# Surface Reconstruction for Three-Dimensional Rockfall Volumetric Analysis

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Surface Reconstruction Background

#### 1.2. Rockfall Objects

#### 1.2.1. Houdini—Synthetic Rockfall Object

^{3}. The volume was calculated by summing the signed volumes of tetrahedra with a common apex based at individual surface triangles. The surface mesh contains 642,573 faces, 632,010 vertices and is watertight manifold. The vertices were subsampled within the open-source software package CloudCompare [17] to minimum point spacings of 1, 2, 5 and 10 cm (Figure 3d).

#### 1.2.2. Natural Rockfalls—White Canyon Rockfall Cases

## 2. Methods

#### 2.1. Surface Reconstruction Background

#### 2.1.1. Convex Hull

#### 2.1.2. Alpha-shapes

#### 2.1.3. Power Crust

#### 2.2. Point to Mesh Comparisons

- H is within the interior of the triangle ABC then the distance is:$$d\left(P,ABC\right)=\left|\right|\overrightarrow{HP}\left|\right|$$
- H is within the interior of the triangle ABC then the distance is:$$d\left(P,ABC\right)=\left|\right|\overrightarrow{KP}\left|\right|$$where, the point K is defined as the point on triangle ABC which minimizes the distance to H. If the triangle ABC has an associated normal vector, as an additional product of the calculation, the calculated distance can be signed (e.g., loss or gain). The normal vector of the triangle provides information if the point is within the interior or exterior of the surface represented locally by the triangle.

## 3. Results

#### 3.1. Houdini—Synthetic Rockfall Object

#### 3.2. Natural Rockfalls—White Canyon Rockfall Cases

## 4. Discussion and Conclusions

- The Convex Hull approach in all cases over-interpolates the input point cloud. The method will generate the minimum smallest polygon that encapsulates all points. Therefore, as demonstrated in all rockfall cases within this study, concave features on all objects are not captured. The volume, using the Convex Hull method, in almost all cases will be an over-estimate.
- The Alpha-shape approach is sensitive to point density, especially when it is non-uniform. As a result, it is difficult and sometimes impossible to choose an Alpha-radius to balance hole-filling against loss of detail.
- The iterative Alpha-shape approach tries to find a balance between hole-filling and loss of detail. It is a robust approach that can be integrated into automated TLS processing workflows. In all cases, analyzed in this study the iterative Alpha-shape over-interpolated the surface in comparison to the input point cloud.
- Power Crust appears to the optimal surface reconstruction algorithm that was tested in the present study. The approach theoretically guarantees that the output surface will be watertight and manifold. The staggering amount of faces in the output surface mesh may present problems for some applications however simplification approaches exist.
- Each approach has its drawbacks, however, it is critical to ensure that the mesh is watertight and manifold for accurate volumetric calculations. The volume of Rockfall 3 in this study rose from approximately 57 m
^{3}(e.g., default Alpha-shape) to around 125 m^{3}(e.g., iterative Alpha-shape & Power Crust) once the object was watertight (e.g., holes were filled). That represents a gain in 68 m^{3}of volume attributed to the rockfall event. This case is an excellent example to show the implications of method selection on frequency-magnitude analyses and using sequential TLS datasets to help design mitigation measures. As more monitoring approaches become automated, the importance of having quality control on the inputs to databases becomes paramount.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Manifold definitions: (

**a**) A 3D triangular mesh of a sphere with 512 triangles; (

**b**) Manifold edge; (

**c**) Non-manifold edge; (

**d**) Non-manifold node; (

**e**) Closed fan; (

**f**) Open fan.

**Figure 2.**Common examples of different forms of point cloud artifacts, shown here in the case of a curve in 2D (Modified after [14]).

**Figure 3.**Generation of the synthetic rockfall object: (

**a**) Procedural geometry generation; (

**b**) Segmented rockfall geometry from a synthetic failure plane; (

**c)**Surface mesh of the rockfall object; (

**d**) Front view of the 3D mesh and resulting subsampled point clouds ranging from 1 cm point spacing to 10 cm point spacing.

**Figure 4.**Overview of the three natural rockfall events used in the study: (

**a**) Rockfall 1: occurred between 2015-10-23 and 2016-02-15; (

**b**) Rockfall 2: occurred between 2015-06-09 and 2015-08-24; (

**c**) Rockfall 3: occurred between 2016-02-15 and 2016-05-01. The black points correspond to the TLS points for the front and back (left to right) of each of the rockfall objects. The colored points correspond to the number of neighbors within a 50 cm diameter sphere. High number of neighbors correspond to the warmer colors while low number of neighbors correspond to the cooler colors.

**Figure 5.**Overview of a Convex Hull in 2D: (

**a**) input points; (

**b**) Convex Hull of the input point cloud.

**Figure 6.**Illustrative example of Alpha-shapes in 2D: (

**a**) The original shape; (

**b**) A Convex Hull fitted to the data points, representing the case when the Alpha-radius is equal to infinity; (

**c**–

**e**) Refined Alpha-shapes fitted to the data points as the Alpha-radius is decreased; (

**f**) Point at which the Alpha-radius can pass internally thought the data points and the Alpha-shape breaks down into smaller shapes.

**Figure 7.**Overview of the Power Crust approach depicted in 2D: (

**a**) A solid object with its medial axis depicted in dark blue; one maximal interior ball is shown (green); (

**b**) The Voronoi diagram of a input point sample from the object surface, with the Voronoi ball surrounding one pole shown; (

**c**) The inner and outer polar balls; (

**d**) The labeled inner and outer power diagram cells of the poles; (

**e**) The Power Crust and the power shape of its interior solid. (Adapted from [21]).

**Figure 8.**Example of Power Crust implementation: (

**a**) Input knot point cloud; (

**b**) Power Crust polygonal surface mesh; (

**c**) Refined triangulated Power Crust surface mesh.

**Figure 10.**Surface reconstruction for the synthetic rockfall object: (

**a**) Triangulated surface meshes for each surface reconstruction algorithm; (

**b**) Cross sections through the triangulated surface meshes. Note that the default Alpha-shape produces a reconstruction with boundary facets within the object.

**Figure 11.**Distance computations for Rockfall 1 using the four reconstruction methods and displaying the four sides of the rockfall objects after 90-degree clockwise rotation.

**Figure 12.**Distance computations for Rockfall 2 using the four reconstruction methods and displaying the four sides of the rockfall objects after 90-degree clockwise rotation.

**Figure 13.**Distance computations for Rockfall 3 using the four reconstruction methods and displaying the four sides of the rockfall objects after 90-degree clockwise rotation.

**Figure 14.**Histograms displaying deviation of the mesh from the input point cloud for each of the surface reconstruction methods for Rockfall 1: (

**a**) Regular Alpha-shape. (

**b**) Iterative Alpha-shape. (

**c**) Power Crust. (

**d**) Convex Hull.

**Figure 15.**Histograms displaying deviation of the mesh from the input point cloud for each of the surface reconstruction methods for Rockfall 2: (

**a**) Regular Alpha-shape. (

**b**) Iterative Alpha-shape. (

**c**) Power Crust. (

**d**) Convex Hull.

**Figure 16.**Histograms displaying deviation of the mesh from the input point cloud for each of the surface reconstruction methods for Rockfall 3: (

**a**) Regular Alpha-shape. (

**b**) Iterative Alpha-shape. (

**c**) Power Crust. (

**d**) Convex Hull.

**Table 1.**Comparison of the surface reconstruction results for the four different subsampled point clouds of the synthetic rockfall object.

Point Spacing | Surface Reconstruction Method | Faces | Vertices | Volume (m^{3}) | Volumetric Error (%) | Alpha-Radius (m) |
---|---|---|---|---|---|---|

1-cm | Default Alpha-shape | 43,110 | 20,888 | 1.69 | −63.90 | 0.3841 |

Iterative Alpha-shape | 21,046 | 10,525 | 5.18 | 10.65 | 0.62619 | |

Power Crust | 954,648 | 464,855 | 4.68 | −0.02 | N/A | |

Convex Hull | 1182 | 593 | 5.87 | 25.39 | Inf. | |

2-cm | Default Alpha-shape | 16,782 | 8063 | 2.08 | −55.63 | 0.4307 |

Iterative Alpha-shape | 10,500 | 5244 | 4.54 | −2.93 | 0.56824 | |

Power Crust | 338,322 | 166,774 | 4.68 | −0.01 | N/A | |

Convex Hull | 782 | 393 | 5.83 | 24.57 | Inf. | |

5-cm | Default Alpha-shape | 5940 | 2866 | 2.86 | −38.82 | 0.493 |

Iterative Alpha-shape | 4392 | 2198 | 5.06 | 8.09 | 0.59193 | |

Power Crust | 71,822 | 35,646 | 4.68 | −0.09 | N/A | |

Convex Hull | 514 | 259 | 5.74 | 22.69 | Inf. | |

10-cm | Default Alpha-shape | 2354 | 1157 | 3.62 | −22.73 | 0.5377 |

Iterative Alpha-shape | 2070 | 1035 | 4.75 | 1.40 | 0.58245 | |

Power Crust | 18,644 | 9261 | 4.66 | −0.40 | N/A | |

Convex Hull | 364 | 184 | 5.62 | 20.00 | Inf. |

Surface Reconstruction Method | Faces | Vertices | Holes | Volume (m^{3}) |
---|---|---|---|---|

Default Alpha-shape | 6746 | 2937 | 30 | 0.25 |

Iterative Alpha-shape | 2922 | 1464 | 0 | 1.34 |

Power Crust | 37,412 | 18,597 | 0 | 1.45 |

Convex Hull | 310 | 157 | 0 | 1.82 |

Surface Reconstruction Method | Faces | Vertices | Holes | Volume (m^{3}) |
---|---|---|---|---|

Default Alpha-shape | 2848 | 1337 | 11 | 0.94 |

Iterative Alpha-shape | 1668 | 836 | 0 | 1.63 |

Power Crust | 28,704 | 14,076 | 0 | 1.39 |

Convex Hull | 222 | 113 | 0 | 2.44 |

Surface Reconstruction Method | Faces | Vertices | Holes | Volume (m^{3}) |
---|---|---|---|---|

Default Alpha-shape | 21,640 | 10,492 | 46 | 57.20 |

Iterative Alpha-shape | 13,368 | 6688 | 0 | 129.63 |

Power Crust | 183,044 | 90,359 | 0 | 123.81 |

Convex Hull | 226 | 115 | 0 | 211.77 |

**Table 5.**Descriptive statistics for the distance computations for each surface reconstruction method for Rockfall 3.

Surface Reconstruction Method | Mean (m) | Standard Deviation (m) | Variance (m) | Skewness |
---|---|---|---|---|

Default Alpha-shape | 0.0017 | 0.0055 | 3.02 * 10^{−5} | 4.22 |

Iterative Alpha-shape | 0.011 | 0.015 | 2.12 * 10^{−4} | 2.36 |

Power Crust | 4.8 * 10^{−5} | 0.0047 | 2.22 * 10^{−5} | 0.10 |

Convex Hull | 0.038 | 0.029 | 8.62 * 10^{−4} | 1.39 |

**Table 6.**Descriptive statistics for the distance computations for each surface reconstruction method for Rockfall 2.

Surface Reconstruction Method | Mean (m) | Standard Deviation (m) | Variance (m) | Skewness |
---|---|---|---|---|

Default Alpha-shape | 0.013 | 0.023 | 5.3 * 10^{−4} | 2.32 |

Iterative Alpha-shape | 0.025 | 0.036 | 0.0013 | 2.12 |

Power Crust | −5.1 * 10^{−5} | 0.0060 | 3.6 * 10^{−5} | −2.35 |

Convex Hull | 0.078 | 0.072 | 0.0052 | 1.17 |

**Table 7.**Descriptive statistics for the distance computations for each surface reconstruction method for Rockfall 3.

Surface Reconstruction Method | Mean (m) | Standard Deviation (m) | Variance (m) | Skewness |
---|---|---|---|---|

Default Alpha-shape | 0.024 | 0.048 | 0.0023 | 2.97 |

Iterative Alpha-shape | 0.051 | 0.088 | 0.0078 | 2.74 |

Power Crust | 0.0062 | 0.040 | 0.0016 | 6.0056 |

Convex Hull | 0.35 | 0.19 | 0.036 | 0.52 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bonneau, D.; DiFrancesco, P.-M.; Hutchinson, D.J.
Surface Reconstruction for Three-Dimensional Rockfall Volumetric Analysis. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 548.
https://doi.org/10.3390/ijgi8120548

**AMA Style**

Bonneau D, DiFrancesco P-M, Hutchinson DJ.
Surface Reconstruction for Three-Dimensional Rockfall Volumetric Analysis. *ISPRS International Journal of Geo-Information*. 2019; 8(12):548.
https://doi.org/10.3390/ijgi8120548

**Chicago/Turabian Style**

Bonneau, David, Paul-Mark DiFrancesco, and D. Jean Hutchinson.
2019. "Surface Reconstruction for Three-Dimensional Rockfall Volumetric Analysis" *ISPRS International Journal of Geo-Information* 8, no. 12: 548.
https://doi.org/10.3390/ijgi8120548