# MATLAB Virtual Toolbox for Retrospective Rockfall Source Detection and Volume Estimation Using 3D Point Clouds: A Case Study of a Subalpine Molasse Cliff

^{1}

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## Abstract

**:**

^{®}environment. The proposed package offers a complete and semiautomatic 3D solution that spans from extraction to identification and volume estimations of rockfall sources using state-of-the-art methods and newly implemented algorithms. To illustrate the capabilities of this package, we acquired a series of high-quality point clouds in a pilot study area referred to as the La Cornalle cliff (West Switzerland), obtained robust volume estimations at different volumetric scales, and derived rockfall magnitude–frequency distributions, which assisted in the assessment of rockfall activity and long-term erosion rates. An outcome of the case study shows the influence of the volume computation on the magnitude–frequency distribution and ensuing erosion process interpretation.

## 1. Introduction

## 2. Toolbox for 3D Point Cloud Processing

^{®}environment, which enables the management of large datasets and complex functions with increased speed [38]. This selection provides a suitable environment for any user to enhance the library of the toolbox owing to the work of the MathWorks

^{®}community. Some properties, such as position, color, or intensity, are obtained as direct information extracted from LiDAR devices or photogrammetric models, whereas other properties, such as normal vectors, curvature, etc., are computed in postprocessing when needed. Other included properties are related to the spatial distribution of the points, such as Delaunay triangulation or voxel structure, Kd-tree [39]. The presented toolbox can be downloaded under link at Supplementary materials.

## 3. Toolbox-Specific Landslide Package: Retrospective Rockfall Source Detection and Volume Estimation Processing

#### 3.1. Step 1: Rockfall Source Location Extract by Thresholding

- Points belonging to topographic changes assumed to result from rockfalls.
- Points belonging to unchanged topography assumed to be stable surfaces.

_{i}= Δ(P

_{i}, S), where (P

_{i}) is, for example, a (i) point in the pre-rockfall event (pre) point cloud (P) of size (n), (S) is the surface built from the post-rockfall event (post) point cloud using the triangulation mesh, and (d

_{i}) is the computed signed distance along the local normal of (S). If no acquisition bias exists in the point cloud, the distribution of distance differences without surface change follows a normal distribution centered on zero. In the locations at which a change in topography occurs, the distance comparison (d

_{i}) must be larger than the standard deviation (σ). According to [5], the standard deviation of the measurements between two epochs can be high and depends on multiple factors, such as the quality of the point cloud datasets, the density of the points, the presence of vegetation, the roughness of the relief, and the quality of the alignment between the point clouds and/or the acquisition locations between the two epochs (LiDAR position or picture positions). According to [41], point cloud points are assumed to be indicative of topographic changes (i.e., here, a rockfall source) without ambiguity when the point-to-surface comparison distances are larger than two times the standard deviation (2σ). As proposed by [41], this threshold can be improved by applying a spatial filter using the mean point-to-surface comparison distance of a point and its 25 nearest neighbors. Thus, the thresholding conditions are defined as follows:

#### 3.2. Step 2: Clustering Rockfall Sources

- A core point, if the neighborhood of radius (ε), has at least k-points (reachable points);
- A border point possesses at least one core point within a radius (ε);
- An outlier is a point with no point or no core point within its radius (ε).

#### 3.3. Step 3: Rockfall Source Volume Estimation

_{α}) formed by a set of points, the value used is an α value corresponding to a research radius in the point cloud ranging from 0 to ∞ and follows the subsequent conditions:

- If α = ∞, S
_{α}is the convex hull of the point cloud; - If α = 0, S
_{α}is each point of the point cloud itself; - If 0 < α < ∞, S
_{α}will be the largest polyhedron or shape connecting m points of the point cloud.

_{α}). As S

_{α}is a complex of DT, we use the DT to decompose volumes defined by the envelope of S

_{α}into tetrahedrons. The volume of a i tetrahedron with a triangular base of a given area, A, and a height, h, are given as follows:

_{i}, inside an α-shaped S

_{α}defined as follows:

## 4. Case Study

_{0}125/14°, joint system J

_{1}234/86° subparallel to the cliff face, and joint systems J

_{2}150/75° and J

_{3}325/80° nearly perpendicular to the slope. Discontinuities are clearly present on sandstone beds. The spacing varies from 0.2 to 1.0 m for J

_{1}and from 0.15 to 2 m for J

_{2}and J

_{3}, respectively.

^{3}(0.1 × 0.2 × 0.15 m) for the minimal volume to 6.0 m

^{3}(3.0 × 1.0 × 2.0 m) for the maximal volume.

^{ER}with theoretical accuracy of 7 mm at 100 m and a standard deviation of 10 mm [53]. The footprint at the cliff range (~300 m) was approximately 50 mm. The effective point surface density was of 860 pts/m

^{2}.

## 5. Results

^{3}. It depends on the size of the cliff, the length of the investigation period, and the geological and geomorphological context. The exponent, b, depends on the geological and geomorphological context only. Thereby, recent studies showed that a and b are correlated to the GSI (Geological Strength Index) of the rock cliff [37]. In addition, the value of b indicates the proportion of small events as compared to larger events. Therefore, this is important in the context of a power-law distribution, where small events in sum could contribute significantly to overall volume loss. The power-law regression has some limits as described in [22], as we have potential temporal and/or spatial resolution bias.

^{3}, which is the expected smallest volume identified without ambiguity from the point cloud. We observed an increase in the number of volumes greater than 1 m

^{3}, or an increasing parameter a for an increase of the α value. In contrast, we observed a decrease in the b exponent. For values α < 0.1, which are a too low research radius, we observed that most of the reconstructed volumes are not filled or contained inner holes, as in Figure 4B. For a value 0.1 ≤ α ≤ 1.25, we can reconstruct volumes conserving a close-to-reality geometry depending on block shape complexity. For a value α > 1.25, we start connecting farther points, increasing shape convexity for all rockfall volumes, and moving away from real geometry up to an infinite α value, which indicates the maximal convex shape and the maximal rockfall volume (i.e., Figure 4D).

^{3}, ranging from 0.0015 to 7.63 m

^{3}with a mean volume of 0.2 m

^{3}(see Figure 8). Less than 1% of the rockfall sources were smaller than 0.003 m

^{3}(0.1 × 0.2 × 0.15 m), which is the smallest volume expected from structural analysis. Less than 25% of the rockfall sources are smaller than 0.01 m

^{3}(0.2 × 0.2 × 0.2 m), which is the expected smallest volume identified without ambiguity from the point cloud. In Figure 8, the regression shown in red represents the rockfall sources with volumes higher than 0.01 m

^{3}, and the other regression shown in blue represents all volumes, including the smallest volumes. Green triangles were not included in the regression analysis because they represent multiple rockfall sources and were too scarce during the period of monitoring; thus, they were not considered to be representative [22,54]. We observed that the choice of the volume interval considered in the regression affects the a and b parameters. With distribution containing all rockfall sources, a increases and b decreases compare to distribution with only rockfall sources >0.01 m

^{3}.

## 6. Discussion

^{3}with a point spacing at cliff range of 3.4 cm.

^{3}) is suspected. The same observation is made on Figure 7. This may have been caused by the limited spatial resolution (one point every 3.4 cm), as described in [35], and the minimal number of points used for cluster detection (see Section 3.2). Some large rockfalls were related to multiple events located in the same area (i.e., triangle symbol in Figure 8) but computed as a single event due to insufficient temporal resolution [35] or too close sources. This presence of multiple unresolved events can be resolved with a higher-frequency data acquisition, such as monthly or permanent acquisition [22], instead of seasonal acquisition. Thus, errors linked to coalescence and superposition of events can be reduced with enough temporal sampling [33].

^{2}in the zone of interest, assuming a mean detachment of material of ~25 m

^{3}per year between October 2011 and October 2013 and of ~21 m

^{3}per year over the full monitoring period from 2010 to 2015 on sandstone beds. In addition to these results, the erosional volume of marls was not considered in this study.

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**The structure of the 3DPointCloudToolBox contains folders labeled as Documents for tutorials, Examples of scripts to start up, Beta for functions in development, and Data to store raw data that can be used in the 3DPointCloudToolBox. The main folder Toolbox contains subfolders for point cloud data treatments (MainLibrary, PointCloudAlignment, PointCloudComparison, SyntheticPointCloud) and a specific subfolder landslide for landslide monitoring, with one for rockfall monitoring and another for landslide surface displacement monitoring (RockfallQuantification, LandslideTracking).

RockfallQuantification Functions | |
---|---|

Step 1: RockfallExtract | Extract point belonging to surface change from two PointCloud objects |

Step 2: RockfallSegment | Individualize single rockfall event by clustering index on PointCloud |

dbscan_optics | Density-Based Spatial Clustering of Applications with Noise [42] and OPTICS improvement [43] |

dist | Compute Euclidean distance between points in the cloud |

epsilon | Compute optimal epsilon radius according to gamma function approximation (Daszykowski et al., 2002) |

Step 3: RockfallVolume | Compute volume and center of mass of PointCloud |

trueboundary | Find boundary points to define shape of PointCloud |

volumes_tetra | Compute volume of single tetrahedron |

alphavol | Compute α-concave hull from PointCloud [45] |

MATLAB Classes—Key Terms | |
---|---|

Class definition | Description of what is common to every instance of a class |

Classes | A class describes a set of objects with common characteristics |

Super classes | Classes that are used as a basis for the creation of more specifically defined classes (i.e., subclasses) |

Subclasses | Classes that are derived from other classes and that inherit the methods, properties, and events from those classes (subclasses facilitate the reuse of code defined in the superclass from which they are derived) |

Objects | Specific instances of a class, which contain actual data values stored in the object’s properties |

Properties | Data storage for class instances |

Methods | Special functions that implement operations that are usually performed only on instances of the class |

Packages | Folders that define a scope for a class and function naming |

PointCloud Methods | |
---|---|

Add | Add the content of a given point cloud to this one |

addNoise | Add simulated noise to the true point positions with following possibility: Gaussian position smearing Outliers to simulate completely wrong position Drop out some points by replacing points position by NaNs |

ComputeBoundaries | Compute the Boundary points |

ComputeCurvature | Compute the curvatures at each point using: Estimation of the curvature based on [64] Variation of the surface from correlation of point clouds based on [65] |

ComputeDelaunayTriangulation | Compute a 3D Delaunay triangulation using built-in MATLAB^{®} function |

ComputeKDTree | Compute a Kd search tree using built-in MATLAB^{®} function |

ComputeNormals | Compute the least squares normal vector estimation of the points based on [64] |

ComputeOptimalNormals | Compute the adaptive normals based on neighbor size, point density, and research radius based on [66] in order to reduce normals dispersion |

ComputeTrueDistance | Compute the mean and root mean squared distances between a PointCloud positions and a given PointCloud true positions |

CopyTrue2MeasPos | Copy the “true” positions to the “measured” ones |

GetMissingPropFromPC | Complete properties of an object PointCloud by getting the missing ones from other PointCloud object |

HasTrueP | Return true if the object PointCloud has true positions |

ImportDataFromASCII | Import data from an ASCII file |

IsEmpty | Is the object PointCloud object empty? |

MeshPointCloud | Create a MeshPointCloud from this PointCloud |

MoveToCM | Move to the center of mass of another given object PointCloud |

NormalsOutTopo | For each point, compute the sign of the normal vector to be oriented toward its indexed sensor using TLSAttribute to have normals orientation to be out of the topography |

Plot3 | Plot the 3D coordinates of each point of the object PointCloud Positions |

PlotCurvature | Plot the computed curvatures |

PlotNormals | Plot the computed normals |

PlotPCLViewer | Plot for large point cloud positions with colors or intensities using Point Cloud Library Viewer [19] |

PlotPositionsWithColors | Plot the point cloud with the colors |

PlotPositionsWithIntensities | Plot the point cloud with the intensities |

RemoveNans | Remove any NaNs values in P and TrueP |

SaveInASCII | Save object PointCloud in ASCII format |

SaveInPCD | Save object PointCloud in PCD format for open Point Cloud Library [19] |

Size | What is the dimension of the object PointCloud? |

Transform | Transform the object PointCloud |

WhatColor | Query: what is the RGB color of the closest point? |

WhatIntensity | Query: what is the intensity of the closest point? |

MainLibrary | |
---|---|

PointCloud | Constructor of the object PointCloud and related methods |

AffinTransform | Apply an affine transformation to object PointCloud |

AlphaBoundary | Determine the convex hull of the object PointCloud using [45] |

EuclDist | Compute the Euclidean distance between two vectors of 3D points. |

HalfWayPoints | Loop on all the possible pairs in the input points and compute the halfway point |

ImportPointCloudFromASCII | Create a PointCloud object from a given input data (in ASCII format), allowing the user selection of the specific point cloud properties |

MeshPointCloud | Class to hold mesh grids as created by functions like GridFit |

PlaneMesh | Create a synthetic planar grid of points |

Plot3DPointClouds | Display one object PointCloud with defined property |

PlotMultiPointClouds | Display several objects PointCloud with defined property |

Quat2Rot | Convert (unit) quaternion representations to (orthogonal) rotation matrices R |

RemoveDuplicate3DPoints | Remove the duplicates in a set of 3D points |

Rot2Quat | Converts (orthogonal) rotation matrices R to (unit) quaternion representations |

RotationMatrix | Compute the rotation matrix given the Eulerian rotation angles |

SubSampling | Create a sub sample of a given object PointCloud |

TransformMatrix | Given the rotation angles and a translation vector, provides a transformation matrix |

TriangularMesh | Decompose a given triangle form mesh into smaller triangles |

Vector | A class to efficiently store any other property or type of data |

PointCloudComparison | |
---|---|

ComparePoint2Point | Compute comparison using the shortest point to point distance. Calculation is made using Euclidean distance between a given point in PointCloud A to the closest point in PointCloud B with output as absolute differences. |

ComparePoint2Surface | Compute comparison using the shortest point to surface distance. Calculation is made between a given point in PointCloud A and the distance parallel to the normal to the closest point in PointCloud B with output as signed differences. |

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**Figure 1.**Sketch of the process of localizing rockfall sources. Where P are the points, S is the reconstructed surface from point cloud at different epochs, and d

_{P}is the difference in distance from the point-to-surface comparison. The subscripts pre and post denote pre- and post-rockfall event acquisition, respectively.

**Figure 2.**(

**A**) Example of the raw dataset required to individualize the different clusters representing the different rockfall sources. (

**B**) The OPTICS (ordering points to identify the clustering structure) density-based clustering algorithm allows classification according to the reachability distance or neighbor radius. (

**C**) The classification allows attributing each point as a border or core point. Moreover, the OPTICS algorithm allows the identification of outlier points to remove.

**Figure 3.**Example of 2D projection of the α-shape hull construction concept. (

**A**) Outer surface Delaunay triangulation (DT, convex shape) with all simplex triangles and circumscribing circles. (

**B**) Gray triangles are defined by circumscribing circles with a radius larger than the defined research radius α. (

**C**) New outer surface from the α shape with concavities formed by the points. The shape is an α-complex compound of multiple simplexes (triangles) from DT.

**Figure 4.**Example illustrating the way to find the optimal research radius that defines the α shape to compute the closer concave hull volume. The results are plotted on a research radius/volume chart. (

**A**) Shows the print of a fossil ammonite. (

**B**) Point cloud with a small research radius with inner connected tetrahedrons. (

**C**) Shows the optimal research radius defined by the flexure of the curve. (

**D**) Results with infinite radius leading to the convex hull.

**Figure 5.**(

**A**)Map of the geological setting of western Switzerland. Red dots indicate the location of the study area in the Molasse Swiss Plateau near Lausanne. (

**B**) La Cornalle cliff is the lateral scarp (green circle) of the slow-moving landslide indicated in yellow. (

**C**) Overview of La Cornalle cliff on top of a slow-moving landslide. (

**D**) A closer view of the cliff shows the lithology of the area composed of alternating metric layers of sandstone and marls. Source map from Swisstopo and orthophoto from Google Earth.

**Figure 6.**(

**A**) An example of the location of identified rockfalls (green) based on field observations and picture analysis between two epochs (Fall 2012–Spring 2013). (

**B**) Identified points belonging to different rockfall sources after using the rockfall source extraction function on the acquired point cloud between Fall 2012 and Spring 2013 with a terrestrial laser scanner in the same area. The segmentation of different individualized rockfall sources plotted with a different color.

**Figure 7.**Magnitude–frequency plot shows the influence of the α value on the computed volumes from identified rockfall sources with different complex shapes between Fall 2012 and Spring 2013 as shown in Figure 6B. Power laws fitted on rockfall sources with volumes >0.01 m

^{3}show different trends according to different α value (from 0.1 to ∞ as a convex shape).

**Figure 8.**Magnitude–frequency plot of the rockfall sources from the full survey period (2010–2015) using the optimal α value according to method described in Section 3.3 for each rockfall source. One power law is fitted on overall rockfall sources and a second on the rockfall sources with volume >0.01 m

^{3}. The green triangles were clearly identified as multiple rockfall sources and are not considered for the regression fit.

Input Parameters | ||
---|---|---|

Threshold for pre- to post-event (T) corresponding to 2σ | 0.074 m | Automatically defined by package |

Minimum number of considered points for a cluster (k) | 34 pts | Manually defined by user |

Neighborhood radius (ε) | 0.251 m | Automatically defined by package |

α value or research radius (α) | 0.25–1.25 m | Manually defined by user |

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## Share and Cite

**MDPI and ACS Style**

Carrea, D.; Abellan, A.; Derron, M.-H.; Gauvin, N.; Jaboyedoff, M. MATLAB Virtual Toolbox for Retrospective Rockfall Source Detection and Volume Estimation Using 3D Point Clouds: A Case Study of a Subalpine Molasse Cliff. *Geosciences* **2021**, *11*, 75.
https://doi.org/10.3390/geosciences11020075

**AMA Style**

Carrea D, Abellan A, Derron M-H, Gauvin N, Jaboyedoff M. MATLAB Virtual Toolbox for Retrospective Rockfall Source Detection and Volume Estimation Using 3D Point Clouds: A Case Study of a Subalpine Molasse Cliff. *Geosciences*. 2021; 11(2):75.
https://doi.org/10.3390/geosciences11020075

**Chicago/Turabian Style**

Carrea, Dario, Antonio Abellan, Marc-Henri Derron, Neal Gauvin, and Michel Jaboyedoff. 2021. "MATLAB Virtual Toolbox for Retrospective Rockfall Source Detection and Volume Estimation Using 3D Point Clouds: A Case Study of a Subalpine Molasse Cliff" *Geosciences* 11, no. 2: 75.
https://doi.org/10.3390/geosciences11020075