# A New Method for Automatic Extraction and Analysis of Discontinuities Based on TIN on Rock Mass Surfaces

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methodology

- (1)
- Data preprocessing. First, remove noise points and outliers from the point clouds of the rock mass, then resample the point clouds and use the Delaunay algorithm to generate a TIN of the rock mass surface. Compared to the regular grid model, TIN has the advantages of reducing data redundancy, better performance of variation characteristics, and easy calculation [9,37].
- (2)
- Discontinuity set recognition. Firstly, calculate the normal vector and centroid of each triangle of the TIN. Secondly, use the DPCA to identify the main potential directions of the discontinuity set. Next, use the K-means algorithm to cluster the discontinuity set. Finally, combine the silhouette coefficient to determine the optimum clustering result. The clustering results can be expressed as Group 1, Group 2, … Group k.
- (3)
- Discontinuity set segmentation. Use the HDBSCAN algorithm to segment the discontinuity set after clustering and identify each discontinuity. Suppose each discontinuity set has m, n, …, p discontinuities, respectively.
- (4)
- Discontinuities fitting. Use the RANSAC method to fit the discontinuities and to obtain its parameters.

#### 2.1. Test Data Set Description

#### 2.2. Data Preprocessing

#### 2.3. Discontinuity Set Recognition

#### 2.3.1. Normal Vector and Centroid Computation

_{i}are P

_{1}(X

_{1}, Y

_{1}, Z

_{1}), P

_{2}(X

_{2}, Y

_{2}, Z

_{2}), and P

_{3}(X

_{3}, Y

_{3}, Z

_{3}), arranged counterclockwise. n = (A, B, C) is the normal vector and C

_{k}is the centroid of the triangle f

_{i}. Calculate the vector between P

_{1}and P

_{2}, and P

_{1}and P

_{3}, which are expressed as

**P**= (X

_{1}P_{2}_{2}− X

_{1}, Y

_{2}− Y

_{1}, Z

_{2}− Z

_{1}) and

**P**= (X

_{1}P_{3}_{3}− X

_{1}, Y

_{3}−Y

_{1}, Z

_{3}− Z

_{1}). The normal vector n = (A, B, C) can be expressed as a cross product of

**P**and

_{1}P_{2}**P**, then A, B, and C can be calculated by Formula (1):

_{1}P_{3}_{2}− Y

_{1})(Z

_{3}− Z

_{1}) − (Z

_{2}− Z

_{1})(Y

_{3}− Y

_{1})

B = (Z

_{2}− Z

_{1})(X

_{3}− X

_{1}) − (X

_{2}− X

_{1})(Z

_{3}− Z

_{1})

C = (X

_{2}− X

_{1})(Y

_{3}− Y

_{1}) − (Y

_{2}− Y

_{1})(X

_{3}− X

_{1})

_{k}can be calculated by Formula (2):

#### 2.3.2. Determination of the Main Direction of Discontinuity Set

_{i}and d

_{j}corresponding to the sample point i and j is defined as Formula (3):

_{i}of the normal vector of the sample point i by Formulas (4) and (5):

_{ij}is the distance between the normal vectors of the sample point i and j, and d

_{cf}is the cutoff distance. ld

_{i}represents the number of sample points with distances to the sample point i less than d

_{cf}. d refers to the difference between d

_{ij}and d

_{cf}; if it is less than 0, the local density of the sample point is increased by 1, whereas if it is greater than 0, the local density of the sample point remains unchanged.

_{i}between sample point i and the other sample points with a local density greater than ld

_{i}using Formula (6):

_{cf}. If d

_{cf}is too large, it will lead to the same ld between the dense sample points. If it is too small, it will lead to the same ld between the sparse sample points. Thus, d

_{cf}is set to 0.05. Figure 4b shows the ld and md corresponding to all the sample points in the rock slope. The main potential directions are located in the discrete area in the decision graph, and the points in the red circle indicate the potential main directions that can be selected.

#### 2.3.3. Determination of the Optimum Number of Discontinuity Set

_{i}is a sample in one of the clusters. Define a(x

_{i}) as the average distance between x

_{i}and all the other samples in the same cluster, and b(x

_{i}) as the minimum average distance between x

_{i}and samples in other clusters. The silhouette coefficient of x

_{i}is defined by Formula (7):

_{i}) is close to 1, it means that x

_{i}is correctly assigned to the appropriate cluster; if the value of S(x

_{i}) is close to 0, it means that x

_{i}can be assigned to this cluster or other clusters, because the average distance from x

_{i}to the other clusters is equal; if the value of S(x

_{i}) is close to −1, it means that xi is misclassified. Calculate the silhouette coefficients of all the sample points and their average value, which is called the average silhouette coefficient, to represent the effectiveness of the current clustering result. In this paper, k is set from 2 to 6. Calculate the average silhouette coefficient corresponding to each k value and use the k value with the larger average silhouette coefficient as the final clustering number of the discontinuity set. It can be seen from Figure 5, the number of clusters corresponding to the maximum average silhouette coefficient for the rock slope is 5, which means that the optimal number of clusters for the rock slope is 5.

#### 2.4. Discontinuity Set Segmentation

#### 2.5. Discontinuities Fitting

#### 2.6. Clustering Results for the Rock Slope

## 3. Workflow Application to an Artificial Quarry Slope

#### 3.1. Clustering Results of the Artificial Quarry Slope

#### 3.2. The Influence of the Triangle Mesh Size

#### 3.3. The Impact of HDBSCAN Algorithm Parameter Min-pts

## 4. Discussion

#### 4.1. Comparison of the Extracting Results of the Rock Slope

#### 4.2. Analysis of the Optimal Triangle Mesh Size

#### 4.3. Relevant Parameters of Proposed New Method

_{cf}and the angle α between the main directions when identifying the main potential directions by DPCA; the only parameter Min-pts when segmenting the discontinuity sets by HDBSCAN; and the minimum number of triangles Min-size in a discontinuity.

_{cf}and the angle α between the main directions need to be set. The cutoff distance d

_{cf}will affect the local density of the sample point. If d

_{cf}is too large, it will reduce the difference of local density and may not be able to effectively identify the main direction. If the d

_{cf}is too small, it will cause the main directions with approach directions to increase. Gao et al. [45] conducted a detailed sensitivity analysis of d

_{cf}and proposed that the lower limit and upper limit of d

_{cf}be set to 0.005 and 0.12, respectively. Generally, if d

_{cf}is greater than 0.12, it will cause the angle between two discontinuities normal vectors to be about 20°. If d

_{cf}is less than 0.005, the angle between two discontinuities normal vectors is about 4°. Therefore, based on the sensitivity analysis of d

_{cf}by Gao et al. [45], this paper sets d

_{cf}to 0.05. The angle α between the main directions will affect the number of discontinuity sets. Due to the normal vector projected into the 3D network, the maximum angle between the normal vectors is 90°, and generally, the rock mass discontinuities can be divided into six clusters at most. If α is greater than 15°, the number of discontinuity sets will be less than six, which is unreasonable for a rock mass with more discontinuity sets.

#### 4.4. Discussion on Fitting Plane by RANSAC

#### 4.5. Discussion on the Applicability of the New Method

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**(

**a**) Isometric 3D network with the projection of the normal vectors of all the sample points; (

**b**) the decision graph of the main potential orientations, which is drawn by the ld and md corresponding to all sample points in the rock slope.

**Figure 8.**Cluster identification in a section of a rock slope. (

**a**) One color per discontinuity set with all clusters labelled. (

**b**) J

_{1}. (

**c**) J

_{2}and J

_{5}. (

**d**) J

_{3}and J

_{4}.

**Figure 9.**(

**a**) Discontinuities marked 13 and 41; enlarged views of discontinuities 13 (

**b**) and 41 (

**c**).

**Figure 12.**Average silhouette coefficients for different numbers of clusters for the artificial quarry slope.

**Figure 13.**Discontinuity sets represented by different colors (J

_{1}—green, J

_{2}—cyan, and J

_{3}—red).

**Figure 15.**Segmenting result of Min-pts = (

**a**) 2; (

**b**) 4; (

**c**) 6; (

**d**) 8; (

**e**) 10; (

**f**) 12; (

**g**) 16; (

**h**) 20; and (

**i**) artificial quarry slope image.

**Figure 18.**Deviation between the discontinuities in each discontinuity set and the main orientation.

Set | Dip Direction/Dip Angle (°) by New Method | Number of Clusters | Dip Direction/Dip Angle (°) by Riquelme et al. [40] | Number of Clusters | Δ|DD| (°) | Δ|DA| (°) |
---|---|---|---|---|---|---|

J_{1} | 248.17/34.74 | 50 | 249.04/36.66 | 59 | 0.87 | 1.92 |

J_{2} | 172.27/82.22 | 14 | 172.29/83.16 | 14 | 0.02 | 0.94 |

J_{3} | 134.38/81.59 | 66 | 137.33/77.87 | 56 | 2.95 | 3.72 |

J_{4} | 93.67/50.82 | 45 | 092.96/48.74 | 34 | 0.71 | 2.08 |

J_{5} | 286.22/65.56 | 55 | 288.45/68.22 | 57 | 2.23 | 2.66 |

Discontinuity | Discontinuity Orientations by | Riquelme et al. in 2014 (°) | Chen et al. in 2016 (°) | New Method (°) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Classical Approach (°) | Riquelme et al. in 2014 (°) | Chen et al. in 2016 (°) | New Method (°) | Δ|DD| | Δ|DA| | Δ|DD| | Δ|DA| | Δ|DD| | Δ|DA| | |

11 | 249.18/40.23 | 246.24/39.02 | 244.62/38.38 | 246.22/38.99 | 2.94 | 1.21 | 4.56 | 1.85 | 2.96 | 1.24 |

12 | 264.23/57.02 | 256.86/52.3 | 256.18/52.16 | 266.09/54.72 | 7.37 | 4.72 | 8.05 | 4.86 | 1.86 | 2.30 |

13 | 263.97/41.91 | 70.26/35.8 | 251.04/36.17 | 250.86/36.09 | 13.71 | 6.11 | 12.93 | 5.74 | 13.11 | 5.82 |

14 | 252.58/36.53 | 252.68/35.48 | 251.44/33.85 | 252.24/37.00 | 0.10 | 1.05 | 1.14 | 2.68 | 0.34 | 0.44 |

15 | 248.71/36.98 | 249.74/35.91 | 250.82/36.83 | 250.43/35.85 | 1.03 | 1.07 | 2.11 | 0.15 | 1.73 | 1.13 |

16 | 254.77/29.86 | 70.47/35.91 | 250.46/35.86 | 250.43/35.85 | 4.30 | 6.05 | 4.31 | 6.00 | 4.34 | 5.99 |

17 | 249.85/35.94 | 255.12/32.82 | 253.19/33.46 | 254.90/32.60 | 5.27 | 3.12 | 3.34 | 2.48 | 5.05 | 3.34 |

21 | 338.68/82.35 | 339.47/83.25 | 157.55/83.81 | 338.63/82.20 | 0.79 | 0.90 | 1.13 | 1.46 | 0.05 | 0.15 |

22 | 347.47/79.01 | 166.33/76.58 | 166.31/78.73 | 348.76/80.77 | 1.14 | 2.43 | 1.16 | 0.28 | 1.29 | 1.76 |

23 | 341.04/89.5 | 160.2/89.86 | 157.52/86.88 | 159.98/88.48 | 0.84 | 0.36 | 3.52 | 2.62 | 1.06 | 1.02 |

24 | 353.5/76.4 | 173.55/76.85 | 353.07/77.82 | 172.64/77.88 | 0.05 | 0.45 | 0.43 | 1.42 | 0.86 | 1.48 |

31 | 314.1/77.18 | 136.59/82.58 | 314.73/80.04 | 136.43/86.25 | 2.49 | 5.40 | 0.63 | 2.86 | 2.24 | 9.07 |

32 | 302.36/75.92 | 131.225/82.67 | 136.52/89.85 | 124.76/79.25 | 8.87 | 6.75 | 14.16 | 13.93 | 2.40 | 3.33 |

33 | 330.19/83.01 | 143.91/89.7 | 145.62/89.85 | 326.47/89.77 | 6.28 | 6.69 | 4.57 | 6.85 | 3.72 | 6.76 |

41 | 286.12/58.91 | 97.55/63.22 | 285.98/59.84 | 98.10/62.34 | 8.57 | 4.31 | 0.14 | 0.93 | 8.02 | 3.43 |

42 | 274.18/51.09 | 91.07/50.19 | 272.57/47.64 | 91.09/50.54 | 3.11 | 0.90 | 1.61 | 3.45 | 3.09 | 0.55 |

43 | 277.22/46.42 | 96.64/47.97 | 277.31/49.31 | 97.24/47.27 | 0.58 | 1.55 | 0.09 | 2.89 | 0.02 | 0.85 |

51 | 305.04/77.62 | 123.42/76.15 | 305.04/77.62 | 304.07/79.79 | 1.62 | 1.47 | 16.25 | 4.41 | 0.97 | 2.17 |

52 | 290.16/66.99 | 105.75/69.94 | 109.29/76.61 | 284.94/69.56 | 4.41 | 2.95 | 0.87 | 9.62 | 5.22 | 2.57 |

Maximum deviation | 13.71 | 6.75 | 16.25 | 13.93 | 13.11 | 9.07 | ||||

Average deviation | 3.87 | 3.03 | 4.26 | 3.92 | 3.07 | 2.81 |

Set | Dip Direction/Dip Angle (°) by New Method | Number of Clusters | Dip Direction/Dip Angle (°) by Manual Method | Number of Clusters | Δ|DD| (°) | Δ|DA| (°) |
---|---|---|---|---|---|---|

J_{1} | 163.64/61.77 | 150 | 160.22/64.58 | 60 | 3.42 | 2.81 |

J_{2} | 198.56/54.61 | 188 | 189.84/44.63 | 81 | 8.72 | 9.98 |

J_{3} | 225.73/66.11 | 172 | 221.93/67.76 | 64 | 3.83 | 1.65 |

Average deviation | 5.32 | 4.81 |

Triangle Mesh Size (cm) | Number of Discontinuities | Time (h) | Precision (°) | |
---|---|---|---|---|

Δ|DD| | Δ|DA| | |||

3 | 736 | 3.25 | 4.97 | 4.53 |

5 | 510 | 1.39 | 5.32 | 4.81 |

7 | 356 | 0.36 | 6.04 | 5.73 |

Discontinuity | Dip Direction/Dip Angle Measured by | RANSAC (°) | The Whole Points (°) | ||
---|---|---|---|---|---|

Classical Approach (°) | RANSAC (°) | The Whole Points (°) | Δ|DD|/Δ|DA| | Δ|DD|/Δ|DA| | |

13 | 263.97/41.91 | 250.86/36.09 | 248.73/34.61 | 13.11/5.82 | 15.24/7.30 |

41 | 286.12/58.91 | 98.10/63.34 | 96.87/64.46 | 8.02/4.43 | 9.25/5.55 |

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## Share and Cite

**MDPI and ACS Style**

Wu, X.; Wang, F.; Wang, M.; Zhang, X.; Wang, Q.; Zhang, S. A New Method for Automatic Extraction and Analysis of Discontinuities Based on TIN on Rock Mass Surfaces. *Remote Sens.* **2021**, *13*, 2894.
https://doi.org/10.3390/rs13152894

**AMA Style**

Wu X, Wang F, Wang M, Zhang X, Wang Q, Zhang S. A New Method for Automatic Extraction and Analysis of Discontinuities Based on TIN on Rock Mass Surfaces. *Remote Sensing*. 2021; 13(15):2894.
https://doi.org/10.3390/rs13152894

**Chicago/Turabian Style**

Wu, Xiang, Fengyan Wang, Mingchang Wang, Xuqing Zhang, Qing Wang, and Shuo Zhang. 2021. "A New Method for Automatic Extraction and Analysis of Discontinuities Based on TIN on Rock Mass Surfaces" *Remote Sensing* 13, no. 15: 2894.
https://doi.org/10.3390/rs13152894