# HRTT: A Hierarchical Roof Topology Structure for Robust Building Roof Reconstruction from Point Clouds

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## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Hierarchical Roof Topology Tree

#### 3.1. Concept and Reconstruction Strategy

#### 3.2. Topology Relations in the HRTT

#### 3.2.1. Plane–Plane Relations

#### 3.2.2. Plane–Model Relations

#### 3.2.3. Model–Model Relations

#### 3.3. Inside Tree Node Determination

## 4. Experiments and Discussion

#### 4.1. Datasets, Reference, and Parameters

#### 4.2. Model Reconstruction Results

#### 4.2.1. Vaihingen

#### 4.2.2. Wuhan University

#### 4.3. Precision Analysis and Limitations

#### 4.3.1. Topology Precision

_{min}of all connections are close to the average point distance and ten connections will all be accepted, either true or false ones. For Type B, some false connections (7–10) even obtain greater point counts (nPt); thus, they can hardly be excluded by simply changing the thresholds. A similar conclusion can also be generated by indices in Type C. As those false connections generate rather statistic values similar to the true ones, a stricter threshold can exclude more false connections (i.e., it can be concluded by comparing the Pro

_{min}values of Connections 5 and 7 in Figure 16e,f) while losing the weak connections (Connection 2). The statistic results prove that using simple local analysis or adjusting the thresholds can never solve the topology problems, and more extra procedures and constraints need to be considered. The final topology precision shows that our strategy is effective.

#### 4.3.2. Model Precision

#### 4.3.3. Limitations

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Example of the error caused by a composite structure: only the intersecting ridges are reflected in the RTG, while the step parts are omitted.

**Figure 3.**Definition of the hierarchical roof topology tree (HRTT). (

**a**) The tree structure. (

**b**) An example of simple RTG.

**Figure 6.**Boundary edge based connection determination. (

**a**) Image, (

**b**) roof edge of the single plane by the alpha-shape boundary, (

**c**) integral edge of the two planes, (

**d**) edge comparison results, (

**e**) compared edges with the intersecting ridge, and (

**f**) mixed edges and possible step parts.

**Figure 7.**Searching for abnormal points when deciding the roof ridges or step edges (same data as we used in Figure 2). (

**a**) Segmented roof points and detected roof ridges, with a (false) roof corner to be decided; (

**b**) the searching region for plane–plane relations; (

**c**) the searching region for the roof corner intersected by multiple planes. A, B, and C are the three adjacent planes, the red lines are the detected ridge lines, and the black dotted lines are where the ridges are supposed to be extended. P1, P2, and P3 are three points that failed to be divided into the target region. We project the ridges into the XOY plane, and a

_{i}, b

_{i}, and c

_{i}are parameters for the equations of corresponding 2D lines.

**Figure 9.**Example of boundary projection and the newly introduced constraints. (

**a**) Integral model boundary, in which O is the point cloud center and the “bias_x” is parallel with the domain orientation; (

**b**) boundary projection analysis; (

**c**) constraints derived from integral boundary projection; (

**d**) constraints derived from integral child structures and plane–model relations.

**Figure 10.**Example of side projection: (

**a**) initial results; (

**b**) side view; (

**c**) the false ridge and segmentation are corrected after the side projection test; points from the child-structure are projected to the main structure when calculating the relationship with other roof planes.

**Figure 13.**Reconstruction results of the three ISPRS test areas in Vaihingen, from (

**a**) to (

**c**) are the results of Areas 1, 2, and 3.

**Figure 15.**Details of the 10 selected error-prone connections in the test data, marked by connection types. In the reference image: red arrows for the intersected ridges, green arrows for the step edges and blue arrows for no connections. In the local scenes: red bold lines for the detected ridges, black bold lines for the step edges and green fine lines for the reference edges.

**Figure 16.**Static results of the ten connections for different local analysis methods. The indices are described in Table 4: (

**a**), d

_{min}for Type A; (

**b**) and (

**c**), nPt and L

_{min}for Type B; (

**d**) and (

**e**), Bd

_{min}and Pro

_{min}for Type C; (

**f**), our robust connections detected by Type C under stricter thresholds. The red lines are the suggested thresholds; connections falsely distinguished by the indices are marked by the red circles.

**Figure 18.**Model boundary precision analysis. (

**a**) The raw point clouds combined extracted roof boundary edges; (

**b**) pixel-level model precision by comparing the DSM generated by the detected roof with the reference DSM, where yellow is used for TP (true positive roof), red for FP (false positive roof/bulged edges), and blue for FN (false negative roof/sunk edges).

**Figure 19.**Buildings could not be well constructed. First row: DSM of our result models, second row: reference DSM, third row: DSM comparison graph at pixel-level, where the yellow regions stand for TP (true positive roof), red for FP (false positive roof), and blue for FN (false negative roof). (

**a**) Improper segmentation results, region b in Figure 14a; (

**b**) light absorbing surfaces, region d in Figure 14c; (

**c**) inaccurate building domain orientation, region c in Figure 14b.

Connection Type | Plane–Plane (Figure 5) | Plane–Model (Figure 8) | Model–Model (Figure 11) |
---|---|---|---|

Robust Step edge | 4 | 2, 3 | 3 |

Main horizontal ridge | 2, 3 | 2, 3, 4 | 2, 3 |

Edges to model boundary | 6, 7 | 2, 3 | 2, 3 |

Sub-structures | 5 | 5 | 3, 4 |

Site | Vaihingen | Wuhan University |
---|---|---|

Acquisition Date | 22 August 2008 | 22 July 2014 |

Acquisition System | Leica ALS 50 | Trimble Harrier 68i |

Fly Height | 500 m | 1000 m (cross flight) |

Point Density | ~4/m^{2} | >15/m^{2} |

Parameters | Vaihingen | Wuhan University |
---|---|---|

Input point cloud examine | ||

Minimum number of points for a building | 50 | 200 |

Minimum number of points for a roof plane | 5 | 20 |

Maximum slope for a roof plane | 80° | 80° |

For ridges/step edges (Section 3.2.1) | ||

Minimum ridge/step edge length (robust) | 1 m | 1 m |

Minimal height difference for step edges (robust) | 0.5 m | 0.5 m |

Buffer width near the roof ridges | 1 m | 1 m |

Small angle (for deciding horizontal/collinear ridges) | 3° | 3° |

For boundary | ||

Scale for alpha-shape edge detection | 3AveDis | 3AveDis |

Minimal spacing between two adjacent parallel edges | 1 m | 1 m |

Minimal number of points in an edge segment | 3 | 5 |

Distance threshold for Douglas–Peucker algorithm (Section 3.2.2) | 0.15 m | 0.15 m |

For abnormal points/area (Section 3.2.1) | ||

searching radius/buffer width | 2 m | 2 m |

Point to plane distance threshold | 0.2 m | 0.2 m |

Distance threshold for adjacent abnormal points | 3AveDis | 3AveDis |

Minimal number of adjacent abnormal points | 3 | 10 |

Minimal overlap for plane-in-plane test (Section 3.2.2) | 60% | 60% |

Method | Roof ridge | Step edge | Parameters (e.g.) |
---|---|---|---|

Type A: by point- point connections [9] | Minimum distance of all point combinations (d_{min}) | Degree of r-vertex (r = 1), detect by boundary tracing | d_{min} = 1 m; |

Type B: by point counts near the feature lines [6,8,10] | Points count within buffer area (nPt) and minimum ridge length projected by points (L_{min}) | N_{min} points within planimetric distance d_{max}, but larger height difference than H_{min} | nPt = 10; d_{max} = 1 m; L_{min} = 0.5 m; H_{min} = 1 m; |

Type C: by roof boundary edge segments [7,31] | Common boundary edge segments within buffer area (Bd_{min}), ridge length projected by edge (Pro_{min}) | Edge pixel chain is split into 2D and test the 2D boundary length | Bd_{min} = 1 m, Pro_{min} = 0.5 m; (Our_initial: Pro_{min} = 1 m) |

Topology Precision | A | B | C | Our_initial | Our_final | |
---|---|---|---|---|---|---|

RTG based | compl (%) | 90.3 | 88.4 | 85.8 | 84.5 | 92.9 |

corr (%) | 85.0 | 92.6 | 93.0 | 94.9 | 96.0 | |

qua (%) | 78.8 | 82.5 | 80.6 | 80.9 | 89.4 | |

ic > 0.3 | compl (%) | 90.3 | 87.1 | 85.2 | 83.9 | 91.6 |

corr (%) | 84.0 | 91.8 | 93.0 | 94.2 | 95.9 | |

Qua (%) | 78.0 | 81.1 | 80.0 | 79.8 | 88.2 |

Area | Pixel Level | Plane Level | Pixel Level | Plane Level | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

TP | FN | FP | TP1 | TP2 | FN | FP | Compl | Corr | Qua | Compl | Corr | Qua | |

1 | 613137 | 39584 | 13445 | 265 | 199 | 23 | 7 | 93.9 | 97.9 | 92.0 | 92.0 | 96.6 | 89.1 |

2 | 422093 | 57492 | 5765 | 48 | 55 | 21 | 1 | 88.0 | 98.7 | 87.0 | 69.6 | 98.2 | 68.7 |

3 | 754427 | 57018 | 18827 | 203 | 143 | 32 | 5 | 93.0 | 97.6 | 90.9 | 86.4 | 96.6 | 83.9 |

sum | 1789657 | 154094 | 38037 | 516 | 397 | 76 | 13 | 92.1 | 97.9 | 90.3 | 87.2 | 96.8 | 84.7 |

Region | Method | Compl | Corr | Compl_10 | Corr_10 | 1:M | N:1 | N:M | RMS(m) | RMSZ(m) |
---|---|---|---|---|---|---|---|---|---|---|

Area 1 | CKU | 86.8 | 98.9 | 88.4 | 99.2 | 10 | 36 | 3 | 0.9 | 0.6 |

ITCX3 | 89.2 | 96.4 | 93.2 | 97.7 | 5 | 39 | 6 | 0.8 | 0.2 | |

TUD2 | 73.3 | 100.0 | 70.7 | 100.0 | 1 | 36 | 3 | 0.8 | 0.2 | |

YOR | 88.2 | 98.5 | 94.6 | 99.2 | 5 | 36 | 14 | 0.8 | 0.3 | |

Ours | 92.0 | 96.6 | 94.6 | 98.5 | 10 | 36 | 8 | 0.8 | 0.2 | |

Area 2 | CKU | 78.3 | 93.1 | 93.8 | 100.0 | 8 | 4 | 0 | 0.5 | 0.7 |

ITCX3 | 71.0 | 100.0 | 89.6 | 100.0 | 3 | 4 | 1 | 0.5 | 0.2 | |

TUD2 | 71.0 | 100.0 | 89.6 | 100.0 | 2 | 3 | 0 | 0.3 | 0.3 | |

YOR | 66.7 | 100.0 | 83.3 | 100.0 | 5 | 3 | 0 | 0.5 | 0.3 | |

Ours | 69.6 | 98.2 | 87.5 | 97.9 | 6 | 3 | 0 | 0.6 | 0.3 | |

Area 3 | CKU | 81.3 | 98.4 | 91.9 | 99.1 | 4 | 48 | 2 | 0.8 | 0.6 |

ITCX3 | 88.1 | 88.2 | 96.8 | 95.8 | 3 | 50 | 2 | 0.7 | 0.1 | |

TUD2 | 73.6 | 100.0 | 81.5 | 100.0 | 0 | 42 | 0 | 0.5 | 0.1 | |

YOR | 84.7 | 100.0 | 97.6 | 100.0 | 2 | 51 | 1 | 0.6 | 0.2 | |

Ours | 86.4 | 96.4 | 95.2 | 100.0 | 9 | 44 | 5 | 0.6 | 0.2 |

^{2}). 1:M, N:1, and N:M are the topology quality of planes. RMS is the planimetric geometric accuracy: average root mean square distance to reference boundary, in XOY plane. RMSZ is the geometric accuracy and the Z component is the average root mean square distance of the detected DSM and reference DSM.

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**MDPI and ACS Style**

Xu, B.; Jiang, W.; Li, L.
HRTT: A Hierarchical Roof Topology Structure for Robust Building Roof Reconstruction from Point Clouds. *Remote Sens.* **2017**, *9*, 354.
https://doi.org/10.3390/rs9040354

**AMA Style**

Xu B, Jiang W, Li L.
HRTT: A Hierarchical Roof Topology Structure for Robust Building Roof Reconstruction from Point Clouds. *Remote Sensing*. 2017; 9(4):354.
https://doi.org/10.3390/rs9040354

**Chicago/Turabian Style**

Xu, Bo, Wanshou Jiang, and Lelin Li.
2017. "HRTT: A Hierarchical Roof Topology Structure for Robust Building Roof Reconstruction from Point Clouds" *Remote Sensing* 9, no. 4: 354.
https://doi.org/10.3390/rs9040354