ 4 }$$

$$\begin{equation}g(x,y){\text{ = }}{e^{{F^{ - 1}}[S(u,v)]}}\end{equation} \tag{ 5 }$$

According to the characteristics of the battery DR image, a Gauss high-pass filter is used to define $H(u,v)$ as expressed by equation (6),

$$\begin{equation}H(u,v) = ({\gamma _H} - {\gamma _L})[1 - {e^{ - c({D^2}(u,v)/D_0^2)}}] + {\gamma _L}\end{equation} \tag{ 6 }$$

where ${\gamma _L} < 1$ and ${\gamma _H} > 1$. $c$ is a constant to control the sharpness of the slope of the function, which transitions between ${\gamma _L}$ and ${\gamma _H}$. ${D_0}$ represents the cut-off frequency of the Gauss high-pass filter. $D(u,v)$ is defined by equation (7),

$$\begin{equation}D(u,v) = {[{(u - P/2)^2} + {(v - Q/2)^2}]^{1/2}}\end{equation} \tag{ 7 }$$

$P = 2M$, $Q = 2N$, where $M$ and $N$ represent the width and height of the given image $f(x,y)$, respectively. The main parameters of the filter function used in this paper are ${\gamma _H}{\text{ = 2}}{\text{.20}}$, ${\gamma _L}{\text{ = 0}}{\text{.25}}$, $c{\text{ = 0}}{\text{.65}}$, and ${D_0}{\text{ = 160}}$ in equation (6).

To apply the homomorphic filtering algorithm online detection, the compute unified device architecture (CUDA) framework has been adopted to accelerate the algorithm on a GPU. The basic idea of GPU acceleration is to make full use of the multiprocessor structure and single-instruction multiple-data characteristics [21, 22]. The filtering process is mapped to multiple threads and runs in parallel. The flow chart of GPU acceleration of the homomorphic filtering algorithm is shown in figure 3.

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Figure 4(A) is a physical image of an IFR 26 650 battery. It will have graphite in the negative electrode, and LiFePO₄ is the active material of the positive one. Figure 4(B) is the DR image of its upper part. All the pole-pieces lie in the ROI as marked by a red rectangular box, except for the central region (invalid region) marked with a yellow rectangular box in figure 4(C).

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In order to achieve accurate detection results, it is necessary to recognize the invalid region in the ROI before detecting pole-piece corners. In this paper, we determine the invalid region recognition algorithm as the following flow:

• (1)  
  On both sides of the dividing point (the center point of the ROI), a segment of pixels (0.05 * W_ROI ) is intercepted respectively, which is recorded as P_line in figure 5(A) (the local region of the ROI).
• (2)  
  On the left and right sides near the dividing point, two minimum values are selected as candidate boundary points of the invalid region severally, such as points 2, 3, 4 and 5 in figure 5(A).
• (3)  
  The gray curve of P_line is plotted in figure 5(B), which spans a total of six pole-pieces. Generally, the distance between two adjacent negative pole-pieces is fixed, which is recorded as R_pole (0.2435 < R_pole < 0.4298 mm). If the horizontal distance between the candidate boundary point and the adjacent negative pole-piece is within R_pole, the candidate point will be determined as the invalid region boundary point.

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In figure 5(B), the horizontal distance between point 2 and point 3 exceeds R_pole, and the horizontal distance between point 1 and point 2 is in R_pole. Therefore, point 3 is invalid, and point 2 is regarded as the left boundary point. Similarly, point 5 is considered as the right boundary point. Finally, the invalid region can be identified according to the left and right boundary points, as shown in figure 5(C).

Corners are considered to be the points with very variable brightness in an image, or the points of maximum curvature on the edge, or the intersection of two or more edges, or the local maximum corresponding pixel of the first derivative (the gray gradient). These points retain critical characteristic information, which plays an important role in the analysis of the image. The gray values of the pole-piece endpoints change significantly in any direction in the ROI. Therefore, the corner detection algorithm can be used to identify the positions of the pole-piece endpoints.

Corner detection methods mainly include gray-level-based, edge outline and corner models [23, 24]. In this application, we choose the Shi-Tomasi corner detection algorithm, which has been improved based on the Harris algorithm [25, 26].

The Shi-Tomasi corner detection algorithm detects corners by calculating the changes of gray levels in a local small window $w(x,y)$ after moving in all directions. In the smooth region (figure 6(A)), the gray values in the window change gently in all directions. In the edge region (figure 6(B)), the gray values in the window only change in one direction. In the corner region (figure 6(C)), the gray values in the window change significantly in all directions.

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When the window $w(x,y)$ moves $(u,v)$ in any direction, the grayscale alteration ${\text{E}}(u,v)$ is:

$$\begin{equation}{\text{E}}(u,v) = {\sum\limits_{x,y} {w(x,y)[I(x + u,y + v) - I(x,y)]} ^2}\end{equation} \tag{ 8 }$$

and ${\text{w}}(x,y) = \exp ( - ({x^2} + {y^2})/{\text{2}}{\sigma ^2})$ is the Gauss kernel function; $I(x,y)$ stands for the gray value of a pixel.

According to the Taylor formula, $I(x + u,y + v) = I(x,y) + {I_x}u + {I_y}v + O({u^2},{v^2})$,

$$\begin{equation}{\text{E}}(u,v) = [u,v]\sum\limits_{x,y} {w(x,y)} \left[ {\begin{array}{*{20}{c}} {I_x^2}&{{I_x}{I_y}} \\ {{I_x}{I_y}}&{I_y^2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} u \\ v \end{array}} \right]\end{equation} \tag{ 9 }$$

Set:

$$\begin{equation}M = \sum\limits_{x,y} {w(x,y)} \left[ {\begin{array}{*{20}{c}} {I_x^2}&{{I_x}{I_y}} \\ {{I_x}{I_y}}&{I_y^2} \end{array}} \right]\end{equation} \tag{ 10 }$$

In equation (10), ${I_x}$ and ${I_y}$ represent the gradient value of the image grayscale in the x and y directions.

Two eigenvalues of the auto-correlation function M are calculated and compared. When the smaller of the two eigenvalues is greater than the threshold, the Shi-Tomasi corner will be retained as a detected corner.

When the Shi-Tomasi corner detection algorithm is used to identify the corners of battery pole-pieces, there are often some false detection corners. If the number of corners set in the Shi-Tomasi corner detection algorithm is the same as the actual number of pole-pieces, there will inevitably be some missing pole-piece corners. Therefore, during the practical application of Shi-Tomasi corner detection algorithm, the number of corners detected is usually more than the actual number of corners, which means some false corners are detected. To improve the detection accuracy, the 1D region growing algorithm is proposed to remove false corners.

When the 1D region growth algorithm is implemented, the seed point should be selected as a starting point of growth. After determining the horizontal growth direction (left or right), the vertical distance between the seed point and the prediction point, as well as the horizontal distance between the prediction point and the adjacent point, need to be compared with the threshold to determine whether the prediction point is the correct corner.

The correct selection of seed points is an important basis for the 1D region growing algorithm. The four seed points are picked up near the invalid region by default corresponding to the positive and negative corners, respectively, in figure 7. Points 1 and 2 are the seed points of the negative pole-pieces on both sides of the invalid region. Correspondingly, points 3 and 4 are the seed points of the positive pole-pieces on both sides of the invalid region.

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The left half of the ROI will be taken as an example to illustrate the selection of seed points under special circumstances. The negative pole-piece near the invalid region of the battery will sometimes be covered, so that two points will appear in the same vertical direction, such as the black rectangular box in figure 8.

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In this case, the position of the negative seed point needs to be further judged. If there is only one point in the vertical direction at the other end of the image, the point is regarded as the seed point, such as the yellow dotted rectangular box in figure 8. According to the characteristics of pole-piece structures, the probability is very small about two points appearing in the same vertical direction on the left side, and the probability is even smaller with two points appearing both on the left or right sides.

Then the seed point is taken as the starting point for growth, and the arrow direction is the orientation of the 1D region growing in figure 8. During the process of forwarding growth, the first step is to determine whether the current point meets the vertical threshold by equation (11). If not, the point is identified as the false corner, as shown in the blue dotted rectangular box in figure 9. Otherwise, it is necessary to determine whether the current point meets the horizontal threshold. If so, it is necessary to use the endpoint prediction method to judge whether the current point is a false corner, as shown in the blue solid rectangular box in figure 9.

$$\begin{equation}Q = \left\{ \begin{array}{@{}c@{}} {TRUE,{{\,\,\,\,\,D}} \le T} \\ {FALSE,{{\,\,\,\,D > }}T} \end{array}\right.\end{equation} \tag{ 11 }$$

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where $TRUE$ represents the corners that need to be retained and $FALSE$ represents the corners that need to be deleted. T is a fixed threshold; in this paper, T =0.2579 mm. D indicates the vertical distance between the current corner and the seed point.

In the endpoint prediction method, the right one, two or more adjacent corners are used to determine whether the current corner is the required corner. When only one point P₁ is used for prediction, the point P_x1 is close to the prediction point P_x in figure 10(A). This method is simple and easy to operate, but there will often be some misjudgments. When two points P₁ and P₂ are used for prediction, the line P₁P₂ equation will be got firstly with these two points in figure 10(B). The prediction point P_x will be calculated according to the straight line P₁P₂ equation so that the point P_x2 is close to the prediction point P_x. This method can meet most of the identification requirements. When three or more points are used for prediction, the amount of computation is large and it is more unstable. Therefore, two points are used to predict whether the current corner is a false corner of the endpoint prediction method.

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In our research, the DR images of 26 650 cylindrical lithium-ion batteries were obtained from a 3D-µCT system as shown in figure 11. The main configuration of the inspection system includes a 225 kV micro-focus ray source, a high-resolution amorphous silicon flat-panel detector, and a four freedom degrees control system.

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The main parameters of the image acquisition experiment are shown in table 1.

The thickness of the copper pole-piece in lithium-ion batteries is usually 8–25 μm, and the aluminum pole-piece is 12–25 μm. The calibration factor is 0.0286 mm (28.6 μm), which can be calculated by equation (12). The thickness of one pole-piece usually occupies one pixel in the image, and the influence of its width on the position difference detection of the pole-piece can be ignored.

$$\begin{equation}B = {S_{pixel}}/k\end{equation} \tag{ 12 }$$

The GPU-accelerated homomorphic filtering algorithm is employed to enhance the original DR images. The algorithm is run on a host PC with a GPU. The basic configuration of the host PC is an Intel (R) Core (TM) i5-3570 3.40 GHz processor, 16 G of RAM and an NVIDIA GeForce GTX 970 graphics card with 4 GB memory. To compare and analyze the acceleration effect of GPU and CPU, the CPU enhancement algorithm of homomorphic filtering was run on the same device.

The original DR image (figure 12(A)) was enhanced through CPU (figure 12(B)) and GPU (figure 12(C)). It can be seen that the enhancement effect is very obvious, and the position of the battery pole-piece can be seen clearly. The enhancement effects of CPU and GPU are basically the same, without obvious differences. To more accurately compare the image enhancement results, the gray curves are plotted in figure 12(D) for the row of pixels shown in figures 12(A)–(C). As can be seen from the gray curves, the enhancement algorithm highlights the details of the original image well. In order to compare the enhanced effect of GPU and CPU more clearly from the gray curve, a section of the gray curve has been enlarged. As can be seen, the gray curves of CPU and GPU remain consistent.

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Single-precision floating-point DR images, whose sizes are 768 × 768, 1024 × 1024, and 2048 × 2048, are used to compare the computational speeds of CPU and GPU. The number of each group image is 500, and the experimental results are shown in table 2.

According to the data in table 2, GPU can achieve a speed-up ratio of about 20 compared with the traditional CPU based on the homomorphic filtering algorithm. The image processing time has greatly reduced so that the image can be enhanced in real time.

When the DR image acquisition parameters are determined, the location of the ROI is usually fixed. As a result, the ROI can be directly captured from the original and enhanced images as shown in figures 13(A) and (B). The local magnification contrast diagrams of the rectangular box in figures 13(A) and (B) are shown in figures 13(C) and (D), which indicate that the details of the pole-pieces are obviously enhanced and the overall contrast is also improved.

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The invalid region is identified in an enhanced ROI by using the invalid region recognition algorithm proposed in this paper. The result is shown in the green rectangular box in figure 14, which is useful to correctly identify the corners in the following analysis.

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When the corner detection algorithm is carried out with respect to the ROI of removing the invalid region, the number of Shi-Tomasi corners should be set more than the actual number of pole-pieces to ensure that there are no leak corners. For example, the number of positive and negative pole-pieces is about 140, and the number of Shi-Tomasi corners is set to 160 in total. Consequently, there will be about 20 false corners.

The corner recognition results of an ROI (figure 14) are shown in figure 15(A). There is non-coincidence between the positive and negative corners, so the positive and negative corners can be separated directly from the average value of all corner vertical coordinates in figure 15(B). The negative and positive corners are denoted by red and green, respectively.

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The 1D region growing algorithm is adopted to delete the false corners, and the diagram of correct corners is obtained as shown in figure 16.

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Sometimes the pole-piece will be inflected by other structures, which will result in some leaking corners in the yellow rectangular box in figure 16. Compared with the negative pole-pieces, the contrast of the positive pole-piece DR images is much lower, and the possibilities of omission and error detection are relatively larger. Moreover, positive corners will interferewith other structures, so there will be some small position deviations between the recognition corners and the practical corners. Because of this situation, the least squares method is used to fit negative and positive corners with the fifth-order polynomial curve, respectively [27, 28]. The curve fitting result is shown in figure 17.

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As a result, the negative leaking corners can be interpolated and the accurate location of positive corners can be redetermined by sampling on the fitting curves. The horizontal position of one positive corner is located between adjacent two negative corners, and the positive and negative corners alternately appear. The final recognition result of a battery's positive and negative corners is shown in figure 18.

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Five 26 650 cylindrical lithium-ion batteries with different pole-piece structural characteristics (figure 19) are selected to calculate the minimum position distance and alignment metrics through the automatic identification method proposed in this paper. The five batteries used in the experiment have the same critical parameters (table 3) except for the pole-piece structures. Finally, the recognition results are shown in figure 20.

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A normal pole-piece DR image is shown in figure 19(A), from which we can find that the pole-pieces are regularly and symmetrically arranged. The positions of the normal positive and negative corners can be correctly identified by the automatic identification method as shown in figure 20(A). There is a curved negative pole-piece which was caused by collision with other objects during production in the yellow rectangular box 1 in figure 19(B). The recognition results are shown in figure 20(B) in the yellow rectangular box 1, which makes it clear that the method can accurately identify the corner of a curved negative pole-piece. The negative pole-pieces are curved and overlapped in the yellow rectangular box 2 in figure 19(C), whose corner features are not very prominent. It is difficult to precisely identify such corners, but the corners can be effectively identified through the method in this paper, as shown in the yellow rectangular box 2 in figure 20(C). During the process of battery production, it usually occurs that the pole-pieces are not consistent in height, especially in the yellow rectangular box 3 in figure 19(D). Under these circumstances, the setting of a threshold is very important for the deletion of false corners. Finally, the recognition result is shown in figure 20(D), which indicates that the recognition effects achieve the requirements. There are smaller position distances between the positive and negative pole-pieces in the yellow rectangular box 4, but larger position distances in the yellow rectangular box 5 near the edge region in figure 19(E). The corner recognition diagram is shown in figure 20(E), which manifests that the method still has good recognition results for the large gap between the ROI center pole-pieces and the edge pole-pieces.

The minimum position distance D_i can be calculated with the positive and negative corners by equation (12). The standard deviations of positive and negative corners are calculated by equation (14) to judge the alignment uniformities of positive and negative pole-pieces,

$$\begin{equation}{D_i} = \min \left(\frac{{{y_{Ni}} - {y_{Pi}}}}{k} \times {s_{pixel}}\right)\end{equation} \tag{ 13 }$$

$$\begin{equation}{S_x}\! =\! \sqrt {\frac{1}{{{M_x}}} \times \sum\nolimits_{i = 1}^{{M_x}} {{{\left(\!\frac{{{y_{xi}}}}{k} \times {s_{pixel}} - \frac{1}{{{M_x}}} \times \sum\nolimits_{i = 1}^{{M_x}} {\frac{{{y_{xi}}}}{k}} \times {s_{pixel}}\!\right)}^2}} } \end{equation} \tag{ 14 }$$

${y_{Pi}}$ and ${y_{Ni}}$ represent the vertical coordinates of the positive and negative corners, respectively. M_x is the number of positive or negative corners ($x$ is P or N). ${s_{pixel}}$ represents detector pixel size. $k$ manifests the amplification ratio.

In the range of 0–180°, a total of six different circumferential angle acquisition positions are set for each battery. The acquisition position interval is 30°. The minimum position distance between positive and negative pole-pieces is shown in table 4. The difference between the maximum and minimum values of minimum position distance obtained at different circular angles of the same battery is taken as the repeated precision. The results of repeated accuracy for five 26 650 batteries are shown in table 5. The minimum of repeated accuracy is 0.01 mm and the maximum is 0.05 mm, which meets the standard that the requirement of specification for the lithium-ion battery industry is not greater than 0.1 mm [29]. It also indicates the effectiveness and stability of the proposed method, which is not affected by the DR image acquisition circumferential angle.

The alignment metrics are calculated by equation (14) with the data of table 4, which are shown in table 5. According to the alignment metrics of positive and negative pole-pieces, we can judge whether the quality of the inspected batteries is acceptable. The vertical positions of the positive corners in battery A have the smallest changes, and the alignment metric is 0.036. The vertical positions of the negative corners in battery D have the greatest changes, and the alignment metric is 0.212. Hence, to ensure an acceptable qualified battery, a threshold can be set to check the alignment metric of pole-pieces.