# Effective Selection of Variable Point Neighbourhood for Feature Point Extraction from Aerial Building Point Cloud Data

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## Abstract

**:**

## 1. Introduction

- In the context of calculating an accurate normal, a new robust method is proposed for automatic selection of neighbouring points of each point in a LiDAR point cloud data. This proposed method can select the optimal minimum number of neighbouring points and, thus, can solve the existing problems of accurate normal calculation of individual points.
- Based on the calculated direction of the normal, we propose an effective method for finding the fold feature points. Maximum angle differences of the neighbouring normal vectors are clustered, and an experimentally selected threshold is adopted to decide fold edge points.
- To find the boundaries of individual objects, a new method for boundary point detection is suggested. This method depends on the distance from a point to the calculated mean of its neighbouring points, selected by the proposed technique of automatic neighbouring point selection.

## 2. Review

#### 2.1. Neighbourhood Selection

#### 2.2. Normal Vector Calculation

#### 2.3. Feature Point Extraction

## 3. Proposed Method

#### 3.1. Estimating Minimal Neighbourhood

- The proposed method first selects a minimal number of neighbouring points (say, $k=3$ since a minimum of 3 points are necessary to calculate a plane normal) for ${P}_{i}$ using the k-NN algorithm. Let the set of neighbouring points be ${S}_{\mathrm{p}}$ including the point ${P}_{i}$.
- A best fit 3D line ${L}_{3}$ is constructed using ${S}_{\mathrm{p}}$. The distance from each point of ${S}_{\mathrm{p}}$ to ${L}_{3}$ is calculated.
- The standard deviation ${\sigma}_{i}$ of the calculated distances is compared with a selected distance threshold ${T}_{\mathrm{d}}$. If ${\sigma}_{i}<{T}_{\mathrm{d}}$, the value of k is increased (say, k = $k+\delta $) and the procedure is repeated with the updated ${S}_{\mathrm{p}}$. Ideally, $\delta =1$ is set to iteratively find a minimal value of k for ${P}_{i}$. However, to avoid a large number of iterations, $\delta =5$ is selected and, once a minimal k is found, a smaller minimal k is obtained by testing its previous $\delta -1$ values.${T}_{\mathrm{d}}$ is equal to the distance between two neighbouring points in the case of regular distribution of LiDAR points and can be calculated using Equation (3) according to Tarsha-Kurdi et al. [59], where $\vartheta $ represents the input point density. The mean area occupied by a single LiDAR point is in a square form, and the area of the square is equal to the inverse of the point density in a regular distributed point cloud data. The side length of the square represents the mean distance between two neighbouring points that satisfies Equation (3).$${T}_{\mathrm{d}}=\frac{1}{\sqrt{\vartheta}}$$
- If ${\sigma}_{i}\ge {T}_{\mathrm{d}}$, ${S}_{\mathrm{p}}$ is the estimated minimal neighbourhood for ${P}_{i}$. The green points in Figure 3a show that the above steps successfully define the minimal neighbourhood for all points on a building roof. However, when there are unexpectedly a large number of points residing along a portion of a scanline, then these steps fail to define the neighbourhood, as in this case, all or most of the points in ${S}_{\mathrm{p}}$ are obtained from the same scanline using the k-NN algorithm (see Figure 3b). Since points are not selected from two or more scanlines, the 3D line is repeatedly formed on the scanline that offers a low ${\sigma}_{i}$ value.
- To avoid the above issue, this paper proposes a new neighbourhood search procedure for ${P}_{i}$ (see Figure 3c). First, depending on the input point density $\vartheta $, when the number of points in ${S}_{\mathrm{p}}$ is larger than $A\vartheta $, where A is the area of the smallest detectable plane, points that are very close (e.g., $\epsilon $ = 0.01 m) to ${L}_{3}$ are removed from ${S}_{\mathrm{p}}$ (blue points remain). Second, a line L passing through ${P}_{i}$ and perpendicular to ${L}_{3}$ (scanline) is generated. Third, a new rectangular neighbourhood ${C}_{1}{C}_{2}{C}_{3}{C}_{4}$ (green shaded in Figure 3c) for ${P}_{i}$ is formed. ${C}_{1}{C}_{2}{C}_{3}{C}_{4}$ is long along L but short along ${L}_{3}$, and thus, the idea is to reduce the neighbouring points from the current scanline (blue points) and to include more points from outside the scanline (green points) and even from the next scanlines (yellow points). Finally, only six points closest to four corners and two midpoints (${C}_{1}$, ${C}_{2}$, ${C}_{3}$, ${C}_{4}$, ${M}_{1}$, and ${M}_{2}$) from within ${C}_{1}{C}_{2}{C}_{3}{C}_{4}$ and ${P}_{i}$ are assigned to an empty ${S}_{\mathrm{p}}$ and ${\sigma}_{i}$ is again estimated to ${L}_{3}$. If the condition (${\sigma}_{i}\ge {T}_{\mathrm{d}}$) is still not satisfied, the rectangle is enlarged (orange shaded) along L to include more points from outside ${L}_{3}$, i.e., four more points closest to corners ${C}_{5}$, ${C}_{6}$, ${C}_{7}$, and ${C}_{8}$ are added to ${S}_{\mathrm{p}}$. It is experimentally observed that, when (mostly in the second iteration) points from the next scanlines are considered in ${S}_{\mathrm{p}}$, the condition is satisfied. Figure 3d shows that all points on the roof now have minimal neighbourhoods.

#### 3.2. Finding Fold Points

- When two planes physically intersect, as shown in Figure 4a, and if ${\theta}_{\mathrm{max}}>{T}_{\theta}$ for ${P}_{i}$ (red dot) but ${\theta}_{\mathrm{max}}$ for its neighbours (green dots) can be clustered into two major groups, where the clusters are not close to each other, ${P}_{i}$ is a fold point.
- When ${P}_{i}$ is a planar point, as shown in Figure 4b, ${\theta}_{\mathrm{max}}\le {T}_{\theta}$ for ${P}_{i}$ and all its neighbours.
- When ${P}_{i}$ is on a curved surface, as shown in Figure 4c, ${\theta}_{\mathrm{max}}>{T}_{\theta}$ for ${P}_{i}$ and ${\theta}_{\mathrm{max}}$ for its neighbours are very close to ${T}_{\theta}$.
- When ${P}_{i}$ is on a step edge, as shown in Figure 4d, there can be one of two situations. The adjacent vertical plane may have no or a small number of points. When there are no points on the vertical plane, then the fold points may be completely undetermined if the two planes (top and bottom) are parallel. If there is a large slope difference between these two planes, then the case in Figure 4a applies and fold points will be determined. When there are points reflected from the vertical plane, the fold points (between vertical and top planes and between vertical and bottom planes) can also be determined using the case in Figure 4a.

#### 3.3. Determining the Threshold ${T}_{\theta}$

#### 3.4. Detection of Boundary Points

## 4. Experimental Results

#### 4.1. Datasets

#### 4.2. Comparison

#### 4.2.1. Fold Points

#### 4.2.2. Boundary Points

#### 4.2.3. Eigenvalue-Based Features

#### 4.2.4. Combined Results

#### 4.3. Applicability in Different Types of Point Clouds

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Light Detection and Ranging (LiDAR) points over a building roof with scanning direction (red arrows).

**Figure 2.**The workflow of the proposed variable neighbourhood selection method: ${T}_{\mathrm{d}}$ is the threshold, ${\sigma}_{i}$ is the standard deviation, $\epsilon $ is the distance error, and $\delta $ is the neighbourhood increment.

**Figure 3.**New neighbourhood search across the scanline: (

**a**) a successfully defined minimal neighbourhood in a building where points are regularly distributed, (

**b**) the red points indicate that minimal neighbourhoods could not be defined, (

**c**) a rectangular neighbourhood is iteratively formed, and (

**d**) a successfully defined neighbourhood after applying the technique described using (

**c**).

**Figure 4.**Cases to decide fold points ${P}_{i}$ (red dots). Their adjacent points are shown by green dots. Arrows indicate normal directions. (

**a**) Gable roof, (

**b**) planar surface, (

**c**) curved surface, and (

**d**) step edge between planes.

**Figure 5.**Calculated ${\theta}_{\mathrm{max}}$ for different neighbourhoods: (

**a**) $k=9$, (

**b**) $k=20$, (

**c**) $k=30$, (

**d**) $k=45$, (

**e**) $k=60$, and (

**f**) the proposed neighbourhood.

**Figure 6.**Average ${\theta}_{\mathrm{max}}$ values for the fold points under different point densities.

**Figure 7.**Boundary point detection examples with (

**a**) the usual point density on all planes and (

**b**) the unexpectedly very high point density on some planes. In the magnified images of (

**a**), ${d}_{\mathrm{i}}$ indicates the distance between ${P}_{i}$ and the mean $\overline{\mathrm{S}}$ of the neighbours ${S}_{\mathrm{p}}$.

**Figure 8.**Used datasets for experiments. The first row shows the selected three sites from Australian datasets: (

**a**) AV1, (

**b**) HB, and (

**c**) AV2. The second row indicates the three areas from the International Society for Photogrammetry and Remote Sensing (ISPRS) benchmark dataset: (

**d**) VH1, (

**e**) VH2 and (

**f**) VH3. Buildings enclosed by the red line areas are taken into consideration for the ISPRS benchmark.

**Figure 11.**Linear (red), planar (green), and fold edge (blue) points using the fixed k-NN and the proposed variable neighbourhood: (

**a**) $k=9$, (

**b**) $k=30$, (

**c**) $k=45$, (

**d**) $k=60$, (

**e**) $k=90$, and (

**f**) the proposed neighbourhood.

**Figure 12.**Combined fold (blue), boundary (red,) and planar (yellow) points compared with the reference 3D building roof [7] for a complex building from the HB dataset: (

**a**) reference 3D building roof and (

**b**) combined result.

**Figure 13.**Combined fold (blue), boundary (red), and planar (yellow) points compared with the reference 3D building roof [7] for all five buildings from the AV1 dataset: (

**a**) reference 3D building roofs and (

**b**) combined results.

**Figure 14.**Combined fold (blue), boundary (red), and planar (yellow) points compared with the reference 3D building roof [7] for several buildings from the test datasets. While the first, third, and fifth rows show the reference information overlaid onto of the building roofs, the second, fourth, and sixth rows show the extracted results by the proposed method.

**Figure 17.**Comparing the results on the “3S” structure of Wuhan University: (

**a**) original point cloud, (

**b**) extracted feature points using AGPN [16], (

**c**) extracted feature points using Chen’s method [17] (

**d**) extracted feature points using the proposed methods. the (

**e**) extracted feature and planar points. Green indicates the planar points, and red represents both the boundary and the fold edge points.

${\mathit{\theta}}_{\mathbf{max}}$ | Number of Neighbouring Points | Proposed Method | ||||
---|---|---|---|---|---|---|

9 | 20 | 30 | 45 | 80 | ||

0–2${}^{\circ}$ | 783 | 2602 | 2793 | 2895 | 2851 | 2465 |

2–10${}^{\circ}$ | 2378 | 620 | 380 | 285 | 331 | 765 |

10–20${}^{\circ}$ | 178 | 109 | 224 | 276 | 371 | 161 |

20–30${}^{\circ}$ | 58 | 167 | 152 | 103 | 6 | 135 |

30–90${}^{\circ}$ | 162 | 61 | 10 | 0 | 0 | 23 |

F1-Score | 0.71 | 0.75 | 0.77 | 0.68 | 0.50 | 0.90 |

**Table 2.**Average processing time (in seconds) of neighbourhood selection techniques for each building.

Datasets | k-NN (k = 30) | k-NN (k = 45) | k-NN (k = 60) | Proposed |
---|---|---|---|---|

VH3 | 0.090 | 0.091 | 0.093 | 3.120 |

AV1 | 0.058 | 0.058 | 0.060 | 0.232 |

AGPN [16] | Chen [17] | Proposed | |
---|---|---|---|

Neighbourhood | Fixed k-NN | Fixed k-NN | Variable |

Extraction approach | Plane fitting and angular gap | Minimal number of clusters of neighbouring normal vectors | Maximum angle difference of the calculated normal vectors |

Normal estimation | RANSAC | Weighted PCA | Weighted PCA |

Geometric property | The RANSAC and angular gap metric | Direction of k-nearest normal vectors | Maximum angle differences among k-nearest normal |

Plane fitting | Required | Not required | Not required |

ISPRS Site (VH3) | Australian Site (AV1) | |||||
---|---|---|---|---|---|---|

Precision | Recall | F1 | Precision | Recall | F1 | |

AGPN [16] | 0.67 | 0.84 | 0.75 | 0.78 | 0.76 | 0.77 |

Chen [17] | 0.74 | 0.79 | 0.77 | 0.75 | 0.73 | 0.74 |

Proposed | 0.79 | 0.87 | 0.83 | 0.84 | 0.85 | 0.84 |

Improved RANSAC [16] | Chen [17] | Proposed | |
---|---|---|---|

Neighbourhood | $kd$-tree | Fixed k-NN | Variable |

Decision of boundary point | Substantial angular gap between vectors in a single plane | Distribution of azimuth angle | Euclidian distance from mean point to the point of interest |

Plane fitting | Required | Required | Not required |

Effect of outliers | High sensitive | Low sensitive | Low sensitive |

ISPRS Site (VH3) | Australian Site (AV1) | |||||
---|---|---|---|---|---|---|

Precision | Recall | F1 | Precision | Recall | F1 | |

Improved RANSAC [16] | 0.80 | 0.73 | 0.76 | 0.85 | 0.80 | 0.82 |

Chen [17] | 0.84 | 0.72 | 0.78 | 0.96 | 0.75 | 0.84 |

Proposed | 0.82 | 0.82 | 0.83 | 0.94 | 0.87 | 0.90 |

Values of k | No. of Linear Points $\mathcal{L}$ ≥ 0.5 | No. of Planar Points $\mathcal{P}$ ≥ 0.5 | F1 (Linearity) | F1 (Planarity) |
---|---|---|---|---|

9 | 2434 | 1115 | 0.19 | 0.15 |

30 | 1739 | 1810 | 0.61 | 0.68 |

45 | 571 | 2978 | 0.84 | 0.88 |

60 | 705 | 2844 | 0.71 | 0.79 |

90 | 843 | 2706 | 0.75 | 0.84 |

Proposed | 409 | 3140 | 0.91 | 0.94 |

Total Extracted Points | Precision | Recall | F1 | |
---|---|---|---|---|

AGPN [16] | 404 | 0.92 | 0.80 | 0.85 |

Chen [17] | 331 | 0.96 | 0.67 | 0.78 |

Proposed | 597 | 0.82 | 1.00 | 0.90 |

**Table 9.**Comparison of the extracted feature (fold and boundary) points using the three methods for “Computer World” building.

Original Point Cloud | Outline Points | Extraction Rate | |
---|---|---|---|

AGPN Method | 29,339 | 5989 | 20.4% |

Chen’s Method | 29,339 | 5097 | 17.4% |

Proposed | 29,339 | 4203 | 14.3% |

**Table 10.**Comparison of the extracted feature (fold and boundary) points using the three methods for the “3S” structure.

Original Point Cloud | Outline Points | Extraction Rate | |
---|---|---|---|

AGPN Method | 53,963 | 5146 | 9.50% |

Chen’s Method | 53,963 | 9150 | 16.95% |

Proposed | 53,963 | 6061 | 11.23% |

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**MDPI and ACS Style**

Dey, E.K.; Tarsha Kurdi, F.; Awrangjeb, M.; Stantic, B. Effective Selection of Variable Point Neighbourhood for Feature Point Extraction from Aerial Building Point Cloud Data. *Remote Sens.* **2021**, *13*, 1520.
https://doi.org/10.3390/rs13081520

**AMA Style**

Dey EK, Tarsha Kurdi F, Awrangjeb M, Stantic B. Effective Selection of Variable Point Neighbourhood for Feature Point Extraction from Aerial Building Point Cloud Data. *Remote Sensing*. 2021; 13(8):1520.
https://doi.org/10.3390/rs13081520

**Chicago/Turabian Style**

Dey, Emon Kumar, Fayez Tarsha Kurdi, Mohammad Awrangjeb, and Bela Stantic. 2021. "Effective Selection of Variable Point Neighbourhood for Feature Point Extraction from Aerial Building Point Cloud Data" *Remote Sensing* 13, no. 8: 1520.
https://doi.org/10.3390/rs13081520