Improved YOLOv11 crack identification and quantification method for strip mine area and slope thereof

Abstract

The invention discloses an improved YOLOv11 crack identification and quantification method for a strip mine area and a slope thereof. The method comprises the steps of data preprocessing, model improvement and adaptability adjustment, initial identification model training, crack detection by applying a target identification model, local area identification and post-processing. According to the method, technical means such as a feature extraction network improvement method based on an AReLU function and an ACmix model and a self-adaptive local threshold segmentation algorithm based on a gray scale standard deviation are introduced, and efficient and accurate recognition of cracks is achieved. According to the method, the automation degree of crack detection is improved, the labor cost and the time cost are reduced, and the method is of great significance to safety monitoring of the strip mine area.

Description

Technical Field

[0001] The present invention belongs to the field of computer vision technology, and in particular relates to an improved YOLOv11 crack recognition and quantification method for open-pit mines and slopes thereof. Background Art

[0002] Mine slope safety is one of the core issues in open-pit mining operations. Slope stability is directly related to the life safety of operators and the normal operation of equipment. Slope instability will cause serious geological disasters. Early identification of cracks is a key link in slope safety management. Once cracks appear in mine slopes, they may lead to catastrophic events such as rock sliding and collapse. Therefore, real-time and accurate identification and monitoring of cracks are of great significance for preventing slope instability and ensuring the safety of mining operations.

[0003] Both classical image processing methods and deep learning-based crack recognition methods are widely used in this field. Classic image processing methods, such as edge detection and morphological processing, have the advantages of low computational complexity and simple implementation, but are prone to missed detection and false detection when dealing with complex backgrounds, illumination changes, and noise interference. In contrast, deep learning-based methods, such as target detection models and image segmentation models, have powerful feature extraction and pattern recognition capabilities and can accurately detect cracks in complex environments. However, deep learning-based methods usually consume large computing resources, have high hardware requirements, and have a relatively limited scope of application.

[0004] Therefore, it is very necessary to develop an improved YOLOv11 crack identification and quantification method for open-pit mines and their slopes that can solve the above problems. Summary of the invention

[0005] The object of the present invention is to provide an improved YOLOv11 crack identification and quantification method for open-pit mines and their slopes.

[0006] The object of the present invention is achieved by comprising the following steps: S100, data preprocessing: first, grayscale conversion is performed on the original image, then Gaussian filtering is applied to smooth the image, and then gamma transformation is applied to enhance the image; since the grayscale mean of the image in the original data set is often uneven, the image is grayscale stretched, and the present invention uses adaptive gamma transformation to perform grayscale stretching, and realizes the adaptation of gamma transformation parameters; the cracks in the image are labeled using the bounding box labeling method, and two labels, "strip cracks" and "net cracks", are used respectively; the labeled data set is divided into a training set, a validation set, and a test set, and the proportions of the training set, the validation set, and the test set are 70%, 15%, and 15%, respectively, and it is ensured that the two labels are evenly distributed in the three data sets; S200, model improvement and adaptive adjustment: use data enhancement methods to simulate various transformations in real scenes; based on the original architecture of YOLOv11, improve its feature extraction network; The original architecture of YOLOv11 includes a backbone network (Backbone) for feature extraction, a neck (Neck) for feature fusion, and a head (Head) for final prediction; the backbone network generates multi-scale feature maps by stacking convolutional layers and modules, in which the C3k2 module is introduced to replace the C2f module of the previous version, and two smaller convolution kernels are used to improve computational efficiency; in addition, YOLOv11 retains the SPPF module and adds a new C2PSA module, which enhances the attention to important areas of the feature map through the spatial attention mechanism; the neck structure aggregates features of different resolutions and passes them to the head for prediction, and is responsible for outputting the positioning and classification results of the object through the Detect layer.

[0007] Data enhancement methods include: image rotation, scaling, translation, cropping, noise addition, and grayscale value adjustment to simulate various transformations in real scenes; adding data enhancement helps to enhance the model's adaptability to different crack morphologies and scene changes; S300, training an initial recognition model: performing model training, model verification and model testing to obtain a target recognition model; S400, applying the target recognition model to perform crack detection: applying data preprocessing to the images collected from the open-pit mine area and the slope; using the target recognition model to perform crack detection on the images, and outputting the detection results to indicate whether cracks exist in the open-pit mine area and the slope; S500, local area recognition: applying an adaptive local threshold segmentation method to the recognition result of the target recognition model, further identifying the crack texture and segmenting the crack area; performing connected domain screening based on the area and the average gray value of the area, wherein the area threshold is positively correlated with the image size; S600, post-processing: first perform morphological processing, then perform crack quantification, and obtain crack parameters.

[0008] Preferably, the gamma transformation formula in step S100 is: Among them, P out (x, y) is the pixel value at (x, y) of the output; A is a constant used to adjust the range of output brightness; P in (x, y) is the pixel value at the input (x, y); γ is the gamma value, which is a positive real number used to control the shape of the gamma curve; The calculation formula of gamma value γ is: where μ is the average gray value of the image; T is the standard gray value, and when T = μ, the image remains unchanged; t is a control parameter used to prevent over-enhancement of the image; when the average gray value of the image is too low, using the calculated gamma value is too small, resulting in too high an average gray value of the image after gamma transformation. Therefore, the parameter t is introduced. When μ < t, using to calculate the gamma value.

[0009] Preferably, in step S200, data augmentation is performed using geometric transformation, noise addition, and gray value adjustment to simulate various transformations in the real scene, so as to enhance the adaptability of the model to different crack morphologies and scene changes; The specific improvement of the feature extraction network in step S200 is as follows: S2021. Replace part of the original SiLU with AReLU, and learn element-level residuals through the attention module, making the network more flexible in the deep feature extraction stage; the original CBS structure of YOLOv11 is replaced with the CBA structure, including a convolutional layer, a batch normalization layer, and an AReLU activation function; AReLU includes: An element-based attention mechanism model (ELement-wise Sign-based Attention, ELSA), and the expression is: where x i is the input of the current layer, α and β are learnable parameters, C() is used to limit the input variable to (0.01, 0.99), and σ is the sigmoid function; The original ReLU function, and the expression is: Add the ELSA attention mechanism model and the Relu function to obtain the AReLU function: Compared with the traditional and more complex attention models, AReLU only contains two learnable parameters, making the structure of the model more concise and the calculation process more efficient; The gradient derivation of the AReLU function is as follows: AReLU can amplify and reduce the gradients of different data, enabling the network to reach the optimal state faster during training; this feature can effectively improve the model performance and accelerate the overall convergence speed; S2022. Embed the ACmix self-attention and convolution hybrid model in the C3 module and the SPPF layer to achieve more efficient local and global feature fusion and further enhance the multi-scale object detection ability; The specific process of step S2022 is: (1) The input features are linearly transformed through three 1×1 convolutional layers, and the projected features are divided into N groups of subsets, each of which contains three channels; (2) The self-attention path and the convolution path are processed in parallel. The self-attention path uses the three feature maps in each subset as query, key, and value, respectively, and introduces a multi-head self-attention mechanism. The convolution path processing includes using lightweight 3×3 convolution to process local features and performing multi-level pooling through the improved SPPF layer to better extract the local texture details of the cracks. (3) The outputs of the self-attention path and the convolution path are weighted and summed according to the following formula: Among them, F att is the output of the self-attention path, F conv is the output of the convolution path, α and β are learnable weight parameters used to balance the contribution of the attention mechanism output and the convolution output.

[0010] Preferably, step S300 specifically includes: S301, Model training: Use the training set to train the model and adjust the model parameters to capture the patterns and relationships in the data; in the early stages of training, try a variety of different model architectures, including neural networks of different depths, the number of convolutional layers, the use of pooling layers, etc., to select the best performing structure; based on prior knowledge of hyperparameters and evaluation results, use Bayesian methods to gradually approximate the best hyperparameter combination; S302, model validation: Use the validation set to evaluate model performance, select the best model structure and hyperparameters, and determine when to stop training to avoid overfitting; after each training cycle, use the validation set to evaluate model performance; performance evaluation indicators include accuracy, precision, recall, and F1 score; S303, model testing: Use the test set to perform a final evaluation of the model, test its generalization ability, and obtain the final target recognition model.

[0011] Preferably, the adaptive local threshold segmentation method in step S500 includes the following steps: S501, calculate the improved difference entropy: S5011, fitting the parameters of the adaptive local threshold segmentation algorithm based on the improved difference entropy; S5012, based on the sliding window and the Gaussian weight matrix, the weighted difference between the central pixel and other pixels is calculated, and the probability density function of the two-dimensional Gaussian distribution is: Where x=[x,y] T is a two-dimensional random variable, μ is the mean vector, Σ is the covariance matrix, and |Σ| represents the determinant of the covariance matrix; S5013, based on the grayscale difference matrix and weight matrix of each pixel, record the weight number of each difference in the grayscale difference matrix, that is, the weighted number of times, and then obtain the distribution X of the grayscale value difference between each pixel of the image and its surrounding pixels; S5014. Substituting the improved difference entropy formula, the calculation formula of the improved difference entropy H(x) is: Where i is the value of the difference, n is the number of gray levels, and P(x) is the distribution rate of the difference; S502, parameter fitting: S5021. Through local threshold segmentation parameter analysis experiments, sufficient optimal sensitivity and optimal minimum standard deviation threshold data are obtained when the crack identification effect is best; S5022. Based on the least square fitting method, the experimental data is fitted with the difference entropy of each image, and a functional relationship is established; S503, adaptive local threshold segmentation: S5031, using a sliding window of definable size, calculating the mean and standard deviation of the grayscale values ​​of pixels in the window; S5032: Segment the foreground and background in the sliding window based on standard deviation segmentation and sensitivity parameters.

[0012] Preferably, the morphological processing in step S600 uses dilation and erosion operations to remove small noise points and connect the broken parts of the cracks.

[0013] Preferably, in step S600, in crack quantification, the crack skeleton direction and length are calculated as the crack direction and length.

[0014] Preferably, in step S600, the width of the crack is calculated in the crack quantification by selecting uniformly spaced seed points along the skeleton, performing directional growth along the normal line, and calculating the average growth length, which is the crack width.

[0015] Beneficial effects of the present invention: 1. Improve crack detection accuracy: The improved YOLOv11 model can accurately and specifically identify slope cracks, and maintain high accuracy even under complex lighting and background conditions; 2. Reduce computing costs and hardware requirements: The improved YOLOv11 model combines the local threshold segmentation algorithm while maintaining high accuracy, which not only optimizes the inference computing efficiency, but also reduces the dependence on high-end computing resources, allowing the model to be deployed on a wider range of devices; 3. Improved automation: The automated crack identification and quantification method of the present invention reduces manual intervention, improves the efficiency of the detection process, and reduces labor and time costs; 4. Significance of safety monitoring: For the safety monitoring of open-pit mines, the present invention can monitor the development of cracks in real time and warn potential safety hazards in time, thereby improving the safety of mining operations; 5. Quantitative analysis capability: The present invention provides a reliable means of crack quantification, which is helpful for detailed analysis and subsequent treatment of cracks; 6. Promote intelligent management of open-pit mines: The technical solution of the present invention provides strong technical support for the intelligent management of open-pit mines, and helps to realize automated and intelligent smart mine safety management. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a schematic diagram of the process of the present invention; Figure 2 This is a schematic diagram of the original network architecture of YOLOv11; Figure 3 Schematic diagram of the CBA structure that introduces AReLU; Figure 4 Schematic diagram of the SPPF and Bottleneck structures introduced into the ACmix model; Figure 5 This is a schematic diagram of the improved YOLOv11 network architecture; Figure 6 It is a flowchart of the adaptive local threshold segmentation method; Figure 7 Schematic diagram for crack quantification. DETAILED DESCRIPTION

[0017] The present invention is further described below in conjunction with the embodiments and drawings, but the present invention is not limited in any way. Any changes or substitutions made based on the teachings of the present invention belong to the protection scope of the present invention.

[0018] Example 1 As attached Figure 1 to Figure 7 As shown, the improved YOLOv11 crack identification and quantification method for open-pit mines and slopes in this embodiment includes the following steps: S100, data preprocessing: first convert the grayscale of the original image, then apply Gaussian filtering to smooth the image, and then apply gamma transformation to enhance the image; The gamma transformation formula is as follows: Among them, P out (x, y) is the pixel value at the output (x, y); A is a constant used to adjust the range of the output brightness; P in (x, y) is the pixel value at the input (x, y); γ, namely the gamma value, is a positive real number used to control the shape of the gamma curve; The calculation formula of the gamma value γ is as follows: Among them, μ is the average gray value of the image; T is the standard gray value. When T = μ, the image remains unchanged; when the average gray value of the image is too low, the gamma value calculated using is too small, resulting in too high average gray value of the image after gamma transformation. Therefore, the parameter t is introduced. t is a control parameter used to prevent over-enhancement of the image. When μ < t, is used to calculate the gamma value; Use the bounding box annotation method to annotate the cracks in the image, and use two labels, namely "strip cracks" and "mesh cracks" respectively; divide the annotated dataset into a training set, a validation set and a test set, and the proportions of the training set, the validation set and the test set are 70%, 15% and 15% respectively, and ensure that the two labels are evenly distributed in the three datasets; S200, Model improvement and adaptability adjustment: Based on the original architecture of YOLOv11 (see Appendix Figure 2 ), use data augmentation methods to simulate various transformations in the real scene. Among them, data augmentation uses geometric transformation, noise addition and gray value adjustment to enhance the adaptability of the model to different crack morphologies and scene changes; then improve the feature extraction network; Appendix Figure 2 In it, the original architecture of YOLOv11 includes a backbone network (Backbone) for feature extraction, a neck (Neck) for feature fusion, and a head (Head) for final prediction; the backbone network generates multi-scale feature maps by stacking convolutional layers and modules. Among them, the C3k2 module is introduced to replace the C2f module in the previous version, and two smaller convolutional kernels are used to improve the calculation efficiency; in addition, YOLOv11 retains the SPPF module and adds a new C2PSA module, and the latter enhances the attention to important regions of the feature map through the spatial attention mechanism; the neck structure aggregates features of different resolutions and passes them to the head for prediction, and the Detect layer is responsible for outputting the localization and classification results of the object; Among them, the improvement of the feature extraction network is specifically: S2021. Replace some of the original SiLU with AReLU (Appendix Figure 3), learning element-level residuals through the attention module makes the network more flexible in the deep feature extraction stage; Figure 3 In the , the original CBS structure of YOLOv11 is replaced by the CBA structure, including convolutional layers, batch normalization layers and AReLU activation functions; AReLU includes: The element-wise Sign-based Attention (ELSA) model is expressed as: Among them, x i is the input of the current layer, α and β are learnable parameters, C() is used to limit the input variables to (0.01, 0.99), and σ is the sigmoid function.

[0019] The original ReLU function is expressed as: Add the ELSA attention mechanism model to the Relu function to get the AReLU function: Compared with the traditional more complex attention model, AReLU only contains two learnable parameters, which makes the model structure simpler and the calculation process more efficient.

[0020] The gradient derivation of the AReLU function is as follows: AReLU can amplify and reduce the gradient of different data, allowing the network to reach the optimal state faster during training. This feature can effectively improve model performance and speed up overall convergence; S2022, embed ACmix self-attention and convolution hybrid model in C3 module and SPPF layer (Appendix Figure 4 ), in order to achieve more efficient local and global feature fusion and further enhance the multi-scale target detection capability; in the attached Figure 4 In the ACmix model, the convolution and self-attention mechanism modules of parallel reasoning are used, and weighted summation is performed to improve the feature extraction efficiency and enhance the model stability. The pre-convolutional layers of the SPPF layer and the Bottleneck module are replaced with ACmix to improve the recognition ability of the model in complex scenarios. Figure 5 It is an improved YOLOv11 network architecture; The specific process of step S2022 is: (1) The input features are linearly transformed through three 1×1 convolutional layers, and the projected features are divided into N groups of subsets, each of which contains three channels; (2) The self-attention path and the convolution path are processed in parallel. The self-attention path uses the three feature maps in each subset as query, key, and value, respectively, and introduces a multi-head self-attention mechanism. The convolution path processing includes using lightweight 3×3 convolution to process local features and performing multi-level pooling through the improved SPPF layer to better extract the local texture details of the cracks. (3) The outputs of the self-attention path and the convolution path are weighted and summed according to the following formula: Among them, F att is the output of the self-attention path, F conv is the output of the convolution path, α and β are learnable weight parameters used to balance the contribution of the attention mechanism output and the convolution output; S300, training an initial recognition model: performing model training, model verification and model testing to obtain a target recognition model; S301, Model training: Use the training set to train the model and adjust the model parameters to capture the patterns and relationships in the data; in the early stages of training, try a variety of different model architectures, including neural networks of different depths, the number of convolutional layers, the use of pooling layers, etc., to select the best performing structure; based on prior knowledge of hyperparameters and evaluation results, use Bayesian methods to gradually approximate the best hyperparameter combination; S302, model validation: Use the validation set to evaluate model performance, select the best model structure and hyperparameters, and determine when to stop training to avoid overfitting; after each training cycle, use the validation set to evaluate model performance; performance evaluation indicators include accuracy, precision, recall, and F1 score; S303, model testing: Use the test set to perform a final evaluation of the model, test its generalization ability, and obtain the final target recognition model.

[0021] S400, applying the target recognition model to perform crack detection: applying data preprocessing to the images collected from the open-pit mine area and the slope; using the target recognition model to perform crack detection on the images, and outputting the detection results to indicate whether cracks exist in the open-pit mine area and the slope; S500, local area recognition: applying an adaptive local threshold segmentation method to the recognition result of the target recognition model, further identifying the crack texture and segmenting the crack area; performing connected domain screening based on the area and the average gray value of the area, wherein the area threshold is positively correlated with the image size; The adaptive local threshold segmentation method includes the following steps: S501, calculate the improved difference entropy: S5011, fitting the parameters of the adaptive local threshold segmentation algorithm based on the improved difference entropy; S5012, based on the sliding window and the Gaussian weight matrix, the weighted difference between the central pixel and other pixels is calculated, and the probability density function of the two-dimensional Gaussian distribution is: Where x=[x,y] T is a two-dimensional random variable, μ is the mean vector, Σ is the covariance matrix, and |Σ| represents the determinant of the covariance matrix; S5013, based on the grayscale difference matrix and weight matrix of each pixel, record the weight number of each difference in the grayscale difference matrix, that is, the weighted number of times, and then obtain the distribution X of the grayscale value difference between each pixel of the image and its surrounding pixels; S5014. Substituting the improved difference entropy formula, the calculation formula of the improved difference entropy H(x) is: Where i is the value of the difference, n is the number of gray levels, and P(x) is the distribution rate of the difference; S502, parameter fitting: S5021. Through local threshold segmentation parameter analysis experiments, sufficient optimal sensitivity and optimal minimum standard deviation threshold data are obtained when the crack identification effect is best; S5022. Based on the least square fitting method, the experimental data is fitted with the difference entropy of each image, and a functional relationship is established; S503, adaptive local threshold segmentation: S5031, using a sliding window of definable size, calculating the mean and standard deviation of the grayscale values ​​of pixels in the window; S5032, segmenting the foreground and background in the sliding window based on standard deviation segmentation and sensitivity parameters; S600, post-processing: first perform morphological processing, then perform crack quantification, and obtain crack parameters; Morphological processing uses dilation and erosion operations to remove small noise points and connect the broken parts of the cracks.

[0022] In crack quantification, the crack skeleton direction and length are calculated as the crack direction and length; In crack quantification, the width of the crack is calculated as follows: uniformly spaced seed points are selected along the skeleton, directional growth is performed along the normal, and the average growth length is calculated, which is the crack width.

Claims

1. An improved YOLOv11 crack identification and quantification method for open-pit mines and their slopes, characterized in that The following steps are involved: S100, data preprocessing: first convert the grayscale of the original image, then apply Gaussian filtering to smooth the image, and then apply gamma transformation to enhance the image; Use bounding box annotation to annotate cracks in the image; Divide the labeled data set into training set, validation set and test set; S200, model improvement and adaptation: use data augmentation methods to simulate various transformations in real-world scenarios; Based on the original architecture of YOLOv11, the feature extraction network is improved; S300, training an initial recognition model: performing model training, model verification and model testing to obtain a target recognition model; S400, crack detection using target recognition model: data preprocessing is applied to the images collected from the open pit mine area and slope; Use the target recognition model to detect cracks in the image and output the detection results to indicate whether there are cracks in the open-pit mine area and slope; S500, local area recognition: applying an adaptive local threshold segmentation method to the recognition result of the target recognition model, further identifying the crack texture and segmenting the crack area; performing connected domain screening based on the area and the average gray value of the area, wherein the area threshold is positively correlated with the image size; S600, post-processing: first perform morphological processing, then perform crack quantification, and obtain crack parameters.

2. The improved YOLOv11 crack identification and quantification method for open-pit mines and their slopes according to claim 1, characterized in that The gamma transformation formula for the S100 step is: Among them, P out (x, y) is the pixel value at (x, y) of the output; A is used to adjust the range of output brightness; P in (x, y) is the pixel value at the input (x, y); γ is the gamma value, which is used to control the shape of the gamma curve; The calculation formula of gamma value γ is: Among them, μ is the average gray value of the image; T is the standard gray value, when T=μ, the image does not change; t is the control parameter.

3. The improved YOLOv11 crack identification and quantification method for open-pit mines and their slopes according to claim 1, characterized in that The feature extraction network improvements in step S200 are as follows: S2021, replace part of the original SiLU with AReLU, and learn the element-level residual through the attention module; S2022. Embed the ACmix self-attention and convolution hybrid model in the C3 module and SPPF layer to fuse local and global features and enhance the multi-scale target detection capability.

4. The improved YOLOv11 crack identification and quantification method for open-pit mines and their slopes according to claim 1, characterized in that The specific steps of S300 are: S301, Model training: Use the training set to train the model and adjust the model parameters to capture the patterns and relationships in the data; in the early stages of training, try a variety of different model architectures, including neural networks of different depths, the number of convolutional layers, and the use of pooling layers to select the best performing structure; based on prior knowledge of hyperparameters and evaluation results, use Bayesian methods to gradually approximate the best hyperparameter combination; S302, model validation: Use the validation set to evaluate model performance, select the best model structure and hyperparameters, and determine when to stop training to avoid overfitting; after each training cycle, use the validation set to evaluate model performance; performance evaluation indicators include accuracy, precision, recall, and F1 score; S303, model testing: Use the test set to perform a final evaluation of the model, test its generalization ability, and obtain the final target recognition model.

5. The improved YOLOv11 crack identification and quantification method for open-pit mines and slopes thereof according to claim 1, characterized in that The adaptive local threshold segmentation method in step S500 comprises the following steps: S501, calculate the improved difference entropy: S5011, fitting the parameters of the adaptive local threshold segmentation algorithm based on the improved difference entropy; S5012, based on the sliding window and the Gaussian weight matrix, the weighted difference between the central pixel and other pixels is calculated, and the probability density function of the two-dimensional Gaussian distribution is: Where x=[x,y] T is a two-dimensional random variable, μ is the mean vector, Σ is the covariance matrix, and |Σ| represents the determinant of the covariance matrix; S5013, based on the grayscale difference matrix and weight matrix of each pixel, record the weight number of each difference in the grayscale difference matrix, that is, the weighted number of times, and then obtain the distribution X of the grayscale value difference between each pixel of the image and its surrounding pixels; S5014. Substituting the improved difference entropy formula, the calculation formula of the improved difference entropy H(x) is: Where i is the value of the difference, n is the number of gray levels, and P(x) is the distribution rate of the difference; S502, parameter fitting: S5021. Through local threshold segmentation parameter analysis experiments, sufficient optimal sensitivity and optimal minimum standard deviation threshold data are obtained when the crack identification effect is best; S5022. Based on the least squares fitting method, the experimental data is fitted with the difference entropy of each image, and a functional relationship is established; S503, adaptive local threshold segmentation: S5031, using a sliding window of definable size, calculating the mean and standard deviation of the grayscale values ​​of pixels in the window; S5032: Segment the foreground and background in the sliding window based on standard deviation segmentation and sensitivity parameters.

6. The improved YOLOv11 crack identification and quantification method for open-pit mines and slopes thereof according to claim 1, characterized in that The morphological processing in step S600 uses dilation and erosion operations to remove small noise points and connect the broken parts of the cracks.

7. The improved YOLOv11 crack identification and quantification method for open-pit mines and slopes thereof according to claim 1, characterized in that In step S600, in crack quantification, the crack skeleton direction and length are calculated as the crack direction and length.

8. The improved YOLOv11 crack identification and quantification method for open-pit mines and slopes thereof according to claim 1, characterized in that Step S600 calculates the width of the crack in crack quantification by selecting uniformly spaced seed points along the skeleton, performing directional growth along the normal, and calculating the average growth length, which is the crack width.

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