# Computational Geometry-Based Surface Reconstruction for Volume Estimation: A Case Study on Magnitude-Frequency Relations for a LiDAR-Derived Rockfall Inventory

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Rockfall Hazard

#### 1.2. Rockfall Magnitude-Frequency Relations in the Digital Age

**Table 1.**Summary of several recent rockfall extraction methodologies, ordered by the most recent publications. Some methods additionally conduct post-database filtering by the point cloud volumes, number of points, or the change detection signature. Site dimensions preceded with a tilde indicate approximate measurements made when the dimensions were not explicitly stated in the text. Data acquisition frequencies were taken as median, or as fact from the texts.

Study | Timespan [Years] | Median Frequency | Site Dimensions | Change Detection Method | Clutter Removal | Clustering | Volume |
---|---|---|---|---|---|---|---|

Guerin et al. [34] | 11 | 365 days | ~160 × 600 m | C2M ^{1} | Neighbour averaging | DBSCAN | SfR ^{2}: Manually w/3D software |

Hartmeyer et al. [35] | 6 | yearly | 5-rockwalls: 234,700 m^{2} | M3C2 ^{3} | - | Region growing | Sum of raster cells |

DiFrancesco et al. [36] | 5.0 | 69 days | 240 × 120 m | M3C2 | Manual | DBSCAN ^{4} | SfR: Alpha Solid |

Benjamin et al. [26] | 2.6 | 309 days | 20.5 km × (30–150 m) | M3C2 | Manual | DBSCAN | SfR: Power Crust |

Guerin et al. [37] | 40 | 34 years–1 month | ~0.8 × 0.6 km; ~1.5 × 0.8 km; ~1.8 × 0.8 km; | C2M | Neighbour averaging | DBSCAN | SfR: Manually w/3D software |

Williams et al. [18,38] | 0.8 | 1 h–30 days | 210 × 60 m | M3C2 (variable search lengths) | Waveform + edge filter; Noise mask | Region growing | Sum of raster cells |

Van Veen et al. [39] | 1.3 | 76 days | 1.1 km × 500 m | M3C2 (nearest neighbors) | - | DBSCAN | SfR: Alpha Shapes * |

Olsen et al. [40] | 1.8 | 11 months | 340 × 170 m; 110 × 11 m; 90 × 8 m | DoD ^{5} | Raster averaging | Region growing | Sum of triangulated raster cells |

Carrea et al. [41] | 3.0 | 6 months | ~80 × 40 m | C2M | - | DBSCAN | SfR: Alpha Shapes * |

^{1}Cloud-to-model comparison [42]. Computes the distance from each point to a neighbouring triangular mesh facet, along the facet normal.

^{2}Surface reconstruction for three-dimensional volume estimation.

^{3}Multiscale model-to-model cloud comparison [43]. Computes locally averaged change between two-point clouds along a locally oriented surface normal.

^{4}Density-based spatial clustering of applications with noise [44]. Clusters data considering a minimum number of points within a minimum search radius.

^{5}Difference of digital elevation models (DEMs). Computes change between rasters, orthogonal to the raster image orientation. * Surface reconstruction mesh hole filling method not specified and/or provided.

#### 1.3. Surface Reconstruction Challenges and Requirements

#### 1.4. Digital Surface Representation and Reconstruction Methods Background

#### 1.5. Study Objectives

## 2. Materials and Methods

#### 2.1. LiDAR-Derived 3D Rockfall Database

#### 2.2. Computational Geometry Tools

#### 2.3. Computational Geometry Surface Reconstruction for Volume Estimation

#### 2.3.1. Convex Hull

#### 2.3.2. Three Dimensional Alpha Shapes

#### 2.3.3. Power Crust

- Poles: A subset of the Voronoi vertices, located on the interior and exterior of the object, but not along the object’s surface;
- Polar balls: The balls centered at the poles, each with radii such that they are touching the nearest input point sample;
- Medial axis: The skeleton of a closed shape, along which, points are equidistant to two or more locations along the shape’s boundary.

#### 2.3.4. Power Crust Mesh Volume Computation

## 3. Results

^{3}.

^{3}in comparison to the estimates of 3851 m

^{3}from the Alpha Solid and 3133 m

^{3}from the Power Crust. The issue causing this underestimation of volume has to do with the definition of the Default Alpha Radius, and the definition of a Delaunay tetrahedron. As noted in the methods section, the Default Alpha Radius is prescribed to be sufficiently high in order to connect all point within a singular the Alpha Shape. A Delaunay tetrahedron is defined by the connection of 4-points, if they can be bounded by an empty circumsphere with a radius less than the value of Alpha (i.e., Figure 4). Restricted Voronoi Cells (i.e., Figure 4d and Figure 5) do not intersect in the interior of the Default Alpha Shape when the Default Alpha Radius is not larger than roughly ½ of the “thickness” of the rockfall point cloud—where the point cloud object “thickness” is the maximum distance which a Delaunay tetrahedron must span in order to bridge occupy the interior of the point cloud. The Default Alpha Radius is a function of the point cloud spacing, and thus has no correspondence with the thickness of the point cloud. Therefore, a point cloud with a thickness greater than roughly two-times the point spacing, will have Delaunay tetrahedra missing from its interior, where the majority of the shape’s volume is contained. The volumetric estimates of the Default Alpha Shape are therefore very poor in the instance of rockfall point clouds which have large thicknesses relative to their point spacing—even though the Default Alpha Shape seems to capture the detailed topological connections along the surface of the shape (Figure 8c). Further, the Default Alpha Shape tetrahedra are unable to bridge across large sections of missing surface information. This is revealed by the presence of holes in the Default Alpha Shape (Figure 8c). Sections of missing information in the point cloud therefore also result in underestimated Default Alpha Shape volumes.

^{−2}m

^{3}.

^{3}(Figure 10b).

^{3}and therefore an Alpha Solid volume substitution dataset was not created for the second set of power-law models.

^{3}cut-off models (Table 3) to estimate the larger magnitude ranges. It is, however, up for discussion as to which model would be better in practice for determining the return period of the larger magnitude events, considering the history of the study site, as we note in the Discussion Section.

## 4. Discussion

^{−2}m

^{3}. As smaller scale rockfalls have surfaces closer to their medial axis, whilst the point spacing is relatively uniform (i.e., subsampled to 10 cm), our survey resulted in the smaller rockfall point clouds not having sufficient sampling for the Power Crust algorithm. This is also partially due to the reality that data resolution is not a typical constraint within the field of surface reconstruction; the Power Crust algorithm was not designed to reconstruct extremely small point clouds with relatively low data resolution. Keep in mind that the typical point spacing for the TLS surveys was subsampled to 10 cm and the limit of detection was 5 cm. Different point clouds causing failed Power Crust reconstructions are to be expected with different data resolutions and limits of detection.

^{3}. In this work, we found similar results, however, the larger scale of analysis (i.e., 3668 rockfalls ranging from approx. 0.001 to 3000 m

^{3}) allow us to make much broader conclusions concerning the development of magnitude–frequency relations. We observed significant overestimates in the Alpha Solid volume, due to larger-scale point clouds with significant concave features. For the largest rockfall tested, Bonneau et al. [50] saw a 5% larger estimate in volume by of the Alpha Solid in comparison to the Power Crust. The largest event presented in the current study had an Alpha Solid reconstruction with a 23% larger estimate (Figure 8). This study therefore demonstrates that the Power Crust is the better method to use on large-scale point clouds.

^{3}rockslide in November of 2012 [60], and an approximate 3000 m

^{3}rockfall in December of 2014 (i.e., Figure 8). The large failures would have destabilized a significant section of the rock mass. The increased proportion of large rockfall events may have been further amplified by the fact that this rock slope was monitored throughout a highly active time period (November 2013 – December 2018). The 50,000m

^{3}rockslide was not included in the magnitude-frequency relations because it is a different classification of landslide with fundamentally different mechanisms [2].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Magnitude-Frequency Visualization and Power-Law Fitting

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**Figure 1.**Point cloud imperfections known to challenge surface reconstruction algorithms: (

**a**) The original surface being sampled; (

**b**) Non-uniform sampling; (

**c**) Noise; (

**d**) Outliers; (

**e**) Misaligned data; (

**f**) Missing data. Adapted from Berger et al. [48].

**Figure 2.**Visual definitions of 2-manifold surfaces and planar patches, modified after Bonneau et al. [50]; (

**a**) A continuous 2-manifold sphere with an outlined neighbourhood which is homeomorphic to a Euclidean disk; (

**b**) A similar 2-manifold surface mesh; (

**c**) A non-manifold mesh edge; (

**d**) A proper manifold mesh edge; (

**e**) A non-manifold mesh self intersection; (

**f**) A non-manifold node connecting two tetrahedra.

**Figure 3.**Rockfall data of the CN Ashcroft Mile 109.4 study site, across a 5-year span from November 2013 to December 2018. Presented in 3 different intervals to reduce overprinting data in the visualization. An 80 m long rock shed constructed in fall of 2014 supports a talus cone which has built up, allowing rockfall fragments to pass overtop of the track. Clustered rockfall point clouds are randomly colourized and projected onto a helicopter photogrammetry point cloud (December 2014); (

**a**) 347 days, from 28 November 2013 to 10 November 2014; (

**b**) 104 days, from 10 November 2014 to 17 February 2015, occurrence of an approximate 3000 m

^{3}rockfall and several other events in the hundreds of cubic meters; (

**c**) 1381 days, from 17 February 2015 to 4 December 2018.

**Figure 4.**Key computational geometry tools presented in 2D. (

**a**) The Voronoi diagram of the point set. Voronoi cells contain all space nearest to its sample; (

**b**) a Power diagram, with weights denoted by the radius of the blue circle surrounding each point. Power cells contain all space, in which the distance subtracted by its cell weight (i.e., the Power distance), is less than competing cells. The cell edges of competing samples are drawn at the distance where the outer circle boundaries intersect. If all weights are equal it would result in the same diagram as in (

**a**); (

**c**) The Delaunay triangulation (blue), which is the dual graph of the Voronoi diagram (black). An example of an empty circumcircle (red), centered at a Voronoi vertex (red), showing that its triangulation is Delaunay; (

**d**) A restricted Voronoi diagram (black), with a uniform weight determining the maximum extent of the cells. The resulting Delaunay triangulation (blue) is made for the trios of adjoining cells. A lack of cell adjacencies results in several points becoming excluded from the triangulation. Visualization adapted from Devert [64] and Royer [65] with the Scipy Spatial [66] and Matplotlib [67] libraries for Python.

**Figure 5.**Restricted Voronoi diagrams and their resulting Delaunay Triangulations producing the triangulated Alpha Shape, adapted from Royer [65]. (

**a**) a zero Alpha Radius producing an empty Alpha Shape; (

**b**–

**h**) an increase in Alpha Radius from left to right, and top to bottom, respectively; (

**i**) an infinite Alpha Radius producing the Convex Hull. Visualization aided by the Scipy Spatial [66] and Matplotlib [67] libraries for Python.

**Figure 6.**A 2D illustration of the Power Crust algorithm, adapted from Amenta et al. [55]; (

**a**) the object with its medial axis; (

**b**) the sampled point cloud, $S$, with an inset image showing the 5-times bounding box points added to the samples; (

**c**) the Voronoi diagram of $S$, with Voronoi vertices, $V$, plotted in blue; (

**d**) maximal balls centered at the Voronoi vertices. The straight lines are a result of far outer polar balls with large radii such that they simplify to half-spaces intersecting the convex hull of $S$, (

**e**) Power diagram of $V$, constructed with the weights of the maximal balls in (

**d**). Polar balls within the envelope of another do not have their own Power cell, as they are overpowered; (

**f**) the labelled Power diagram with the medial axis approximation. The boundary between the interior and exterior Power diagram cells is the Power Crust. Visualization adapted from Devert [64] and aided by the Scipy Spatial [66] and Matplotlib [67] libraries for Python.

**Figure 7.**A mesh facet expanded into a signed tetrahedron with its apex at the origin. Adapted from Lien and Kajiya [77].

**Figure 8.**Surface reconstruction results for the largest rockfall point cloud, with a side view (top) and a front view (bottom), rendered in Blender [78]. (

**a**) The rockfall point cloud visualized by its change detection signature. Missing surface information in the point cloud is observed where the back (cold-coloured) points are visible through the gaps in the front (warm-coloured). Missing information occurs between the boundaries of the back and front data, and across surfaces with high incidence angles relative to the scanner; (

**b**) The Convex Hull reconstruction; (

**c**) The Default Alpha Shape reconstruction; (

**d**) The Alpha Solid reconstruction; (

**e**) The Power Crust reconstruction.

**Figure 9.**The rank-frequency (cumulative) distributions generated by each of the surface reconstruction methods, with cumulative frequencies normalized by the 5-year monitoring period. The Power Crust dataset does not include the rockfalls for which the reconstruction failed, hence its more significant rollover and unique cumulative frequency. The majority of failed reconstructions were for small-scale point clouds and thus for small volumes.

**Figure 10.**(

**a**) the maximum likelihood estimation of power-law models, with the datasets truncated at their estimated values of ${x}_{min}$ found by minimizing the Kolmogorov–Smirnov statistic; (

**b**) the maximum likelihood estimation of the power-law models, truncated at 1m

^{3}in order to better fit the tail-end of the distribution.

**Table 2.**

**Upper**: power-law PDF fits for each of the datasets utilizing the Kolmogorov–Smirnov statistic for estimating ${x}_{min}$ and the MLE of the scaling parameter;

**Lower**: annual rockfall frequency calculations.

Convex Hull | Default Alpha Shape | Alpha Solid | Power Crust | Power Crust Substituted ^{1} | |
---|---|---|---|---|---|

${x}_{min}$ [m^{3}] | 0.0501 | 0.0272 | 0.0339 | 0.0279 | 0.0217 |

$b$ | 1.6315 | 1.7193 | 1.6808 | 1.7012 | 1.7080 |

$f\left({V}_{1}<V<{V}_{2}\right)$ [Rockfalls per Year] | |||||

$0.01{\mathrm{m}}^{3}V0.1{\mathrm{m}}^{3}$ | 494 | 327 | 374 | 366 | 382 |

$0.1{\mathrm{m}}^{3}V1{\mathrm{m}}^{3}$ | 115 | 62.4 | 77.9 | 72.8 | 74.8 |

$1{\mathrm{m}}^{3}V10{\mathrm{m}}^{3}$ | 26.9 | 11.9 | 16.2 | 14.5 | 14.6 |

^{1}Failed Power Crust reconstructions substituted with the Alpha Solid volumes.

**Table 3.**

**Upper**: Power-law PDF fits for each of the datasets utilizing MLE with a defined ${x}_{min}$ of 1 m

^{3};

**Lower**: annual rockfall frequency calculations.

Convex Hull | Default Alpha Shape | Alpha Solid | Power Crust ^{1} | |
---|---|---|---|---|

${x}_{min}$ [m^{3}] | 1 | 1 | 1 | 1 |

$b$ | 1.6001 | 1.6471 | 1.5743 | 1.5757 |

$f\left({V}_{1}<V<{V}_{2}\right)$ [Rockfalls per Year] | ||||

$1{\mathrm{m}}^{3}V10{\mathrm{m}}^{3}$ | 25.6 | 11.3 | 14.3 | 12.6 |

$10{\mathrm{m}}^{3}V100{\mathrm{m}}^{3}$ | 6.41 | 2.54 | 3.82 | 3.35 |

$100{\mathrm{m}}^{3}V1000{\mathrm{m}}^{3}$ | 1.61 | 0.573 | 1.02 | 0.890 |

$1000{\mathrm{m}}^{3}V10,000{\mathrm{m}}^{3}$ | 0.403 | 0.129 | 0.272 | 0.236 |

^{1}No Power Crust failed reconstructions occurred above the 1 m

^{3}value of ${x}_{min}$.

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**MDPI and ACS Style**

DiFrancesco, P.-M.; Bonneau, D.A.; Hutchinson, D.J. Computational Geometry-Based Surface Reconstruction for Volume Estimation: A Case Study on Magnitude-Frequency Relations for a LiDAR-Derived Rockfall Inventory. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 157.
https://doi.org/10.3390/ijgi10030157

**AMA Style**

DiFrancesco P-M, Bonneau DA, Hutchinson DJ. Computational Geometry-Based Surface Reconstruction for Volume Estimation: A Case Study on Magnitude-Frequency Relations for a LiDAR-Derived Rockfall Inventory. *ISPRS International Journal of Geo-Information*. 2021; 10(3):157.
https://doi.org/10.3390/ijgi10030157

**Chicago/Turabian Style**

DiFrancesco, Paul-Mark, David A. Bonneau, and D. Jean Hutchinson. 2021. "Computational Geometry-Based Surface Reconstruction for Volume Estimation: A Case Study on Magnitude-Frequency Relations for a LiDAR-Derived Rockfall Inventory" *ISPRS International Journal of Geo-Information* 10, no. 3: 157.
https://doi.org/10.3390/ijgi10030157