# Investigation on the Weighted RANSAC Approaches for Building Roof Plane Segmentation from LiDAR Point Clouds

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. RANSAC-based Segmentation

_{U}is the number of points in U, and T(P

_{i}) is the inlier indicator:

_{i}is the point-plane distance, θ

_{i}is the angle between point P

_{i}’s normal and plane’s normal [7], and d

_{t}and θ

_{t}are the corresponding thresholds.

#### 2.2. Spurious Planes

_{t}can be well estimated beforehand according to the precision of the point clouds, then a proper segmentation is achieved if the hypothesis planes π

_{1}and π

_{2}receive the largest inlier ratio. However, poorly estimated planes may be detected, such as plane π

_{3}in Figure 1b, whose point count is much larger than that of π

_{1}or π

_{2}, thus leading to false segmentation. The RANSAC method extracts planes one after the other from LiDAR points so these mistakes may occur at plane transitions. The situation in Figure 1b can further intensify such competitions as the inaccurate hypothesis tends to generate more supports from roof points.

**Figure 1.**An example of spurious planes. (

**a**) The well estimated hypothesis planes (π

_{1}and π

_{2}); the two green parallel lines are the boundary of the point-to-plane distance threshold; (

**b**) A spurious plane (π

_{3}) is generated under the same thresholds; (

**c**) A detail view of (

**b**), where

**n**is the normal vector of the plane π

_{3}, and

**e**and

_{1}**e**are the point normal vectors. The

_{2}**d**,

_{1}**d**,

_{2}**θ**,

_{1}**θ**are the corresponding observed values of point P

_{2}_{1}and P

_{2}in Equation (2).

**θ**is the angle between π

_{0}_{3}and the real roof surface (π

_{0}).

#### 2.3. Existing Weighted RANSAC Methods

_{d}= d

_{t}/1.96), and ν is a constant which reflects the size of available error space.

## 3. Weighted RANSAC for Point Cloud Segmentation

#### 3.1. Improvements Consideration of the Weighted Function

**Figure 2.**Comparison of point-to-plane distance distribution between proper and improper hypotheses. (

**Top**) Plots of the weight functions (d

_{t}= 1.96σ

_{d}, MLESAC: γ = 0.3, ν = 3σ

_{d}); (

**Bottom**) Examples of distance distribution for the proper hypothesis and the spurious plane. A, B, and C are a rough division of the distance range: A for regions where the proper planes are dominant in the point count, B for regions where the point counts are similar, and C for regions where spurious planes generate more inliers. The red region represents the lost roof points when using a stricter threshold and the yellow region indicates that more points are excluded from the spurious planes than the proper hypothesis. BDSAC is a newly designed weight curve.

_{t}threshold, MSAC and MLESAC suppress the spurious planes by assigning smaller weights to the inliers with larger distances so that the inliers in area C of Figure 2 will contribute less to the evaluation of the hypothesis plane. The inadequacy of MSAC and MLESAC are mainly caused by its slow decrease of the weight curves. Generally, the weighted methods are expected to suppress the spurious plane as far as possible without excessively penalizing the proper planes. Under such consideration, we expect the curve of the weight function to decrease rapidly in area B and gradually with small weight values in area C (i.e., the curve of BDSAC in Figure 2). However, as shown in Figure 2, there are still a great deal of inliers that have large weight values and gradients in area C for MSAC and MLESAC. MSAC has the largest absolute gradient at the threshold boundary, and the MLESAC has a boundary weight value of over 0.2, which limit their suppressing to spurious planes. To overcome the drawbacks of these two methods, we attempted to modify the weight functions, and the improved versions of weight functions are shown in Section 3.2.

#### 3.2. Modified Weight Functions and New Weight Functions

_{t}in the weight function. For example, the MSAC with a reduction ratio μ is expressed by (denoted by MSAC

_{μ}):

_{u}and MLESAC

_{u}. RANSAC

_{u}uses smaller threshold $\mu \cdot {d}_{t}$ for inlier determination; and the σ

_{d}in Equation (5) of MLESAC

_{u}is reduced to $\mu \cdot {d}_{t}/1.96$.

_{1}and d

_{2}are the selected thresholds between 0 and d

_{t}( i.e., 0.2d

_{t}and 0.7d

_{t}in our test).

_{d}into pre-calculated tables, the efficiency of all the methods are similar.

**Figure 3.**Weight functions of various RANSAC methods: (

**Left**) Plots of the weight functions; (

**Right**) Plots of the absolute value of gradient.

#### 3.3. Joint Weight Function Regarding Angular Difference

_{0}in Figure 1c) can exist between the hypotheses plane and the real roof surface. As the normal of the points turns out to be in accord with the real roof surface, this deviation will reflect in most roof points. As a result, the angular difference between the points and the hypothesis plane (θ in Equation (2)) has long been used to evaluate the quality of inliers, either as constant thresholds in [7] or as a normal vector consistency validation in [4]. It is very natural for us to consider adding the angular difference into the weight definition.

_{θ}. Then, the weight of the angular difference can be defined by using the same form as the distance (simply replace d by θ). For instance, the weight functions of BDSAC for angular difference θ can be defined as:

_{nv}” is added to the methods that take the angular difference into account (e.g., BDSAC

_{nv}for the improved method of BDSAC).

#### 3.4. Weight Function Evaluation

_{ref}stands for the total weight of the reference plane (the plane fitted by all the inliers), and W

_{test}is the total weight of the test hypothesis plane.

**Figure 4.**Buildings with both positive and negative hypotheses. The deep blue triangle is a negative hypothesis as it is athwart the two roof planes, and the cyan triangle is a positive hypothesis which can produce a correct segmentation.

- (1)
- For all the weighted methods, the evaluation of the positive hypotheses (planes 1, 2, and 3) are stable as the ratios in Figure 5a are close to 1.0 and the ratio reductions in Figure 5b are close to 0. Meanwhile, all the weighted methods can significantly decrease the ratios of the negative hypotheses when compared to RANSAC, but their suppressing ability are different.
- (2)
- By comparing the results between the modified weight functions and the original functions (i.e., MSAC
_{0.7}and MSAC), it can be concluded that reduction of the inlier threshold can suppress the outliers effectively. The newly designed LDSAC and BDSAC functions have the best performances, which verifies our considerations in Section 3.1. - (3)
- From Figure 5c, it can be seen that all the methods can be affected by the threshold in some degree, but the newly designed weighted methods are least influenced.
- (4)
- Figure 5b,d illustrate the improvements after taking the angular differences into the weight functions. All the weighted methods gain positive effects and the effects are not sensitive to the thresholds.

**Figure 5.**Suppressing ability and threshold sensitivity test. (

**a**) Suppressing ratios for planes under different weight forms; (

**b**) Ratio reductions after considering the angular difference (i.e., the ratio reduction of BDSACnv is the ratio of BDSAC minus the ratio of BDSACnv); (

**c**) Mean ratio of the ten planes under different dt thresholds; (

**d**) Mean ratios reduction after considering the angular difference (the reduction approach is similar to (

**b**)).

_{t}thresholds. For data with a larger σ, the d

_{t}needs to be larger in order to include all the plane inliers; otherwise, over-segmentation may occur. As a result, nearly all the methods fail when d

_{t}is smaller than 2σ in Figure 6c (the value need to be even larger in real applications). The value of ∆d reflects the separability of the two planes and stricter thresholds are needed for a successful separation. The setting of thresholds needs to consider both factors and find a proper value between the two limitations, finally forming the acceptable areas for different weighted methods in Figure 6. For classical RANSAC, the results are rather disappointing and a proper threshold is difficult to generate. However, for the weighted methods, as the spurious planes are suppressed, much looser thresholds are allowed which result in larger areas in Figure 6. It also can be seen that both adding new weight forms and considering the angular difference in the weights produce positive effects on the acceptable areas. This decreases the difficulty of threshold selection and allows the possibility of processing more complex data. For instance, when ∆d equals 0.15 m and 0.2 m in Figure 6b or when σ equals 0.03 m and 0.04 m in Figure 6c, the classical RANSAC methods will always fail while our new weighted methods can produce a correct segmentation. Intuitively, a spurious plane that passes through the middle of the two planes will include all the points if d

_{t}is larger than ∆d/2 for classical RANSAC and cannot distinguish the two planes well when d

_{t}is larger than ∆d/3 in our experiments. In comparison, proper results are produced by BDSAC

_{nv}even when d

_{t}is larger than 2∆d/3.

**Figure 6.**Data sensitivity test. (

**a**) Simulated data, with changeable ∆d and σ; (

**b**) Segmentation results under different ∆d; (

**c**) Segmentation results under different σ. The colored regions depict the range of d

_{t}that can produce a correct segmentation.

## 4. Experiments and Evaluation

#### 4.1. Datasets and Fundamental Algorithm

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

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} |

MinPt | MinLen | Angle | d_{t} | θ_{t} | Ncc | Dcc | P_{0} | NbPt1 | NbPt2 | |
---|---|---|---|---|---|---|---|---|---|---|

Vaihingen | 5 | 1 m | 15° | 0.15 m | 10° | 5 | 1.5 m | 0.99 | 5 | 20 |

Wuhan | 20 | 1 m | 15° | 0.20 m | 10° | 5 | 0.75 m | 0.99 | 10 | 30 |

_{0}is the confidence probability to select the positive hypotheses at least once. NbPt1 and NbPt1 are the two parameters (nearest n points) used in the tensor voting-based method [10] (two rounds of voting).

#### 4.2. Evaluation Metrics

**i**ntegral

**c**onsistence is set as the product of the three values:

**Figure 7.**Definition of ridge similarity. Line

**AB**: the reference ridge (Ref); line

**CD**: the detected ridge (Test), where C1 and D1 are the corresponding projection points of C and D; and α is the intersect angle.

#### 4.3. Experiments

#### 4.3.1. Local Data

**Figure 8.**Results of segmentation and ridge detection for error-prone buildings. (

**a**–

**h**) are eight selected buildings containing error-prone regions. From left to right: reference images, results by classical RANSAC, results by RG, and results by BDSAC

_{nv}.

**Figure 9.**Suppressing ratios comparison. The spurious planes detected by RANSAC in Figure 8 (regions 1–11). (

**a**) Suppressing ratio for methods that only consider point-plane in weight functions; (

**b**) Ratios for methods considering both distance and angular difference.

**Table 3.**Quality of segmentation results for data in Figure 8.

ID | nPls | nRidges | Method | Segmentation | Ridges (RTG) | Ridges (ic > 0.3) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

%Cm | %Cr | %Qua | %Cm | %Cr | %Qua | %Cm | %Cr | %Qua | ||||

a | 10 | 7 | RANSAC | 80 | 100 | 80 | 71.4 | 55.5 | 45.5 | 57.1 | 44.4 | 33.3 |

RG | 100 | 100 | 100 | 71.4 | 100 | 71.4 | 71.4 | 100 | 71.4 | |||

BDSAC_{nv} | 100 | 100 | 100 | 100 | 100 | 100 | 71.4 | 100 | 71.4 | |||

b | 5 | 3 | RANSAC | 80 | 57.1 | 50 | 66.7 | 50.0 | 40.0 | 0 | 0 | 0 |

RG | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |||

BDSAC_{nv} | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |||

c | 7 | 5 | RANSAC | 85.7 | 60 | 51.5 | 60.0 | 37.5 | 30.0 | 40.0 | 25.0 | 18.2 |

RG | 85.7 | 75 | 66.7 | 100 | 71.4 | 71.4 | 100 | 71.4 | 71.4 | |||

BDSAC_{nv} | 100 | 100 | 100 | 100 | 83.3 | 83.3 | 100 | 83.3 | 83.3 | |||

d | 10 | 11 | RANSAC | 80.0 | 100 | 80.0 | 90.9 | 100 | 90.9 | 90.9 | 100 | 90.9 |

RG | 80.0 | 100 | 80.0 | 90.9 | 100 | 90.9 | 90.9 | 100 | 90.9 | |||

BDSAC_{nv} | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |||

e | 9 | 7 | RANSAC | 88.9 | 100 | 88.9 | 71.4 | 100 | 71.4 | 57.1 | 80 | 50 |

RG | 60 | 100 | 60 | 71.4 | 100 | 71.4 | 57.1 | 80 | 50 | |||

BDSAC_{nv} | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |||

f | 12 | 12 | RANSAC | 66.7 | 66.7 | 50 | 33.3 | 50.0 | 25.0 | 33.3 | 50 | 25.0 |

RG | 83.3 | 90.9 | 76.9 | 41.7 | 55.5 | 31.2 | 41.7 | 55.5 | 31.2 | |||

BDSAC_{nv} | 91.7 | 91.7 | 84.6 | 91.7 | 91.7 | 84.6 | 66.7 | 66.7 | 50.0 | |||

g | 23 | 5 | RANSAC | 69.6 | 64.0 | 50.0 | 100 | 62.5 | 62.5 | 100 | 62.5 | 62.5 |

RG | 78.3 | 72.0 | 60.0 | 100 | 71.4 | 71.4 | 100 | 71.4 | 71.4 | |||

BDSAC_{nv} | 69.6 | 66.7 | 51.6 | 100 | 50.0 | 50.0 | 100 | 50.0 | 50.0 | |||

h | 11 | 10 | RANSAC | 90.9 | 71.4 | 66.7 | 90.0 | 64.3 | 60.0 | 80.0 | 57.1 | 50.0 |

RG | 81.8 | 69.2 | 60.0 | 70.0 | 46.7 | 38.9 | 60.0 | 40.0 | 31.6 | |||

BDSAC_{nv} | 90.9 | 66.7 | 62.5 | 90.0 | 69.2 | 64.3 | 80.0 | 61.5 | 53.3 | |||

sum | 87 | 60 | RANSAC | 78.2 | 73.9 | 61.3 | 71.7 | 65.2 | 51.8 | 61.7 | 56.1 | 41.6 |

RG | 82.8 | 83.7 | 71.3 | 75.0 | 73.8 | 59.2 | 71.7 | 70.5 | 55.1 | |||

BDSAC_{nv} | 89.7 | 84.8 | 77.2 | 96.7 | 84.1 | 81.7 | 86.7 | 77.6 | 69.3 |

_{test}, and the total weight of the largest reference plane in the corresponding region is W

_{ref}. Regions 1–11 in Figure 8 are evaluated. Region 12 is omitted because the roof surface is nonplanar.

_{nv}. Meanwhile, the extent of the improvements by angular difference in the weights may be different for the planes. For planes with distinct biases in both the distance and normal vectors, such as regions 2 and 8, the suppressing of the total weights can be larger than in other regions. Again, BDSAC

_{nv}provides the best results. In addition, although our methods fail in region 11, the ratio of the BDSAC

_{nv}is still smaller than the other weighted methods.

#### 4.3.2. Vaihingen (Germany)

_{nv}method generates significant improvements compared to the traditional methods RANSAC and MSAC. Higher scores are achieved by our methods when using either the segmentation-based metrics or the two ridge-based metrics. The improvements of MSAC and MLESAC to RANSAC are not evident in the test data, and many spurious planes are still detected. It should be noted that some error-prone regions are also difficult for the RG method because regions with small angular or height differences often have very smooth transitions (e.g., B and E). Besides, the RG methods seem to be unstable in a few regions, such as the over-segmentation that unexpectedly occurs in F (see Appendix I).

**Figure 10.**Segmentation of Vaihingen data. (

**a**–

**f**) are six selected areas from the data. (top: image, bottom: results of BDSAC

_{nv}).

**Figure 11.**Quantitative results of the Vaihingen data. (

**a**–

**f**) are the six areas selected in Figure 10. Three metrics are used, from left to right: quality of segmentation and quality of two ridge based metrics.

#### 4.3.3. Wuhan University (China)

**Figure 12.**Segmentation results of Wuhan University data. (

**g**–

**l**) are six selected areas from the data. (top: image, bottom: results of BDSAC

_{nv}).

**Figure 13.**Quantitative results of the Wuhan University data. (

**g**–

**l**) are the six areas selected in Figure 12. Three metrics are used, from left to right: quality of segmentation and quality of two ridge based metrics.

_{nv}generate significant improvements compared to both classical RANSAC and the existing weighted methods. Compared to RANSAC, BDSAC

_{nv}improves the overall segmentation quality from 85.7% to 90.1%, as well as the two ridge-based metrics from 75.9% to 83.6% and 68.9% to 80.2%. The quality of the RG method is lower than the RANSAC-based methods, mainly due to their instability in areas (c), (g) and (h).

**Figure 14.**Integral quantitative results. Three metrics are used, from left to right: quality of segmentation and quality of two ridge based metrics.

## 5. Conclusions

_{nv}method is able to effectively suppress the outliers from spurious planes. As a result, we chose BDSAC

_{nv}for the further experiments and compare its performance with other existing segmentation methods, including original RANSAC, MSAC, MLESAC, and a representative RG method. A set of local data with error-prone regions and two large area datasets of varying densities are used to evaluate the performance of the different methods. The quantitative results of both the segmentation-based metrics and the ridge-based metrics indicated that the different weighted methods improve the segmentation quality differently, but BDSAC

_{nv}significantly improve the segmentation accuracy and topology correctness. When compared with RANSAC, BDSAC

_{nv}improved the overall segmentation quality from 85.7% to 90.1%; and the two ridge-based metrics also improved from 75.9% to 83.6% and 68.9% to 80.2%. Moreover, the robustness of BDSAC

_{nv}is better compared to the RG method. As a result, we believe there is potential for the wide adoption of BDSAC

_{nv}as an upgrade to or replacement of classical RANSAC in roof plane segmentation.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix I

## References

- Baltsavias, E.P. Object extraction and revision by image analysis using existing geodata and knowledge: Current status and steps towards operational systems. ISPRS J. Photogramm. Remote Sens.
**2004**, 58, 129–151. [Google Scholar] [CrossRef] - Haala, N.; Kada, M. An update on automatic 3D building reconstruction. ISPRS J. Photogramm. Remote Sens.
**2010**, 65, 570–580. [Google Scholar] [CrossRef] - Rottensteiner, F.; Sohn, G.; Gerke, M.; Wegner, J.D.; Breitkopf, U.; Jung, J. Results of the ISPRS benchmark on urban object detection and 3D building reconstruction. ISPRS J. Photogramm. Remote Sens.
**2014**, 93, 256–271. [Google Scholar] [CrossRef] - Chen, D.; Zhang, L.Q.; Li, J.; Liu, R. Urban building roof segmentation from airborne Lidar point clouds. Int. J. Remote Sens.
**2012**, 33, 6497–6515. [Google Scholar] [CrossRef] - Tarsha-Kurdi, F.; Landes, T.; Grussenmeyer, P.; Koehl, M. Model-driven and data-driven approaches using Lidar data analysis and comparison. In Proceedings of the ISPRS, Workshop, Photogrammetric Image Analysis (PIA07), Munich, Germany, 19–21 September 2007.
- Sampath, A.; Shan, J. Segmentation and reconstruction of polyhedral building roofs from aerial Lidar point clouds. IEEE Trans. Geosci. Remote Sens.
**2010**, 48, 1554–1567. [Google Scholar] [CrossRef] - Awwad, T.M.; Zhu, Q.; Du, Z.; Zhang, Y. An improved segmentation approach for planar surfaces from unconstructed 3D point clouds. Photogramm. Rec.
**2010**, 25, 5–23. [Google Scholar] [CrossRef] - Fan, T.J.; Medioni, G.; Nevatia, R. Segmented descriptions of 3-D surfaces. IEEE Trans. Rob. Autom.
**1987**, 3, 527–538. [Google Scholar] - Alharthy, A.; Bethel, J. Detailed building reconstruction from airborne laser data using a moving surface method. Int. Arch. Photogramm. Remote Sens.
**2004**, 35, 213–218. [Google Scholar] - You, R.J.; Lin, B.C. Building feature extraction from airborne Lidar data based on tensor voting algorithm. Photogramm. Eng. Remote Sens.
**2011**, 77, 1221–1231. [Google Scholar] [CrossRef] - Vosselman, G. Automated planimetric quality control in high accuracy airborne laser scanning surveys. ISPRS J. Photogramm. Remote Sens.
**2012**, 74, 90–100. [Google Scholar] [CrossRef] - Lagüela, S.; Díaz-Vilariño, L.; Armesto, J.; Arias, P. Non-destructive approach for the generation and thermal characterization of an as-built BIM. Constr. Build. Mater.
**2014**, 51, 55–61. [Google Scholar] [CrossRef] - Hoffman, R.; Jain, A.K. Segmentation and classification of range images. IEEE Trans. Pattern Anal. Mach. Intell.
**1987**, 9, 608–620. [Google Scholar] [CrossRef] [PubMed] - Filin, S. Surface clustering from airborne laser scanning data. Int. Arch. Photogramm. Remote Sens.
**2002**, 34, 119–124. [Google Scholar] - Filin, S.; Pfeifer, N. Segmentation of airborne laser scanning data using a slope adaptive neighborhood. ISPRS J. Photogramm. Remote Sens.
**2006**, 60, 71–80. [Google Scholar] [CrossRef] - Biosca, J.M.; Lerma, J.L. Unsupervised robust planar segmentation of terrestrial laser scanner point clouds based on fuzzy clustering methods. ISPRS J. Photogramm. Remote Sens.
**2008**, 63, 84–98. [Google Scholar] [CrossRef] - Dorninger, P.; Pfeifer, N. A comprehensive automated 3D approach for building extraction, reconstruction, and regularization from airborne laser scanning point clouds. Sensors
**2008**, 8, 7323–7343. [Google Scholar] [CrossRef] - Awrangjeb, M.; Fraser, C.S. Automatic segmentation of raw Lidar data for extraction of building roofs. Remote Sens.
**2014**, 6, 3716–3751. [Google Scholar] [CrossRef] - Fischler, M.A.; Bolles, R.C. Random sample consensus—A paradigm for model-fitting with applications to image-analysis and automated cartography. Commun. ACM
**1981**, 24, 381–395. [Google Scholar] [CrossRef] - Schnabel, R.; Wahl, R.; Klein, R. Efficient RANSAC for point-cloud shape detection. Comput. Graph. Forum
**2007**, 26, 214–226. [Google Scholar] [CrossRef] - Choi, S.; Kim, T.; Yu, W. Performance evaluation of RANSAC family. In Proceedings of the British Machine Vision Conference, London, UK, 7–10 September 2009.
- Frahm, J.-M.; Pollefeys, M. RANSAC for (Quasi-) degenerate data (QDEGSAC). In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York, NY, USA, 17–22 June 2006.
- Chum, O.R.; Matas, J.R.I. Matching with PROSAC-progressive sampling consensus. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Miami, FL, USA, 20–25 June 2005.
- Chum, O.; Matas, J. Optimal randomized RANSAC. IEEE Trans. Pattern Anal. Mach. Intell.
**2008**, 30, 1472–1482. [Google Scholar] [CrossRef] [PubMed] - Berkhin, P. A Survey of Clustering Data Mining Techniques; Springer Berlin Heidelberg: New York, NY, USA, 2006. [Google Scholar]
- Gallo, O.; Manduchi, R.; Rafii, A. CC-RANSAC: Fitting planes in the presence of multiple surfaces in range data. Pattern Recognit. Lett.
**2011**, 32, 403–410. [Google Scholar] [CrossRef] - Yan, J.X.; Shan, J.; Jiang, W.S. A global optimization approach to roof segmentation from airborne Lidar point clouds. ISPRS J. Photogramm. Remote Sens.
**2014**, 94, 183–193. [Google Scholar] [CrossRef] - Xiong, B.; Elberink, S.O.; Vosselman, G. A graph edit dictionary for correcting errors in roof topology graphs reconstructed from point clouds. ISPRS J. Photogramm. Remote Sens.
**2014**, 93, 227–242. [Google Scholar] [CrossRef] - Elberink, S.O.; Vosselman, G. Building reconstruction by target based graph matching on incomplete laser data: Analysis and limitations. Sensors
**2009**, 9, 6101–6118. [Google Scholar] [CrossRef] [PubMed] - Hesami, R.; BabHadiashar, A.; HosseinNezhad, R. Range segmentation of large building exteriors: A hierarchical robust approach. Comput. Vis. Image Underst.
**2010**, 114, 475–490. [Google Scholar] [CrossRef] - Torr, P.H.S.; Zisserman, A. Mlesac: A new robust estimator with application to estimating image geometry. Comput. Vis. Image Underst.
**2000**, 78, 138–156. [Google Scholar] [CrossRef] - Bretar, F.; Roux, M. Hybrid image segmentation using Lidar 3D planar primitives. In Proceedings of the ISPRS Workshop Laser Scanning, Enschede, The Netherlands, 12–14 September 2005.
- López-Fernández, L.; Lagüela, S.; Picón, I.; González-Aguilera, D. Large-scale automatic analysis and classification of roof surfaces for the installation of solar panels using a multi-sensor aerial platform. Remote Sens.
**2015**, 7, 11226–11248. [Google Scholar] [CrossRef] - Wang, C.; Sha, Y. A designed beta-hairpin forming peptide undergoes a consecutive stepwise process for self-assembly into nanofibrils. Protein Pept. Lett.
**2010**, 17, 410–415. [Google Scholar] [CrossRef] [PubMed] - Girardeau-Montaut, D. Detection de Changement sur des Données Géométriques 3D. Ph.D. Thesis, Télécom ParisTech, Paris, France, 2006. [Google Scholar]
- Awrangjeb, M.; Fraser, C.S. An automatic and threshold-free performance evaluation system for building extraction techniques from airborne Lidar data. IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens.
**2014**, 7, 4184–4198. [Google Scholar] [CrossRef] - Rutzinger, M.; Rottensteiner, F.; Pfeifer, N. A comparison of evaluation techniques for building extraction from airborne laser scanning. IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens.
**2009**, 2, 11–20. [Google Scholar] [CrossRef] - Perera, G.S.N.; Maas, H.G. Cycle graph analysis for 3D roof structure modeling: Concepts and performance. ISPRS J. Photogramm. Remote Sens.
**2014**, 93, 213–226. [Google Scholar] [CrossRef]

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## Share and Cite

**MDPI and ACS Style**

Xu, B.; Jiang, W.; Shan, J.; Zhang, J.; Li, L.
Investigation on the Weighted RANSAC Approaches for Building Roof Plane Segmentation from LiDAR Point Clouds. *Remote Sens.* **2016**, *8*, 5.
https://doi.org/10.3390/rs8010005

**AMA Style**

Xu B, Jiang W, Shan J, Zhang J, Li L.
Investigation on the Weighted RANSAC Approaches for Building Roof Plane Segmentation from LiDAR Point Clouds. *Remote Sensing*. 2016; 8(1):5.
https://doi.org/10.3390/rs8010005

**Chicago/Turabian Style**

Xu, Bo, Wanshou Jiang, Jie Shan, Jing Zhang, and Lelin Li.
2016. "Investigation on the Weighted RANSAC Approaches for Building Roof Plane Segmentation from LiDAR Point Clouds" *Remote Sensing* 8, no. 1: 5.
https://doi.org/10.3390/rs8010005