Underwater dredging pipeline intelligent identification method based on multi-temporal remote sensing image
Through multi-time phase remote sensing image processing technology, the underwater silting pipeline is accurately positioned, which solves the problem of time-consuming, labor-intensive and insufficient accuracy of traditional methods, and achieves efficient and safe identification and management of silting pipelines.
Patent Information
- Application Number
- CN202510101017.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-22
- Publication Date
- 2025-05-06
- Estimated Expiration
- 2045-01-22
AI Technical Summary
The existing reservoir silt pipeline identification method is time-consuming and labor-intensive and has limited accuracy, especially in complex underwater environments, which is difficult to achieve efficient and accurate identification.
An intelligent recognition method based on multi-time phase remote sensing images is adopted. By obtaining multi-time phase remote sensing images, a specific band is extracted, the optical remote sensing water index is calculated, and the threshold is determined using the rainbow color map display and the maximum inter-class variance method. Combined with Lee filter and dual-threshold Sobel edge detection, a silting pipeline distribution map is formed.
It realizes efficient and accurate identification of underwater silting pipelines, improves the efficiency and safety of silting management work, reduces interference to the reservoir ecosystem, and provides accurate pipeline condition data support.
Smart Images

Figure CN119942318A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of dredging pipeline identification, and in particular to an underwater dredging pipeline intelligent identification method based on multi-temporal remote sensing images. Background Art
[0002] As an important water conservancy project facility, reservoirs undertake multiple functions such as flood control, water supply, irrigation and power generation. However, over time, reservoirs inevitably face the problem of siltation. The deposition of silt and organic matter will lead to a reduction in reservoir capacity and affect its function. In order to maintain the normal function of the reservoir, regular silt removal becomes a necessary maintenance work. Accurately identifying the location of silt removal pipelines is crucial to understanding the progress and location of silt removal. It is related to the efficiency and safety of silt removal work and directly affects the protection of reservoir infrastructure.
[0003] Traditional pipeline identification methods mainly rely on manual detection or sonar technology, which are often time-consuming and labor-intensive, and have limited accuracy in complex underwater environments. With the development of remote sensing technology, water body identification methods based on satellite remote sensing images have provided new possibilities for reservoir management. Among them, the Normalized Difference Water Index (NDWI), as an effective water body extraction index, has shown great potential in water body boundary identification and underwater object detection. The NDWI index is used to enhance water body information in remote sensing images while suppressing vegetation and soil information. The calculation formula of NDWI is: NDWI = (Green - NIR) / (Green + NIR); where Green represents the reflectance of the green light band and NIR represents the reflectance of the near infrared band. NDWI values are usually between -1 and 1, water bodies usually have positive values, while other objects such as vegetation and soil are usually negative values. This index is mainly used for water body range identification. Different researchers have proposed a variety of water extraction methods based on the NDWI index, such as patents CN 201510272030, CN 201810520933.1, CN202011272866.X.
[0004] As a pipeline for sucking and transporting silt, dredging pipelines usually extend from the bottom of the water to the surface, spanning the above-water and underwater environments, and require coordinated work between land and water. Considering the complexity of the environment and the special requirements for identification, remote sensing technology innovation is needed to improve the accuracy and applicability of identification for this specific object. Summary of the invention
[0005] The present invention provides an underwater desilting pipeline intelligent identification method based on multi-temporal remote sensing images, so as to solve the problem that the accuracy and applicability of the existing technology in identifying the desilting pipeline need to be improved.
[0006] The present invention provides an underwater dredging pipeline intelligent identification method based on multi-temporal remote sensing images, comprising: Step 1: Obtain multi-temporal remote sensing images of the same study area; Step 2: extracting the blue light band, green light band and near infrared band, or the blue light band, green light band and short-wave infrared band of the remote sensing images of each time phase; Step 3, calculating the optical remote sensing water index based on the extracted bands; Step 4, displaying the optical remote sensing water index using a rainbow color map; Step 5, the maximum inter-class variance method is used to determine the optimal threshold for water body segmentation of the optical remote sensing water index; Step six, processing the optical remote sensing water index according to the optimal threshold value, clarifying the water body and non-water body areas, and obtaining a processed optical remote sensing water index image; Step 7, using Lee filter to enhance the processed optical remote sensing water index image and remove noise to obtain the optical remote sensing water index image after noise filtering; Step 8: Use a custom convolution kernel to perform directional filtering enhancement on the noise-filtered optical remote sensing water index image; Step nine, using a double-threshold Sobel edge detection method on the optical remote sensing water index image enhanced by directional filtering to obtain an edge detection result map; Step ten, superimposing the edge detection result map with the rainbow color map, repairing the discontinuous parts at the middle position of the continuous line of the edge detection result map, and removing the non-segment edges of the debris to form a complete dredging pipeline distribution display map.
[0007] Furthermore, in step 1, the source of the remote sensing image is MODIS, Landsat or Sentinel, GF series, and cloud-free satellite remote sensing images of the required time period are downloaded from the selected satellite data platform, and the remote sensing images are preprocessed in combination with the preset area range.
[0008] Furthermore, in step one, the remote sensing image source is Sentinel-2; in step two, the blue light band, green light band and near infrared band are extracted; or, in step one, the remote sensing image source is Landsat-8; in step two, the blue light band, green light band and short-wave infrared band are extracted.
[0009] Furthermore, in step 3, for the multi-temporal remote sensing images containing the green band and the near-infrared band, the NDWI index is calculated: NDWI = (R(GREEN)-R(NIR)) / (R(GREEN)+R(NIR)); For multi-temporal remote sensing images containing green bands and short-wave infrared bands, calculate the MNDWI index: MNDWI=(R(GREEN)- R(MIR)) / (R(GREEN)+ R(MIR)); In the above formula, NDWI is the NDWI index, MNDWI is the MNDWI index, R(GREEN)- is the green light band, R(NIR) is the near infrared band, and R(MIR) is the short-wave infrared band.
[0010] Furthermore, in step 4, when the NDWI index is displayed using a rainbow color map, the NDWI values are displayed using a rainbow color map from large to small.
[0011] Furthermore, in step 5, the maximum inter-class variance method is used to automatically determine the optimal threshold for segmenting the image into two classes. For each threshold t, the method calculates the inter-class variance; the inter-class variance is the weighted sum of the variances of the foreground and background, and the formula is:
[0012] in, is the between-class variance, and are the weights of foreground and background, respectively. and are the average gray values of the foreground and background respectively, so the threshold t that maximizes the inter-class variance is selected as the optimal threshold t1.
[0013] Furthermore, in step six, when the NDWI index is processed, when NDWI ≥ t1, it is a water body area, and the area when NDWI < t1 is processed, and the area when NDWI < t1 is assigned a value of 0, thereby clarifying the water body and non-water body areas.
[0014] Furthermore, in step 7, the output pixel value of the Lee filter Calculated by the following formula:
[0015] in, is the input pixel value, is the mean value in the local window, W is the weight factor, defined as:
[0016] in, is the variance within the local window, is the variance of the noise.
[0017] Furthermore, in step eight, the optimal kernel parameters are determined by adjusting the parameters of the filter, and the optimal kernel parameters include the convolution template and the gradient direction of the convolution kernel.
[0018] Furthermore, step nine includes: For the optical remote sensing water index image after directional filtering, the OTSU threshold t2 is calculated using the maximum inter-class variance method; The sobel operator is used to detect the edge of the optical remote sensing water index image after directional filtering enhancement. The horizontal convolution kernel as follows:
[0019] Vertical convolution kernel as follows:
[0020] By performing convolution operation on the optical remote sensing water index image after directional filtering, the gradient of each pixel in the horizontal and vertical directions is calculated, and the final edge intensity Calculated by the following formula: +
[0021] Based on the OTSU threshold t2, determine the low threshold T low and high threshold T high Since t2 has effectively segmented the foreground (high-value pixels) and background (low-value pixels) of the image, the weight k of the high threshold is 1, that is, the high threshold T high =k*t2=t2; At the same time, setting a low threshold lower than the high threshold can capture possible weak edges, which is obtained by multiplying the high threshold by the weight, T low =k*T high , the weight of the low threshold in the example is k=0.5; the high threshold is used to determine strong edges, and any pixel above this threshold is considered to be an edge; the low threshold is used to connect edges, and any pixel below the high threshold but above the low threshold is also considered to be an edge if it is connected to a strong edge, and other pixels are suppressed as non-edges; Output a binary image where the pixel value is 0 or 1, representing non-edge and edge respectively.
[0022] The present invention has the following beneficial effects: the present invention provides an intelligent identification method for underwater dredging pipelines based on multi-temporal remote sensing images, which can accurately locate underwater dredging pipelines through satellite remote sensing technology, and can significantly improve the efficiency of dredging management. Compared with traditional manual inspection methods, satellite remote sensing identification technology can efficiently and safely perform large-scale scanning and accurate positioning without interfering with the normal operation of the reservoir. By reducing the investment and application of manpower and equipment, the work efficiency and safety are greatly improved, the risk of personnel entering potential dangerous areas is reduced, the dredging work can be more accurate and efficient, and the interference with the reservoir ecosystem is minimized. Through accurate pipeline distribution data, decision makers are helped to more accurately determine the pipeline status, facilitate the formulation of more accurate dredging plans, make up for the lack of management data, and provide valuable data support for the long-term management of the reservoir. Through the innovative application of satellite remote sensing technology in the field of reservoir management, the efficiency and accuracy of reservoir management are significantly improved, and more scientific and sustainable reservoir management is achieved. BRIEF DESCRIPTION OF THE DRAWINGS
[0023] In order to more clearly illustrate the technical solution of the present invention, the drawings required for use in the embodiments are briefly introduced below. Obviously, for ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.
[0024] Figure 1 A flow chart of an underwater dredging pipeline intelligent identification method based on multi-temporal remote sensing images provided by the present invention; Figure 2 It is RGB remote sensing image; Figure 3 is the NDWI image after threshold processing; Figure 4 is the NDWI image after Lee filtering; Figure 5 It is the NDWI image enhanced by directional filtering; Figure 6 This is the double threshold sobel edge detection result image; Figure 7 It is the overlay of edge detection result and NDWI rainbow color map; Figure 8 This is the effect of superimposing the modified pipeline distribution map and the NDWI rainbow color map. DETAILED DESCRIPTION
[0025] In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be clearly and completely described in conjunction with the specific embodiments of the present invention and the corresponding drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention. The technical solutions provided by the embodiments of the present invention are described in detail below in conjunction with the drawings.
[0026] See also Figures 1 to 8 The embodiment of the present invention provides an intelligent identification method for underwater dredging pipelines based on multi-temporal remote sensing images, comprising: Step 1: Obtain multi-temporal remote sensing images of the same study area.
[0027] Specifically, the source of the remote sensing image can be a Moderate-resolution Imaging Spectroradiometer (MODIS), Landsat or Sentinel satellite, or GF series satellite. Cloud-free satellite remote sensing images of the required time period are downloaded from the selected satellite data platform, and conventional pre-processing such as cropping is performed on the remote sensing image in combination with a preset area range.
[0028] Step 2: extract the blue light band, green light band and near infrared band, or the blue light band, green light band and short-wave infrared band of the remote sensing images of each phase.
[0029] Specifically, the bands extracted mainly depend on the satellite source. If the remote sensing image source is Sentinel-2, the blue band, green band and near-infrared band are extracted; if the remote sensing image source is Landsat-8, the blue band, green band and short-wave infrared band are extracted. The bands extracted in this step are guaranteed to achieve a combination of blue, green and near-infrared or short-wave infrared bands.
[0030] Step three: calculate the optical remote sensing water index based on the extracted bands.
[0031] For multi-temporal remote sensing images containing green bands and near-infrared bands, calculate the NDWI index: NDWI = (R(GREEN)-R(NIR)) / (R(GREEN)+R(NIR)); For multi-temporal remote sensing images containing green bands and short-wave infrared bands, calculate the MNDWI index: MNDWI=(R(GREEN)- R(MIR)) / (R(GREEN)+ R(MIR)); In the above formula, NDWI is the NDWI index, MNDWI is the MNDWI index, R(GREEN)- is the green light band, R(NIR) is the near infrared band, and R(MIR) is the short-wave infrared band.
[0032] The idea of using the NDWI index in the identification of desilting pipelines comes from the following observation: the area of the desilting pipeline that is not covered by water has obvious spectral characteristics, while the area covered by water may still produce different reflection and scattering of incident light from the surrounding water due to the special material and shape. This difference may be reflected in the NDWI index, thus providing the possibility of developing remote sensing to identify desilting pipelines.
[0033] Step 4: Display the optical remote sensing water index using a rainbow color map.
[0034] Taking NDWI images as an example, when the NDWI index is displayed using a rainbow color map, it is displayed according to the NDWI values from large to small. The rainbow color map can better judge the subtle differences in local areas and the changes in different value segments. Different values are reflected by gradient colors, making it possible to observe the distribution of specific values in space.
[0035] Step five, use the maximum inter-class variance method to determine the optimal threshold for water body segmentation of optical remote sensing water index.
[0036] Specifically, the maximum inter-class variance method is used to automatically determine the optimal threshold for segmenting an image into two classes. For each possible threshold t, the inter-class variance is calculated; the inter-class variance is the weighted sum of the variances of the foreground and background, and the formula is:
[0037] in, is the between-class variance, and are the weights of foreground and background, respectively. and are the average gray values of the foreground and background respectively, so the threshold t that maximizes the inter-class variance is selected as the optimal threshold t1.
[0038] Step six, processing the optical remote sensing water index according to the optimal threshold value, clarifying the water body and non-water body areas, and obtaining a processed optical remote sensing water index image.
[0039] Specifically, the NDWI or MNDWI image can be segmented into water and non-water areas based on the determined optimal threshold t1. Taking NDWI as an example, when NDWI ≥ t1, it is a water area, and the area when NDWI < t1 is processed, and the area when NDWI < t1 is assigned a value of 0, so that the water and non-water areas can be clearly defined.
[0040] Step seven, use Lee filter to enhance the processed optical remote sensing water index image and remove noise to obtain the optical remote sensing water index image after noise filtering.
[0041] This method adjusts pixel values by local statistical characteristics, smoothing uniform areas while preserving image edges and details. A typical method for filtering using local statistical characteristics of an image, based on a weighted average of the input pixel value and the local mean. Specifically, the output pixel value of the Lee filter is Calculated by the following formula:
[0042] in, is the input pixel value, is the mean value in the local window, is the weight factor, defined as:
[0043] in, is the variance within the local window, is the variance of the noise.
[0044] In the example, the size of the Lee filter is determined to be a 3×3 matrix.
[0045] Step eight, use a custom convolution kernel to perform directional filtering enhancement on the optical remote sensing water index image after noise filtering.
[0046] Highlighting the edges and details with specific directional features in the image is helpful for identifying linear structures in the image. By adjusting the parameters of the filter, the optimal kernel parameters are determined, including the appropriate convolution template and the gradient direction of the convolution kernel. The specified direction can enhance the edges in the image that change along that direction.
[0047] In this example, the appropriate convolution kernel size is determined to be a 3×3 matrix, the direction of the convolution kernel is 30°, and the convolution kernel is:
[0048] Step nine, using the double threshold Sobel edge detection method on the optical remote sensing water index image enhanced by directional filtering to obtain an edge detection result map.
[0049] Specifically, the OTSU threshold t2 is calculated using the maximum inter-class variance method for the optical remote sensing water index image after directional filtering; The Sobel operator is used to detect the edge of the optical remote sensing water index image after directional filtering enhancement. The Sobel operator is a discrete differential operator, which is mainly used to calculate the gradient amplitude and direction of the image and highlight the area with intensity changes in the image through convolution operation. as follows:
[0050] Vertical convolution kernel as follows:
[0051] By performing convolution operation on the optical remote sensing water index image after directional filtering, the gradient of each pixel in the horizontal and vertical directions is calculated, and the final edge intensity Calculated by the following formula: +
[0052] Using the double threshold detection technology, the double threshold technology is applied to the result of Sobel edge detection to enhance the continuity and accuracy of the edge. Based on the OTSU threshold t2, the low threshold T is determined. low and high threshold T high Since t2 has effectively segmented the foreground (high-value pixels) and background (low-value pixels) of the image, the weight k of the high threshold is 1, that is, the high threshold T high =k*t2=t2; At the same time, setting a low threshold lower than the high threshold can capture possible weak edges, which is obtained by multiplying the high threshold by the weight, T low =k*T high , the weight of the low threshold in the example is k=0.5; the high threshold is used to determine strong edges, and any pixel above this threshold is considered to be an edge; the low threshold is used to connect edges, and any pixel below the high threshold but above the low threshold is also considered to be an edge if it is connected to a strong edge, and other pixels are suppressed as non-edges; Output a binary image where the pixel value is 0 or 1, representing non-edge and edge respectively.
[0053] In the example, the size of the determined Sobel convolution kernel is 3×3 pixels, and the weights of the low threshold and the high threshold are 0.6 and 1 respectively.
[0054] Step ten, superimposing the edge detection result map with the rainbow color map, repairing the discontinuous parts at the middle position of the continuous line of the edge detection result map, and removing the non-segment edges of the debris to form a complete dredging pipeline distribution display map.
[0055] The present invention uses the remote sensing optical water index, which can also be replaced by other indices that can reflect the difference in characteristics obtained by optical or SAR images, etc. The image denoising method can select other suitable denoising methods in combination with image features, not limited to the Lee method. The threshold can be determined using the OTSU threshold, or it can be determined in combination with artificial image features. The edge detection and recognition method can also introduce new image processing algorithms or machine learning models in combination with image characteristics.
[0056] It can be seen from the above embodiments that the present invention is an efficient non-invasive identification method, which uses remote sensing image processing algorithms to accurately identify dredging pipelines without the need for on-site manual surveys, greatly improving monitoring efficiency and coverage. This remote monitoring capability enables pipelines in large areas and difficult-to-reach areas to be monitored. The present invention uses image processing algorithms to automatically identify dredging pipelines, reduces the subjectivity and errors of manual interpretation, and improves the accuracy and consistency of identification. This automated processing method can quickly process a large amount of remote sensing data and significantly improve work efficiency. Compared with traditional field survey methods, the present invention greatly reduces manpower and time costs. By analyzing remote sensing images, the pipeline area that needs to be dredged can be quickly located, unnecessary field surveys can be reduced, and resource allocation can be optimized. By reducing the need for field surveys, the present method minimizes interference with the environment, especially in applications in ecologically sensitive areas. It has significant advantages. The method framework of the present invention has good scalability, and the accuracy and efficiency of identification can also be improved by introducing new image processing algorithms or machine learning models.
[0057] The above-described embodiments of the present invention do not limit the protection scope of the present invention.
Claims
1. An intelligent identification method for underwater dredging pipelines based on multi-temporal remote sensing images, characterized in that: include: Step 1: Obtain multi-temporal remote sensing images of the same study area; Step 2: extracting the blue light band, green light band and near infrared band, or the blue light band, green light band and short-wave infrared band of the remote sensing images of each time phase; Step 3, calculating the optical remote sensing water index based on the extracted bands; Step 4, displaying the optical remote sensing water index using a rainbow color map; Step 5, the maximum inter-class variance method is used to determine the optimal threshold for water body segmentation of the optical remote sensing water index; Step six, processing the optical remote sensing water index according to the optimal threshold value, clarifying the water body and non-water body areas, and obtaining a processed optical remote sensing water index image; Step 7, using Lee filter to enhance the processed optical remote sensing water index image and remove noise to obtain the optical remote sensing water index image after noise filtering; Step 8: Use a custom convolution kernel to perform directional filtering enhancement on the noise-filtered optical remote sensing water index image; Step nine, using a double-threshold Sobel edge detection method on the optical remote sensing water index image enhanced by directional filtering to obtain an edge detection result map; Step ten, superimposing the edge detection result map with the rainbow color map, repairing the discontinuous parts at the middle position of the continuous line of the edge detection result map, and removing the non-segment edges of the debris to form a complete dredging pipeline distribution display map.
2. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images according to claim 1, characterized in that: In step 1, the source of the remote sensing image is MODIS, Landsat or Sentinel, GF series, and cloud-free satellite remote sensing images of the required time period are downloaded from the selected satellite data platform, and the remote sensing images are preprocessed in combination with the preset area range.
3. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images as claimed in claim 2, characterized in that: In step one, the remote sensing image source is Sentinel-2; in step two, the blue light band, green light band and near infrared band are extracted; or, in step one, the remote sensing image source is Landsat-8; in step two, the blue light band, green light band and short-wave infrared band are extracted.
4. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images as claimed in claim 3, characterized in that: In step 3, for multi-temporal remote sensing images containing green light bands and near-infrared bands, the NDWI index is calculated: NDWI = (R(GREEN)-R(NIR)) / (R(GREEN)+R(NIR)); For multi-temporal remote sensing images containing green bands and short-wave infrared bands, calculate the MNDWI index: MNDWI=(R(GREEN)- R(MIR)) / (R(GREEN)+ R(MIR)); In the above formula, NDWI is the NDWI index, MNDWI is the MNDWI index, R(GREEN)- is the green light band, R(NIR) is the near infrared band, and R(MIR) is the shortwave infrared band.
5. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images as claimed in claim 4, characterized in that: In step 4, when the NDWI index is displayed using a rainbow color map, the NDWI values are displayed from large to small using a rainbow color map.
6. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images as claimed in claim 5, characterized in that: In step 5, the maximum inter-class variance method is used to automatically determine the optimal threshold for segmenting the image into two categories. For each threshold t , calculate the inter-class variance; the inter-class variance is the weighted sum of the variances of the foreground and background, and the formula is: in, is the between-class variance, and are the weights of foreground and background, respectively. and are the average grayscale values of the foreground and background, respectively, so as to select the threshold that maximizes the inter-class variance t As the optimal threshold t1.
7. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images as claimed in claim 6, characterized in that: In step six, when the NDWI index is processed, when NDWI ≥ t1, it is a water body area, and the area when NDWI < t1 is processed, and the area when NDWI < t1 is assigned a value of 0, thereby clarifying the water body and non-water body areas.
8. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images as claimed in claim 7, characterized in that: In step 7, the output pixel value of the Lee filter Calculated by the following formula: in, is the input pixel value, is the mean value in the local window, W is the weight factor, defined as: in, is the variance within the local window, is the variance of the noise.
9. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images as claimed in claim 8, characterized in that: In step eight, the optimal kernel parameters are determined by adjusting the parameters of the filter, and the optimal kernel parameters include the convolution template and the gradient direction of the convolution kernel.
10. The method for intelligent identification of underwater dredging pipelines based on multi-temporal remote sensing images according to claim 9, characterized in that: Step nine includes: For the optical remote sensing water index image after directional filtering, the maximum inter-class variance method is also used to calculate the OTSU threshold t2; The sobel operator is used to detect the edge of the optical remote sensing water index image after directional filtering enhancement. The horizontal convolution kernel as follows: Vertical convolution kernel as follows: By performing convolution operation on the optical remote sensing water index image after directional filtering, the gradient of each pixel in the horizontal and vertical directions is calculated, and the final edge intensity Calculated by the following formula: + Based on the OTSU threshold t2, determine the low threshold T low and high threshold T high The weight k, high threshold T high =k*t2=t2; at the same time, set a low threshold lower than the high threshold, and obtain the low threshold by multiplying the high threshold by the weight, T low =k*T high ; Among them, the high threshold is used to determine the strong edge, and any pixel above the threshold is considered to be an edge; the low threshold is used to connect the edge, and any pixel below the high threshold but above the low threshold is also considered to be an edge if it is connected to the strong edge, and other pixels are suppressed as non-edges; Output a binary image where the pixel value is 0 or 1, representing non-edge and edge respectively.
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