1. Introduction

2.2. Algorithm for Corrugated Board Geometrical Feature Identification

Figure 3 presents the flow diagram of the proposed algorithm. RGB image obtained from the device is first subjected to different preprocessing operations. Various versions of the input image are then utilized in order to identify several geometrical features of a 5-layer corrugated board sample, such as its height, flutes’ heights, periods and phase shifts, liners’ and flutes’ thickness. The algorithm has been implemented in Python using the OpenCV and geneticalgorithm libraries.

2.2.1. Image Preprocessing

The input of the system was a single frame from the camera. It was an RGB image with dimensions of 3264 × 2448 pixels. The first preprocessing operation is a grayscale conversion into range <0,255>. Next, an 800 × 800 pixels subset of a grayscale image is cut out of the central acquisition area. Figure 4a presents the final image acquired as a result of described actions. In order to remove small noise from the image (caused by the presence of cellulose fibers), two blurring methods were applied: averaging with a normalized box filter and with a kernel size of 3 × 3 and bilateral filter. Figure 4b presents the result of these operations. Finally, the blurred image was converted into a binary image (Figure 4c) by applying lower threshold binarization with a threshold value equal to 75. All the parameters in the preprocessing stage were chosen empirically.

2.2.2. Corrugated Cardboard Thickness Estimation

The thickness of 5-layer corrugated cardboard can be estimated with the same method used for 3-layer samples that was presented in [25]. The boundary points of outer liners can be identified by applying column-wise scanning of the binary image (Figure 4c). Pixels in each column of the image are analyzed. The y coordinate of the first white pixels in each column is saved to ULEP matrix for scanning from the top of the image towards the bottom. Next, scanning is continued until the first black pixel is recognized. Its y coordinate is written to ULIP matrix. As a result, external points of the upper liner are written in the ULEP matrix, whereas ULIP contains upper liner internal points. Analogically, in order to determine lower liner boundary points, the direction of column-wise scanning is reversed, starting from the bottom of the image towards the top. In that way, new matrices LLEP and LLIP, which respectively store external and internal pixels of the lower liner, are created. Figure 5 presents the result of this operation.

In order to determine the corrugated board sample height $d$, the average distance between external points of the upper and lower liner are calculated. It can be expressed as

$$d=\frac{1}{NC}\sum _{x=0}^{NC-1}\left|ULEP\left(x\right)-LLEP\left(x\right)\right|,$$

(1)

where $x$ denotes the column index, and $NC=800$ is the total number of columns.

For the purpose of further geometrical features identification, the external boundaries of both upper and lower liners are approximated using linear functions and coordinates from ULEP and LLEP matrixes. The resulting linear equations can be expressed as

$$y_U=a_{U}x+b_U,$$

(2)

$$y_L=a_{L}x+b_L,$$

(3)

where $a_U$ and $b_U$ denote the parameters of the upper liner approximation, while $a_L$ and $b_L$ are the parameters of the lower liner approximation.

2.2.3. Flutes Center Lines and Heights Estimations

At this stage, the corrugated cardboard flutes center lines and heights are estimated. The binary image row sum curve is plotted to find the localization of liners and flutings regions. Analyzing the number of white pixels in each row of the image, as presented in [25], allows us to determine the approximate location in the image of the bottom and upper liner. The sample is placed horizontally, and curve local maximums are related to the presence of flat layers. Therefore, the occurrence of an additional liner in the middle of the sample should create one additional extremum visible on the curve. In the case of the 5-layered corrugated board sample, which consists of 3 liners and 2 flutes, 3 local maximums should always be detected at this stage. In order to smooth the row sum curve and highlight the maximum resulting from liners, the same version of the Savitzky–Golay filter with 30 interpolation points and a first-degree polynomial was applied. Furthermore, the distance between maximums has to be larger or equal to 20, and the minimal value of the local maximum was equal to 0.4 of the global maximum value. Figure 6a depicts the original row sum and smoothed curves with 3 local maximums detected. It is also worth noting that the peak values can differ significantly for both the ideal and the creased samples. Their values mainly depend on the overall arrangement of layers and their thickness.

At this point, the row sum curve can be further analyzed. Based on local maximums, the curve is divided into 3 ranges. Each range corresponds to the area of one liner. Ranges borders (black bold dashed line) are determined as middle points between two adjacent local maximums marked as blue bold dots in Figure 6a. In Figure 6b, the bottom, middle, and upper liner ranges are marked in green, blue, and red, respectively. In each of these intervals, the subsequent actions are carried out:

• The local maximum $S_{max}$ is identified.
• The vertical line with an ordinate equal to the value of 0.9$S_{max}$ (for bottom and upper liner regions, or 0.9$5S_{max}$ for middle liner region) is now plotted. Two intersection points of the curve and plotted line are determined and marked by bold dots visible in Figure 6b.
• The distance between intersection points within each range is calculated and denoted as $b_{US}$, $b_{mS}$ and $b_{BS}$ for the upper, middle, and bottom liners, respectively.

For the 5-layer corrugated board samples, the center line and height estimations are calculated separately for each fluting. First, the middle liner approximate location in the image must be determined. Another column-wise scanning of the binary image (Figure 4c) is carried out. Scanning is limited to the rows with coordinates $y\in <y_{int1}-20,y_{int2}+20>$, where $y_{int1}$ and $y_{int2}$ are consecutive intersection points of the middle liner visible in Figure 6b. The coordinates of the first white pixels in each column are written into the MLUP matrix for scanning from the top to the bottom of the image and for reversed direction into MLBP. A linear approximation of a line passing through the center of the area bounded by MLUP and MLBP pixels is performed. It can be expressed as:

$$y_m=a_{m}x+b_m,$$

(4)

where $a_m$ and $b_m$ denote the parameters of the middle liner approximation. Figure 7 shows results of above operations.

Both flutings searching can be now limited based on the boundaries of the liners expressed in Equations (2), (3), and (4). The boundary lines limiting the ${fluting}_1$searching can be written in the following forms

$$y_{1UBL}=a_{U}x+b_U+b_{US},$$

(5)

$$y_{1LBL}=a_{m}x+b_m-b_{mS},$$

(6)

whereas for limiting the ${fluting}_2$

$$y_{2UBL}=a_{m}x+b_m+b_{mS},$$

(7)

$$y_{1LBL}=a_{L}x+b_L-b_{LS}.$$

(8)

The boundary lines are depicted in Figure 8 in red and blue, while the center lines are presented in yellow. The center lines are approximated as central lines between two boundary lines, for each flute, respectively, and can be expressed as:

$$y_{center1}=a_{center1}x+b_{center1},$$

(9)

$$y_{center2}=a_{center2}x+b_{center2},$$

(10)

where $a_{1center}$, $a_{2center},b_{1center},b_{2center}$ are the parameters of the center lines.

The heights of the flutes can be approximated using the following formulas

$$H_1=\frac{1}{NC}\sum _{x=0}^{NC-1}\left|y_{1UBL}\left(x\right)-y_{1LBL}\left(x\right)\right|=\frac{1}{NC}\sum _{x=0}^{NC-1}\left|\left(a_U-a_m\right)x+b_U+b_{US}-b_m+b_{mS}\right|,$$

(11)

$$H_2=\frac{1}{NC}\sum _{x=0}^{NC-1}\left|y_{2UBL}\left(x\right)-y_{2LBL}\left(x\right)\right|=\frac{1}{NC}\sum _{x=0}^{NC-1}\left|\left(a_m-a_L\right)x+b_m+b_{mS}-b_L+b_{LS}\right|.$$

(12)

It is worth noting, that flutes height is equal to two times the amplitude of the sinusoidal function.

Figure 8. The binary image with boundary lines for limiting the searching areas for ${fluting}_1$ (blue lines) and ${fluting}_2$ (red lines), and the central lines (yellow lines).

Figure 8. The binary image with boundary lines for limiting the searching areas for ${fluting}_1$ (blue lines) and ${fluting}_2$ (red lines), and the central lines (yellow lines).

2.2.4. Flute Period Searching Range

Next, the binary image is skeletonized. The result of this operation is presented in Figure 9a. The presence of protruding fibers in the cross-section of the sample can cause some disturbances in the form of side branches in the skeleton. A custom filtering function is applied to the skeletonized image to filter out the unwanted offshoots. As a result, all side branches with contour lengths less than 50 pixels are removed, see Figure 8b.

The limitation of period searching is necessary for genetic algorithm to provide credible solutions. Values $T_{\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}}$ and $T_{\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{x}}$ (the range of the period searching in pixels for $i$-th fluting) are based on calculating the distances between the intersection points of the skeleton contours, with 3 lines parallel to the flutings center line, drawn through the flute area, see Figure 8c. The average distance between successive intersection points and the maximum distance is determined for each fluting. The values $T_{\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}}=0.5{di}_{\mathrm{i}\mathrm{a}\mathrm{v}}$, and $T_{\mathrm{m}\mathrm{a}\mathrm{x}}=3{di}_{\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{x}}$ are taken, where ${di}_{\mathrm{i}\mathrm{a}\mathrm{v}}$ is the average distance between the intersection points for the 3 parallel lines, and ${di}_{\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{x}}$ denotes the maximal value of the distances between these intersection points for $i$-th fluting. In case of too many disturbances, e.g., due to the presence of long side branches or critical deformation of the sample, the default values $T_{\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}}=50$, $T_{\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{x}}=800$ are set.

Figure 8. (a) Results of the skeletonization process; (b) results after skeletonization process and removing side branches; (c) three parallel green lines for each fluting period limitations.

Figure 8. (a) Results of the skeletonization process; (b) results after skeletonization process and removing side branches; (c) three parallel green lines for each fluting period limitations.

2.2.5. Application of the Genetic Algorithm for Flutes Parameters Approximation

Genetic algorithm implemented in [25] enabled the authors to determine sinusoidal function parameters, such as period and phase shift, to asses fluting layer geometrical features. In the case of double-wall corrugated board samples, the genetic algorithm ought to be used for each flute independently.

In the searching process for the parameters of flutes, the following formulas for its approximation were taken into account:

$$y_{{fluting}_i}=a_{centeri}x+b_{centeri}-A_i\sin \left({\phi }_i+\frac{2\pi }{T_i}x\right),$$

(8)

where the parameters $a_{centeri}$, $b_{centeri}$, and amplitude $A_i=H_i/2$ are fluting parameters determined in the previous stages of the proposed algorithm. The solution from the genetic algorithm is a set of two values: phase shift ${\phi }_i$ and the period $T_i$, where i denotes the fluting index ($i=\mathrm{1,2}$). Two separate genetic algorithm runs provided an independent set of parameters for each fluting.

In each run, the genetic algorithm takes the eroded version of the binary image as an input, as shown in Figure 9a. The main reason for utilizing erosion is to narrow down the flute region, so that the sine function approximation is more precise. The phase shift ${\phi }_i$ and period $T_i$ search is limited by ${\phi }_{\mathrm{m}\mathrm{i}\mathrm{n}}=0$, ${\phi }_{\mathrm{m}\mathrm{a}\mathrm{x}}=2\pi$, $T_{\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}}$, and $T_{\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{x}}$, where i denotes the fluting index. The objective function is defined as a total sum of the common pixels for the eroded image and function expressed in Equation (7) or (8) (depending on the analyzed flute) for the given ${\phi }_i$ and $T_i$.

Applying the genetic algorithm, the following parameters were utilized:

• Maximal number of iterations: 500;
• Population size: 100;
• Mutation probability: 0.15;
• Elite group ratio: 0.01;
• Crossover probability: 0.2;
• Parents portion: 0.2;
• Crossover type: uniform.

An example of the genetic algorithm result is presented in Figure 9b.

Figure 9. (a) The eroded image; (b) an example of the result obtained after the optimization processes using the genetic algorithm (red lines).

Figure 9. (a) The eroded image; (b) an example of the result obtained after the optimization processes using the genetic algorithm (red lines).

2.2.6. Estimation of the Corrugated Cardboard Layers Thicknesses

Figure 10 depicts a graphic representation of the idea for measuring the thickness of paper layers. The approximate location of flutes in the image is determined based on the sinusoidal function approximations performed in the previous stages of the algorithm. It provides enough information to choose the regions for layers thickness measurement. Areas around layer bonding points are more distorted and have usually higher number of disturbances such as protruding fibers. The area of the upper liner, where the thickness can be measured, is marked in blue. A similar region for the middle and bottom liner is marked in orange and turquoise, respectively. The pink and green colors indicate the areas in which the thickness of the flutings is determined. Figure 11 shows an example of the result.

3. Results

4. Discussion

Figure 17. Example of the error in recognizing the number of layers in the corrugated board sample: (a) a single-wall C-flute sample with a visible hank of cellulose fibers; (b) a smoothed row-sum curve of the image with wrongly localized peaks potentially reflecting the liners of the corrugated board.

Figure 17. Example of the error in recognizing the number of layers in the corrugated board sample: (a) a single-wall C-flute sample with a visible hank of cellulose fibers; (b) a smoothed row-sum curve of the image with wrongly localized peaks potentially reflecting the liners of the corrugated board.

Figure 17. (a) The original example (very weak effect) and (b) the results on the same image after limiting the layers thickness measurement areas.

Figure 17. (a) The original example (very weak effect) and (b) the results on the same image after limiting the layers thickness measurement areas.

5. Conclusions

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