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2020,  2 (3):   261 - 275   Published Date:2020-6-20

DOI: 10.1016/j.vrih.2020.05.004
1 Introduction2 Methods 2.1 The visual perception rules 2.2 Construction of roof structure graph (RSG) 2.3 Generation of hierarchical roof structures 2.4 Intelligent perception visualization of RSG models 3 Results and discussion 3.1 Results of roof plane segmentation 3.2 Structured reconstruction results 3.3 Comparison and evaluation 4 Conclusion

Abstract

Background
Three-dimensional (3D) building models with unambiguous roof plane geometry parameters, roof structure units, and linked topology provide essential data for many applications related to human activities in urban environments. The task of 3D reconstruction from point clouds is still in the development phase, especially the recognition and interpretation of roof topological structures.
Methods
This study proposes a novel visual perception-based approach to automatically decompose and reconstruct building point clouds into meaningful and simple parametric structures, while the associated mutual relationships between the roof plane geometry and roof structure units are expressed by a hierarchical topology tree. First, a roof plane extraction is performed by a multi-label graph cut energy optimization framework and a roof structure graph (RSG) model is then constructed to describe the roof topological geometry with common adjacency, symmetry, and convexity rules. Moreover, a progressive roof decomposition and refinement are performed, generating a hierarchical representation of the 3D roof structure models. Finally, a visual plane fitted residual or area constraint process is adopted to generate the RSG model with different levels of details.
Results
Two airborne laser scanning datasets with different point densities and roof styles were tested, and the performance evaluation metrics were obtained by International Society for Photogrammetry and Remote Sensing, achieving a correctness and accuracy of 97.7% and 0.29m, respectively.
Conclusions
The standardized assessment results demonstrate the effectiveness and robustness of the proposed approach, showing its ability to generate a variety of structural models, even with missing data.

Content

1 Introduction
Buildings are a dominant type of man-made object and 3D building models are required to visualize and analyze different applications in urban environments (e.g., visibility analysis, solar potential estimation, and infrastructure planning)[1,2,3]. Due to the vast range of applications, the reconstruction of 3D building models have long been an active topic in many research fields, such as computer vision and photogrammetry. With the rapid development of sensor technology, point clouds are easily obtained and have become the third type of standard geographical data, after maps and images[4,5]. These huge point clouds with high accuracy can be used to describe the real world and render it possible to reconstruct models of 3D buildings in a large area. In recent decades, a large number of algorithms and systems have been proposed to reconstruct building models and promising progress has been made with the associated automation[6,7,8], reconstruction scale[9,10,11,12], and output models[13,14,15,16]. However, the reliable and automatic generation of detailed building models, especially for the roof structures and topology, remains a challenging issue because of the input data quality and complexity of the reconstructed structures[17,18,19,20,21]. Furthermore, the methods used to reconstruct the 3D building models with unambiguous geometry-structure-topology, which can meet the present requirements for visualization and spatial computing, are still in the development phase.
Among the huge variety of reconstruction methods proposed in the literature, we can identify two basic methods: model-driven and data-driven. Interested readers are referred to the previous studies of Musialski et al. and Wang et al.[18,22]. The model-driven approaches search the most appropriate models with the best-fit point cloud[23,24,25] and a predefined model library should be established to generate a number of simple roof blocks in the early stages. Recently, these methods have focused on the optimization and reconstruction of the model libraries. Rychard et al. proposed a novel approach that can automatically interpret the roof structural units using the reconstructed building model database and thus avoid multiple matching of the same roof plane[15]. It is sufficiently robust to create a simple and watertight building model with prior scene knowledge such as parallelism and symmetry, but it is difficult to generate a satisfactory result for various building styles using the limited predefined building model library.
Most of the early data-driven methods[10,26,27,28,29] extract the roof planes and then reconstruct an assumed polyhedral building model using the intersecting information and regularization rules. Applying these methods, the flat Manhattan buildings are always reconstructed into mesh models using the airborne laser scanning (ALS) point cloud[30,31,32,33]. Considering the various assumptions, Lin et al. have successfully achieved the componentized model with the terrain laser scanning point cloud[10]. In general, these approaches are prone to errors because of the incomplete point clouds caused by occlusions and complex roof styles, and the incorrect topologies between roof elements. To improve the roof topology, many methods have been proposed to generate precise and reliable planes[34,35,36,37], but incorrect segmentation often occurs in the areas of intersection and transition with sparse points. To generate precise 3D building models, roof topology is gradually included to identify the roof features and track its reliability[28,38,39]. The semantic features are introduced to building reconstruction using machine learning-based methods[40,41,42]. To achieve semantically rich 3D models, support vector machines are proposed to extract the main structural components such as floors, ceilings, and roofs, among others[41]. The proposed semantic labeling of the objects is based on features, which encode both local (such as surface area, orientation, dimensions) and contextual (such as normal similarity, coplanarity, parallelism, proximity) information, achieving both high precision and recall in a variety of existing buildings. In addition, a deep learning-based approach has been developed to reconstruct the lightweight building models with Level of Detail 2 (LOD2), which can handle significant amounts of missing data in the roof area through the edge-aware resampling algorithm[43]. The main advantage of the data-driven approach is that it can reconstruct polyhedral buildings with complex shapes, while the drawback is the sensitivity to incomplete input data. Furthermore, it cannot interpret the semantic structures of the building and maintain valid topology.
The objective of this study is to generate structural building models from a raw ALS point cloud. Our method establishes a hierarchical topology tree among the generated structures and reconstructs its unambiguous geometry and topology. The paper is presented as follows. After the introduction, the proposed method is described in Section 2. Experimental results, including the quality assessment by the International Society for Photogrammetry and Remote Sensing (ISPRS), are analyzed and discussed in Section 3, followed by the conclusion and future work in Section 4.
2 Methods
The proposed structural reconstruction scheme encompasses the following key components: the basics of the visual perception rules (Section 2.1), construction of a roof structure graph using the optimized planes and perception rules (Section 2.2), generation of a hierarchical roof structure (Section 2.3), and generation of different roof structure graph (RSG) models using intelligent perception (Section 2.4). An illustration of the proposed building reconstruction method is given in Figure 1.
2.1 The visual perception rules
The recognition rules for the human visual system have long been studied and are summarized by the well-known Gestalt principles, which are a set of laws arising from 1920s psychology. The Gestalt principles describe how humans typically see objects by grouping similar elements, recognizing patterns, and simplifying complex images[44,45,46]. This suggests that each complex building can be divided into different simple roof structures that conform to the basic geometry and visual perception rules with each roof structure composed of various parameterized planes. Similar ideas can also be found in previous studies[10,15]. Therefore, a hierarchical-tree-based representation was introduced for a building structural model, as shown in Figure 2. The root of the hierarchy-tree is a whole structural model, while the child and leaf nodes are structural roof units and planes, respectively.
A building roof structure in a hierarchical tree is an unambiguous meaning box, which contains considerable prior knowledge with regard to artificial design. Therefore, the visual perception rules derived from the well-known Gestalt rules are introduced and listed as follows:
Rule-1: Proximity (F d ) — planes that are closer together can be grouped. We defined two planes as adjacent if the associated boundaries have an intersection line with a certain length. Its connection weight can be set as a function of the Euclidian distance.
Rule-2: Similarity (F con ) — planes that share visual characteristics, such as shape and convexity/concavity, can form a perceptive group. A visual and vivid image for the similarity rule is illustrated in Figure 3.
As psycho-physical studies suggest that the lowest level decomposition of objects into parts is closely intertwined with 3D concave/convex relationships, which can be conducive to calculate the similarity by
F c o n = F c o n v e x = θ 1 < θ 2 n 1 d > n 2 d F c o n c a v e = θ 1 > θ 2 n 1 d < n 2 d
where the normal and centroid of two planes are
n 1 ,   n 2
and
x 1 ,   x 2
,
d = x 1 - x 2
.
Rule-3: Continuity (F cc ) — continuous concavity/convexity of shapes is established as a preference. Such shapes will be aligned as one group. As presented in Figure 4, the adjacent planes A and B can be grouped only if the following are satisfied: (a) planes A and B are linked with a convex edge as described in the similarity rule; (b) a shared plane S exists between current planes ( A-S , B-S ) or a neighboring group of planes ( C-S, B-S ). The similarity F con between the two pairs should be the same.
Rule-4: Closure (F o ) — whole closed structures can be obtained by filling in missing data, which is mainly used to generate a meaningful and attractive visualization model.
2.2 Construction of roof structure graph (RSG)
To achieve the building structural model and its topology, an RSG node is proposed. The RSG, as illustrated in Figure 5, is a logical model of building structures and planes, which can be represented as the undirected weighted graph C . Both the roof structures and planar primitives in a hierarchical structural tree (Figure 1) can be expressed by the RSG .
In general, the RSG is firstly constructed by roof planes as presented in Figure 5a and Figure 5b. Each vertex in C is a roof planar primitive, while an edge between two vertices represents that the two roof primitives are spatially connected. Especially, the proximity and similarity rules are also marked with the adjacent edges. Furthermore, each structure in Figure 5c and Figure 5d is a set of grouped roof planes that can be further enhanced to become a meaningful convex box using the visual perception rules.
To generate an RSG for roof planes, the roof planar primitives will first be extracted and then used to construct the RSG nodes. Thus, a multi-label graph energy optimization framework was proposed to obtain accurate roof planes. Different from the most common locally used methods, such as RANSAC and Hough. Transform, the energy-based method is a global solution that formulates the segmentation as an optimization problem. Taking the ALS point cloud as input, initial over-segmented roof planes ({L p}) are first generated and then optimized by assigning a point (Data) to the best-matched input plane (Label). The initial planes can be extracted by a simple region growing randomly sampled minimum subsets of points and other over-segmented methods. The plane segmentation model, as Equation (2), utilizes different energy costs to balance the geometric errors (data cost), spatial coherence (smooth cost), and number of planes (label cost).
E L = p P t s D p D a t a C o s t + p , q T i n E d g e ω p q δ S m o o t h C o s t + l L p L l L a b e l C o s t
The data cost D (p) is used to measure the sum of the geometric errors using a quadratic perpendicular deviation between a point p and plane lp . The smooth cost is a prior smoothness and assumes some specific neighborhood edge for the point pairs with the neighborhood system based on 3D Triangulated Irregular Networks. A closer point pair is a priori more likely to fit the same plane, thus, the Potts model and predefined spatial distance monotone decreasing model between adjacent points are set to the indicator function
δ ( · )
and weight
ω p q
, respectively. The label cost L (l) is used to penalize the number of planes (Labels) and fewer planes are encouraged to represent the point cloud compactly. Thus, it can be set to a function of plane inlier points (|l p |). The final optimization framework for assigning points-to-planes can be determined as follows:
E L = p P t s p - L p 2 + p ,   q T i n E d g e e - p - q 2 δ l p l q + l p L p e - l p
The proposed planes optimization is an iterative process which can be solved by the ɑ-expansion algorithm[47,48]. It can achieve satisfactory plane segmentation results even with noisy, incomplete data, and poor topology in the transition region. The final generated roof planes for RSG nodes can be fitted with the optimized labels. In addition, the proximity and similarity rules are calculated for each RSG edge.
2.3 Generation of hierarchical roof structures
To achieve the roof structures and generate their hierarchical tree, a progressive iterative decomposition of the RSG is introduced, as illustrated in Figure 6. We decompose the RSG in each iteration and update the hierarchical structure tree with the extracted planes and eventually reach a plausible state. For each iteration, the aim is to find and search the best set of roof planes that potentially includes a common roof structure, which conforms to the visual perception rules of proximity, similarity, and continuity.
Moreover, an enhancement is performed to clarify incomplete roof structures caused by acquisition devices and scene occlusions. It covers an adjacent plane symmetry and a closed structure hull inspection, and finally constructs a closed convex hull polyhedron. The adjacent plane symmetry operator is to rotate one plane relative to the other and is performed to determine whether the projected normal vectors are mutually parallel with respect to the intersection line of the adjacent planes. While the closed structure hull inspection is an add and union primitive operation on the concave loop by stitching the projected plane primitives together in a sequence. These refinements are performed on each grouped structure and RSG nodes are synchronously updated, generating a meaningful roof structure and hierarchical topology, as shown in Figure 7.
2.4 Intelligent perception visualization of RSG models
In this section, a visually constrained process is introduced to generate RSG models with a reasonable level of detail. This operation is performed by one of the two key visual factors, namely, plane fitted residuals and areas. The changes in the RSG node are inversely proportional to one factor while the other is fixed. Therefore, the visual parameters in different urban applications, for example the calculation of a building stress field, the simulation analysis of wind pressure, or the buildings style transformation, are used to complete the plane merge and delete operations for each RSG node (roof plane and structure), resulting in RSG models with different levels of detail (LOD). This process is conducted within each independent roof structure and the key to this operation is to preserve the roof structure and topology. An example of different RSG models under the plane area constraint is illustrated in Figure 8.
A roof dormer structure, as in Figure 8a, usually contains three slanted planes. The traditional transfor-mation by CityGML will delete the two smaller planes, resulting in an isolated roof plane that requires manual interactive editing. In addition, the building topology and structure are always ignored. Thus, the proposed operation is executed on each roof structure and the roof hierarchical topology and RSG models with different levels of detail can be automatically constructed as shown in Figure 8.
3 Results and discussion
We have implemented the proposed algorithms and tested them mainly on two datasets that differ in point density and urban characteristics. Various internal consistency metrics[49], such as accuracy and quality, are used to evaluate the generated 3D building structural models. These ALS datasets were collected from the city of Vaihingen in 2008. Area 1 has 37 historic buildings with different normalization orientations, irregular outside boundaries, and large windows covering the main structure. Area 3 is a purely residential area that contains 56 buildings with small superstructures. The coverage areas are 120×200 and 150×220m2, respectively; an overview of these datasets is shown in Figure 9.
3.1 Results of roof plane segmentation
To ensure the adequacy of candidate planes for the plane optimization model, a plane fitting operation by randomly sampling at least four points was introduced and the evaluation was performed on a per-roof plane level by ISPRS. The proposed method achieved 201 correctly reconstructed roof planes out of 202 that were fully reconstructed in Area 1, while 130 were correctly reconstructed for Area 3. Moreover, the correctness of the roof plane reconstruction in Areas 1 and 3 was 99.5% and 97.7%, respectively. Interestingly, the most common reason for false roof reconstruction is the insufficient number of points which is caused by the unsuccessful preprocessing of the building extraction. The extracted planes shown in Figure 10 will be used to construct the RSG.
Figure 10 shows that the global energy-optimized approach can be effective to extract roof planes. Compared with a traditional multi-model fitting method like RANSAC, the proposed approach can overcome inconsistencies such as noise and missing data between adjacent planes, and is more beneficial to construct the adjacent relationship between roof planes.
3.2 Structured reconstruction results
The final structured models are illustrated in Figure 11 and the heights of the building models are extracted from the ground point cloud. Since the visual perception rules of a roof structure are the only assumptions, the proposed approach can model a variety of complex buildings and deal with missing data.
The 37 buildings in Area 1 and 56 buildings in Area 3 were successfully reconstructed, while the number of correctly reconstructed roof planes for Areas 1 and 3 is 130 and 201, respectively, on a per-roof plane level. The dominant structures contain roof planes whose mutual covered area is greater than 97.5%, based on the ISPRS reference data, and the approximate number of planes located in the 2.5% area of overlap on a per-roof area level is 31 and 21, respectively. The main reason is that the missed input data that results from the building point cloud detection errors can easily lead to a degree of unsuccessful small patch modeling. The proposed building structural reconstruction is not sensitive to the failed segmentation since it is highly regularized by various visual perception rules. Furthermore, a visualized perception area factor, calculated by the map scale and human visual resolution, was used to generate various RSG models. The human visual resolution is always set to 0.2mm, while the map scale is usually determined by the degree of detail of the contents. For example, a map scale of 1:25000 will lead to a 5m identification accuracy through the human eye. The initial RSG model in LOD2 will be gradually simplified into the LOD1 model, as shown in Figure 12.
3.3 Comparison and evaluation
The reconstructed structural models shown in Figure 11 were assessed by the topology correctness and model precision, and compared to four state-of-the-art algorithms by the ISPRS. The assessment of topological consistency achieved from ISPRS is a process based on a mutual overlap between the reference and reconstructed, which is an expansion of the method presented by Rutzinger et al.[49]. The evaluation results on topology correctness were obtained on a per-roof plane level with a complete correctness for the topological reconstruction of approximately 66.8% (135 roof instances out of 202) and 60.15% (80 roof planes in 133) in Areas 1 and 3, respectively. A reasonable approach to overcome the limitation of topological differences is to improve the preprocessing step in the building points extraction, which is beyond the scope of this study.
For the model precision, the quantitative metrics on a per-area level (pixel size: 0.100m) is evaluated based on the mutual overlapping with reference data. These metrics, completeness, correctness, level error, and height error, for example, have been defined by Rutzinger et al.[49], and compared to the four state-of-the-art methods, as listed in Table 1.
Comparison of model precision with different metrics
Methods Completeness (%) Correctness (%) Level error (m) Height error (m)
Area 1 CKU 86.8 98.9 0.9 0.6
ITCX3 89.2 96.4 0.8 0.2
YOR 88.2 98.5 0.8 0.3
TUD2 73.3 100 0.8 0.2
Proposed 92.4 98.8 0.6 0.3
Area 3 CKU 81.3 98.4 0.8 0.6
ITCX3 88.1 88.2 0.7 0.1
YOR 84.7 100 0.6 0.2
TUD2 73.6 100 0.5 0.1
Proposed 85.5 97.7 0.5 0.2
Notes: the method name used is the same as that in the description on the ISPRS website, where both evaluation results and methods can be easily found. CKU[50], ITCX3[8], TUD2[26], and YOR[51] are the abbreviated method names.
Taking the measured completeness metrics in Table 1 as an example, approximately 7.6% (Area 1) of roof planes have been missed for construction, while 14.5% of those in Area 3 are missing. The main reason is that the missed input data resulting from errors in the detection of building point clouds can easily lead to a number of small unsuccessful extracted patches. Additionally, an excellent reconstruction algorithm is used to find a balance between completeness and correctness, and four state-of-the-art methods from the ISPRS website were also compared and visualized to check our balance, as shown in Figure 13.
From the comparison metrics, the completeness and correctness indicators are similar to the median, which means that we have achieved a balance for the constructed model. Compared to the four state-of-art methods, the average errors of the level and height are approximately equal, which indicates that the proposed approach has high geometric accuracy. Moreover, the clear difference between the proposed structural modeling approach and others is that the structures and hierarchical topology tree can be simultaneously reconstructed in this study. An evaluation with a visual inspection for the reconstructed models is conducted with two state-of-the-art reconstructed algorithms, as illustrated in Figure 14.
Figure 14 shows that the different models generated from the same building demonstrate the effectiveness of the proposed approach, which can produce more unambiguous and meaningful roof structures. The reconstructed model by Verma et al. is based on searching and matching from a fixed building model database and forms a whole building model[52]. The minimum cycle matching approach by Xiong et al. is a well-known roof graph matching method, which can result in redundancy because the roof planes and their associated intersection lines may match multiple targets[8]. These generated sub-models can be assembled by the constructive solid geometry model, but cannot be interpreted as meaningful roof structures. In addition, these two method types are hindered by the limited number of building model styles in a predefined library. Furthermore, the specific hierarchical structural tree generated by the proposed approach has unambiguous planes, structures, and topology relationships, as shown in Figure 15.
4 Conclusion
In this paper, we present a novel approach for roof structural modeling from the ALS point cloud based on the RSG and the visual perception rules. The output of the building reconstruction is a visually pleasing structural model and the consistent geometry-structure-topology is organized by a hierarchical topology tree. The key idea is to reconstruct the potential roof structures through the progressive RSG decomposition and enhancement, while an analysis-and-enhancement scheme is developed to improve the basic structure best fit to human vision constraints. The procedures assessed by ISPRS have proven the effectiveness of building structural models. However, the proposed solution also has limitations, which include missing neighboring planes, non-planar objects segmentation, and detailed semantics interpretation of the roof planes or the structures. In our future work, we will reconstruct detailed 3D building models by using the improved point clouds obtained from different platforms, with a mobile or ground laser scanner, for example, and extending the plane segmentation approach to the urban scene with multiple geometric primitives, such as planes, spheres, and cones.

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