Airborne LiDAR Point Cloud Building Reconstruction Based on Planar Optimal Combination and Feature Line Constraints

Abstract

This paper proposes a building reconstruction framework for airborne LiDAR data to address the challenge of automated modeling under conditions of uneven point cloud density and missing vertical walls, generating high-precision and structurally compact 3D building models. The method first combines adaptive resolution hypervoxels with a global graph cut optimization strategy to extract precise roof plane primitives from sparse point clouds of buildings. Subsequently, it infers building facades and internal vertical walls based on point cloud projection contours and height change detection, thereby completing the wall structures commonly missing in airborne LiDAR data. Finally, a feature line constraint term is introduced into the hypothesis-and-selection-based reconstruction framework to guide the structural optimization of candidate planes, ensuring the reconstructed model closely matches the actual building geometry. The proposed method was evaluated on multiple public airborne LiDAR datasets, demonstrating its effectiveness through qualitative and quantitative comparisons with various state-of-the-art approaches.

1. Introduction

The digitization of urban scenes represents a significant research direction in fields such as computer vision, remote sensing, and computer graphics [1,2]. As a critical infrastructure for digital twin cities, 3D building models have been widely applied in numerous domains, including urban planning [3], navigation and positioning [4,5], disaster early warning [6], energy assessment [7], and environmental analysis [8].

Among various data sources, airborne LiDAR scanning offers distinct advantages: it efficiently captures three-dimensional structural information of building roofs over large areas, while its emitted pulses can penetrate vegetation or shadow-obscured areas, thereby supplementing building information that is difficult to detect and reconstruct using optical imagery [9]. Consequently, urban 3D building reconstruction based on airborne LiDAR point clouds has garnered extensive research attention [10,11,12,13,14]. However, existing technical methods still face significant challenges in achieving automated reconstruction of highly accurate and structurally compact building models. First, the overall distribution of airborne LiDAR point clouds is sparse and unevenly dense, complicating the precise extraction of roof plane primitives. Second, due to sensor field-of-view and flight path limitations, critical structural information, such as vertical building walls, is often incomplete or even absent in airborne LiDAR point clouds. Furthermore, the diverse forms and complex structures of urban buildings further increase the difficulty of model topological reconstruction.

To address these issues, this paper proposes a building reconstruction method based on planar optimization combinations and feature line constraints for automatically generating compact and effective 3D building models from airborne LiDAR point clouds. The method first converts the discrete point cloud into a voxel, then combines a global graph cut strategy to extract and optimize the roof plane primitives. Subsequently, projected contours are extracted from the building point cloud, and height change point pairs are detected to infer the building’s exterior walls and internal vertical walls. Finally, the method incorporates roof feature lines as structural prior into the hypothesis-and-selection-based reconstruction framework [15] to guide the screening and optimization of candidate surface sets, thereby improving topological accuracy and enhancing detail recovery. The main contributions of this paper are as follows:

• To address the issues of uneven density and unstable boundaries in airborne LiDAR point clouds, this study proposes a roof plane segmentation method combining adaptive resolution supervoxels with graph cut re-labeling. This approach enhances adaptability to density variations while effectively suppressing jagged boundaries and oversegmentation problems.
• Introduces a vertical plane inference strategy based on height change detection to effectively restore missing vertical wall information prevalent in airborne LiDAR data.
• Introduces a feature line constraint term within the hypothesis-and-selection-based reconstruction framework, encouraging the prioritization of planar combinations connected to roof feature lines to achieve reconstruction models that better align with actual building geometries.

2. Related Work

To date, a substantial number of building reconstruction approaches based on airborne LiDAR point clouds have been proposed. These approaches can be broadly categorized into data-driven, model-driven, hybrid-driven approaches, and deep learning–based approaches that have gained prominence in recent years.

2.1. Data-Driven Approaches

Data-driven approaches adopt a bottom-up strategy, typically beginning with planar detection methods (such as the Hough Transform [16], RANSAC [17,18,19], and region growing [20,21,22]) to extract initial planar primitives from airborne LiDAR point clouds. These primitives are then analyzed for their topological relationships to extract building geometric elements (e.g., edges and key vertices), which are subsequently assembled into building models.

Zhou and Neuman [23] pioneered a simplified approach based on 2.5D irregular triangular networks (TINs). Subsequent work [24] extended this by extracting minimum bounding boxes of roof segments to achieve topological control and global regularization. While this method can generate building models of arbitrary shapes, overall model quality remains an area for improvement. Sampath and Shan [25] employed a fuzzy k-means approach to cluster planar points into multiple planar fragments, reconstructing the building model through the intersection lines of these adjacent planar fragments. Chen et al. [26] proposed a multi-level LOD reconstruction framework. It first extracts roof planar primitives using an improved probability density clustering algorithm, then extracts and regularizes primitive boundaries based on Voronoi diagrams, and finally generates building models using a topology-aware strategy. Li et al. [11] introduced a point cloud segmentation strategy that combines TIN and region growing to extract initial planar primitives, then projects these primitives onto a 2D grid map. Subsequent refinement of roof regions and their boundaries ultimately generates structurally compact building models. Xie [27] proposed a rule-and-hypothesis-based reconstruction method, introducing pairwise, ternary, and proximity constraints to resolve ambiguous topological relationships and reconstruct building models. Huang [28] utilized point cloud height maps to infer missing building facade information from airborne LiDAR point clouds, generating final building models through an extended “hypothesis-selection” framework. Peters et al. [29] proposed a building reconstruction method based on line elements and graph cut optimization, maintaining high efficiency in large-scale scenes. However, it relies on classified point clouds and 2D building outlines as prior information, limiting its generalization capability in real-world scenarios.

Overall, data-driven methods effectively preserve architectural details and are not constrained by fixed geometric forms, making them theoretically applicable to modeling any building type. However, these approaches rely on the quality of the point cloud and are sensitive to local point density, noise, and data gaps.

2.2. Model-Driven Approaches

Model-driven approaches employ a top-down strategy, typically pre-constructing a model library containing typical roof structures (such as flat roofs, single-slope roofs, gable roofs, hipped roofs, and pyramid roofs). Based on the input point cloud, they search within this library for the optimal model combination and its parameters.

Haala and Brenner [30] defined four fundamental building primitives—hipped roof, flat roof, gabled roof, and shed roof—then reconstructed the final building model by combining these primitives and optimizing their parameters. Henn et al. [31] proposed a model-driven framework that first identifies roof types using Support Vector Machines (SVM), then robustly estimates geometric parameters for each roof model using the RANSAC algorithm. The “Manhattan World Hypothesis” [32,33,34] constrains building surface orientations to three mutually orthogonal principal directions, representing structures using multicubes aligned with coordinate axes. Addressing the high dependency of model-driven methods on precise roof plane segmentation, Zhang et al. [35] transformed the reconstruction task into an optimization problem. By minimizing the distance between point clouds and TIN models, they achieved a more robust generation of semantic building models. Li and Shan [36] proposed a multi-primitive reconstruction method. It employs a two-step RANSAC strategy for planar primitive segmentation, combined with holistic primitive fitting and 3D Boolean operations, to partition complex buildings into multiple predefined primitives, thereby generating regularized building models.

Overall, model-based approaches generate more regular building models with certain advantages in reconstruction efficiency. However, due to the difficulty of predefined templates covering all real-world building morphologies, these methods often face insufficient generalization capabilities when handling complex urban scenes.

2.3. Hybrid-Driven Approaches

Hybrid reconstruction methods aim to integrate data-driven and model-driven approaches to achieve superior reconstruction outcomes.

A common hybrid approach involves partitioning complex roofs into multiple subsets and matching each subset with a corresponding parametric model from a model library. For instance, Lin et al. [37] proposed a reconstruction method for low-rise buildings. This method first semantically classifies point clouds into distinct categories, then transforms the reconstruction problem into a multi-level assembly of small, structurally distinct blocks. Bizjak et al. [38] introduced a novel automatic reconstruction method based on a half-space representation. By partitioning buildings into sub-structures free of height discontinuities and modeling them separately, this approach enhances reconstruction accuracy for complex roof configurations. Another hybrid approach leverages the Roof Topology Graph (RTG) to transform the reconstruction problem into a matching task between the topology graph and defined models. For instance, Xiong et al. [39] decomposed the semantic structure of buildings based on RTG fundamental elements and physical semantics, then assembled complete building models through CSG-like subgraph pairing. Xu et al. [40] proposed a hierarchical roof topology tree (HRTT) representation method. By prioritizing robust adjacency relationships and integrating hypothesis-testing strategies with global constraints, they effectively mitigated RTG’s susceptibility to errors and structural expression limitations. Hu et al. [41] represent roof structural topology through roof attribute graphs and design a decomposition and refinement process based on Gestalt principles to achieve complete reconstruction of complex roof models.

Overall, hybrid-driven methods partially combine the advantages of data-driven and model-driven approaches but inevitably inherit their respective shortcomings. In practical applications, these methods still face challenges in computational complexity and automation levels.

2.4. Deep Learning–Based Approaches

In recent years, with the rapid advancement of deep learning technology, deep learning-based 3D building reconstruction methods have gradually gained attention. Chen et al. [42] proposed a building reconstruction method based on deep implicit fields, achieving high-fidelity and compact model reconstruction by representing building surfaces implicitly through neural networks. Chen et al. [43] decomposed building spaces into a set of polyhedra and utilized graph neural networks to learn the combinatorial relationships among polyhedral primitives, thereby achieving topologically closed and structurally compact building reconstruction. Liu et al. [44] proposed an autoregressive generative model that directly predicts vertex and face sequences of polygonal meshes from aerial LiDAR point clouds, thereby reducing error propagation and enhancing reconstruction fidelity.

Additionally, some studies recover building structures through wireframe reconstruction. For instance, Li et al. [45] decomposed the roof wireframe extraction task into vertex prediction and edge prediction subproblems, proposing an end-to-end neural network to directly predict roof model elements from point clouds. Jiang et al. [46] utilized graph neural networks to extract 3D building lines from airborne LiDAR point clouds, restoring structurally meaningful edges by incorporating corner information.

Overall, deep learning-based methods offer advantages in feature representation and model flexibility but typically rely on large amounts of labeled data and computational resources. They still face challenges in geometric detail, topological consistency, interpretability, and cross-scenario generalization capabilities.

3. Methodology

3.1. Overview

This method takes point clouds of individual buildings captured by airborne LiDAR as input and outputs a three-dimensional polygonal building surface model that satisfies watertight and 2-manifold properties. The current implementation exports results in the standard OBJ polygon mesh format. The overall workflow is illustrated in Figure 1.

Figure 1. Overall Methodology Flowchart. (a) Input point cloud; (b) Roof segmentation plane; (c) Roof feature lines; (d) Inference of vertical walls; (e) Hypothesis of candidate surface set; (f) Reconstructed model.

The method primarily consists of three stages: First, a planar segmentation strategy based on adaptive resolution voxels and globally optimized graph cuts is employed to extract roof plane primitives. Subsequently, building facades and internal vertical walls are inferred using projected contours and height change information from the point cloud. Finally, under the hypothesis-and-selection-based reconstruction framework, a hypothesis set of building surfaces is generated based on the extracted roof planes and inferred vertical planes. The optimal subset is then selected through optimization to produce the final model. In this process, a feature line constraint term is further introduced. Roof feature lines are generated by intersecting adjacent roof planes, and a matching relationship is established between candidate plane intersections and the feature lines, thereby guiding the optimization process to prioritize candidate plane intersections that match the feature lines and their corresponding plane combinations.

3.2. Roof Plane Segmentation

Due to the uneven density distribution of airborne LiDAR point clouds, traditional planar detection methods such as RANSAC and region growing are prone to oversegmentation and undersegmentation and struggle to preserve planar boundaries effectively. To extract complete and accurate planar primitives from building roof point clouds for subsequent reconstruction, this paper proposes a roof plane segmentation method combining adaptive resolution hypervoxels with global graph cut optimization: First, a structurally robust coarse segmentation is obtained via adaptive resolution hypervoxels. Then, hierarchical clustering enhances planar completeness. Finally, global graph cuts are applied to relabel point-level planar tags, correcting jagged boundaries and local oversegmentation. The overall workflow is illustrated in Figure 2.

Figure 2. Roof Planar Segmentation. (a) Input building point cloud; (b) Hypervoxel segmentation result; (c) Clustering and merging result; (d) Graph cut optimization result.

3.2.1. Adaptive Resolution Hypervoxel Coarse Segmentation

Given the disordered nature of point cloud data, this paper introduces a representative point-driven hypervoxel coarse segmentation method [47]. By incorporating local distance constraints and an adaptive balancing mechanism for representative point counts into the energy function, this approach enables hypervoxel resolution to automatically adjust based on local point density. This achieves boundary-friendly and structurally robust hypervoxel coarse segmentation, thereby reducing the risk of cross-plane undersegmentation. The energy function is defined as follows:

$$\min E(Z)={\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^N{z}_{ij}D\left(p_i,p_j\right)+\lambda | C(Z)-K |, s.t. z_{ij}=\{0,1\}, {\displaystyle \sum }_{i=1}^N{z}_{ij}=1,\forall j$$

(1)

where $z_{ij}$ indicates whether the neighboring point $p_j$ is represented by the representative point $p_i$; $D\left(p_i,p_j\right)$ represents the distance between the representative point and its neighboring points; $\lambda$ is the parameter balancing distance and voxel count; $C(Z)$ represents the number of representative points; $K$ denotes the preset voxel count.

3.2.2. Hierarchical Clustering Merge

As shown in Figure 2b, a roof surface is divided into multiple super-voxels of varying scales. Merging these super-voxels yields the complete roof surface. First, an adjacency graph $G=(V,E)$ is constructed among the super-voxels, where each node $v\in V$ represents a super-voxel and each edge $e\in E$ connects two adjacent super-voxels. Subsequently, hierarchical clustering is performed on this graph to iteratively merge hypervoxels. In each iteration, the edge $e(p,q)$ with the minimum mean squared error (MSE) is selected, merging its corresponding two hypervoxels into a new node while updating the graph structure. This process is repeated until the minimum MSE exceeds a preset threshold $T_m$, at which point the iterative aggregation process terminates. By progressively enhancing the integrity of roof planes through hierarchical clustering merging, more reliable initial planes are provided for subsequent global graph cut optimization.

3.2.3. Global Graph Cutting Optimization

As indicated by the red rectangle in Figure 2c, after voxel coarse segmentation and cluster merging, some roof surfaces still exhibit jagged boundaries and over-segmentation issues. This paper addresses these problems by performing point-level global relabeling on the clustered plane labels through modeling and solving a global energy minimization problem. The global energy function is defined as:

$$E(L)={\displaystyle \sum }_{p\in P}{D}_p\left(L_p\right)+{\displaystyle \sum }_{p,q\in N}{\delta }_{p,q}\left(L_p\ne L_q\right)$$

(2)

where data item $D_p\left(L_p\right)$ represents the geometric distance from point $p$ to plane $L_p$, ensuring that each point tends to be assigned to the plane with the shortest distance:

$$D_p\left(L_p\right)={\displaystyle \frac{\left|a_{L_P}{x}_p+b_{L_P}{y}_p+c_{L_P}{z}_p+d_{L_P}\right|}{\sqrt{{a_{L_P}}^2+{b_{L_P}}^2+{c_{L_P}}^2}}}$$

(3)

The plane equation corresponding to $L_p$ is $a_{L_P}{x}_p+b_{L_P}{y}_p+c_{L_P}{z}_p+d_{L_P}=0$.

The smoothing term ${\delta }_{p,q}\left(L_p,L_q\right)$ penalizes cases where adjacent points are assigned to different planes, thereby enforcing spatial continuity and eliminating jagged boundaries.

$${\delta }_{p,q}\left(L_p,L_q\right)=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} L_p=L_q\\ \exp (-\sqrt{{\left(x_p-x_q\right)}^2+{\left(y_p-y_q\right)}^2+{\left(z_p-z_q\right)}^2}) L_p\ne L_q\end{array}\right.$$

(4)

After constructing the optimization function, the $\alpha -\beta$ exchange technique based on multi-label cut minimizes the objective function $E(L)$, thereby obtaining an approximate global optimum solution. Figure 2d illustrates the optimization results, demonstrating that jagged boundaries and over-segmentation issues have been effectively resolved.

Through the aforementioned coarse-to-fine planar clustering and optimization workflow, the proposed method effectively mitigates issues such as oversegmentation, undersegmentation, and boundary jaggedness. This enhances the accuracy and completeness of aerial LiDAR point cloud planar segmentation, thereby providing a more reliable topological planar foundation for subsequent building reconstruction.

3.3. Vertical Wall Inference

In airborne LiDAR point clouds, due to the limited scanning field of view of the sensors, vertical building walls are often sparsely sampled or even completely absent. However, vertical walls play a crucial role in representing building structures, encompassing not only the main exterior walls connecting the roof to the ground but also internal vertical walls located between multi-level roofs. To address this, this paper combines projected contours and height change information from point clouds to jointly infer both internal and external vertical walls of buildings. The overall workflow is illustrated in Figure 3.

Figure 3. Vertical Plane Inference. (a) Input point cloud; (b) Alpha-shape outer boundary; (c) Regularized outer contour; (d) 2D footprint; (e) Height change points; (f) Inner wall segments; (g) Inner and outer wall segments; (h) Inner and outer vertical planes.

3.3.1. Vertical Exterior Wall Inference

First, the point cloud of the single building is projected onto a two-dimensional horizontal plane, and its initial outer boundary is extracted using the α-shape method, as shown in Figure 3b. Subsequently, segmented least-squares line fitting is performed on the boundary point sequence, and collinear points within a distance threshold are merged using a region-growing strategy to obtain an initial set of line segments. Building upon this, directional clustering is performed on the extracted line segments, and the orientations within each cluster are adjusted to the average direction. After completing the orientation adjustment, approximately collinear and approximately orthogonal line segments are further refined to be exactly collinear and orthogonal, respectively, ultimately yielding a regularized building outer footprint, as illustrated in Figure 3c.

Based on this regularized contour, the vertical exterior walls of the building can be obtained by extrapolating along the vertical direction. Additionally, the contour polygon represents the building’s two-dimensional footprint, as illustrated in Figure 3d. Therefore, this paper enables direct model reconstruction without requiring the building footprint as prior information.

3.3.2. Vertical Interior Wall Inference

In airborne LiDAR point clouds, interior vertical walls of buildings typically manifest as significant height changes. Based on this characteristic, height change point pairs are first detected from the original point clouds, and the spatial distribution of interior vertical walls is subsequently inferred. Let the input building point cloud be $P=\left\{p_i=\left(x_i,y_i,z_i\right)\right\}$. For any two points $p_i,p_j\in P$, their two-dimensional horizontal distance and height difference are defined as:

$$\begin{array}{c}{d}_{ij}=\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}\\ h_{ij}=\left|z_i-z_j\right|\end{array}$$

(5)

when the following conditions are satisfied, point $\left(p_i,p_j\right)$ is identified as a height change point pair:

$$d_{ij}<{\epsilon }_d,\hspace{1em}{h}_{ij}>{\epsilon }_h$$

(6)

where ${\epsilon }_d$ denotes the horizontal neighborhood search radius, and ${\epsilon }_h$ represents the height change threshold.

By performing a neighborhood search on the building point cloud under the aforementioned conditions, a set of height change point pairs $H$ can be obtained. Figure 3e illustrates the detection results: red points indicate upper-layer height change points, while blue points denote corresponding lower-layer height change points. Subsequently, the upper-layer height change points are projected onto a two-dimensional plane. Piecewise least-squares line fitting is then applied to extract two-dimensional line segments representing the positions of internal walls, as shown in Figure 3f. Considering that interior walls in building layouts are typically parallel or perpendicular to the outer footprint, an orientation regularization constraint is further imposed on interior wall segments to enforce parallelism or orthogonality with the principal directions of the projected outer footprint, as shown in Figure 3g.

Finally, by extrapolating the regularized outer contour and internal wall segments along the perpendicular direction, the building’s inner and outer vertical planes are obtained, as shown in Figure 3h. This method effectively supplements missing vertical wall information in airborne LiDAR point clouds, providing reliable planar primitives for subsequent closed reconstruction and topological optimization of building 3D models.

3.4. Building Model Reconstruction

This paper employs the hypothesis-and-selection-based framework for building model reconstruction: first, roof segmentation planes and internal vertical planes are extended in space and intersected pairwise to generate a set of hypothetical candidate surfaces (see Figure 1e); subsequently, an optimal subset is selected from this set, and the result is clipped by external vertical planes to obtain a closed building surface model. To further encourage the generation of physically plausible building models, this paper introduces a novel energy term. The optimization objective for the candidate surface patch set $F$ is defined as:

$$F=\arg \underset{X}{\min } {\lambda }_d{E}_d+{\lambda }_c{E}_c+{\lambda }_l{E}_l$$

(7)

where $X=\left\{x_i\mid x_i\in \{0,1\}\right\}$ is the binary selection variable for candidate surfaces. $E_d$ is the data fitting term, which encourages the selection of surface patches supported by more points; $E_c$ is the model complexity term, used to suppress overly complex surface combinations (see [15] for details). $E_l$ is the feature line constraint term proposed in this paper, designed to enhance the topological rationality of roof reconstruction at the structural level.

In complex architectural scenarios, the hypothesis-and-selection framework generates a vast number of candidate surfaces. Relying solely on the optimization process of and can easily lead to topological errors, resulting in surface combinations that “appear plausible but fail to match the actual roof geometry.” To address this, this paper establishes a matching relationship between candidate plane intersections and roof feature lines, leveraging the structural prior of roof feature lines to guide the screening and optimization of candidate planes.

3.4.1. Matching Relationship Construction

Let the set of candidate plane intersections be denoted as $e=\left\{e_1,e_2,\dots ,e_m\right\}$, and the set of roof feature lines as $l=\left\{l_1,l_2,\dots ,l_n\right\}$. As shown in Figure 4a, the roof feature lines are obtained by intersecting adjacent roof segmentation planes and are trimmed via point cloud projection to better approximate their actual lengths. This approach can partially reflect key geometric structures such as ridge lines and valley lines. The candidate plane intersection lines participating in matching are generated by extending the roof segmentation planes and intersecting them in pairs. These lines are then clipped by a vertical plane into segments of varying lengths, as shown in Figure 4b. Theoretically, the roof feature line should be a subset of the candidate plane intersection lines, being completely collinear with some candidates in space but differing only in length. Figure 4c shows the overlay effect of the two, where the green line represents the roof feature line. Based on this, if the candidate intersection line $e_i$ and feature line $l_k$ simultaneously satisfy the following joint conditions, $e_i$ is considered to match $l_k$:

$$e_i\sim l_k\hspace{1em}\iff \hspace{1em}\left(e_i\parallel l_k\right)\wedge \left({\displaystyle \frac{\left|e_i\cap l_k\right|}{\left|e_i\right|}}\ge {\tau }_o\right)$$

(8)

where $e_i\parallel l_k$ denotes that the two lines are collinear (i.e., share the same direction); ${\displaystyle \frac{\left|e_i\cap l_k\right|}{\left|e_i\right|}}$ represents the proportion of the candidate intersection line covered by the feature line; ${\tau }_o$ denotes the overlap rate threshold, set at 0.9 in this study’s experiments to ensure only high-confidence matches are retained.

Figure 4. Schematic diagram of roof feature lines and candidate plane intersections. (a) Roof feature lines; (b) Candidate plane intersections; (c) Overlay of both (green indicates roof feature lines).

3.4.2. Feature Line Constraint Item

$E_l$ is defined based on the aforementioned matching relationship and is used to constrain the selection process of candidate planes at the structural level:

$$E_l={\displaystyle \frac{1}{\left|E\right|}}{\displaystyle \sum }_{i=1}^{\left|E\right|}intersect\left(e_i\right)$$

(9)

where $\left|E\right|$ denotes the number of all pairwise intersecting surfaces within the candidate surface set; $intersect\left(e_i\right)$ is a binary indicator function denoting that an edge connects two intersecting surfaces $e_i$: if $e_i$ matches a specific roof feature line $l_k$, this indicator takes 0; otherwise, it takes 1. This term is designed to increase the likelihood of topologically correct roof plane selection by biasing the cost function toward plane combinations connected to roof feature lines, thereby reducing structural misselections caused by purely data-driven fitting and complexity control. It should be noted that this feature line constraint, as a soft constraint, does not enforce the selection of planes. Instead, it imposes a structural bias on the optimization process by reducing the corresponding energy cost. The final outcome remains jointly determined by the data fitting term and the model complexity term.

Figure 5 illustrates the difference in local reconstruction when the feature line constraint term $E_l$ is introduced. The roof appendage within the red rectangle is prone to being discarded when the feature line constraint is not applied, due to its limited support points on the segmentation plane and significantly increased model complexity. However, introducing $E_l$ preserves this appendage.

Figure 5. Effect of Feature Line Constraint $E_l$ on Reconstruction Results. (a) The input consists of segmented roof planes and the feature lines generated by their intersections; (b) Reconstruction result without $E_l$; (c) Reconstruction result with $E_l$ incorporated.

In summary, this paper introduces structural priors based on roof feature lines within the hypothesis-selection framework. This approach enables the reconstruction process to balance point cloud fitting with model simplicity while closely following critical structural edges such as ridges and valleys. Consequently, it yields architectural surface models that more accurately reflect the true geometric and topological relationships of the roof.

4. Experiments

4.1. Experimental Setup

The methodology of this study was implemented in C++ (C++17), with each module’s functionality developed using CGAL 5.4.1 and PCL 1.14.0 libraries. Experiments were conducted on a device equipped with an Intel Core i9-12900H processor at 2.50 GHz and 16 GB of memory.

4.1.1. Test Dataset

We tested the proposed method on three publicly available airborne LiDAR point cloud datasets for qualitative and quantitative evaluation.

• AHN3 [48]: AHN3 is a large-scale airborne LiDAR point cloud dataset covering the entire Netherlands, collected by the Dutch National Topographic Service with an average point density of 8 points/m². The dataset provides multi-class semantic labels, including buildings, trees, ground, and water bodies, enabling direct extraction of building point clouds.
• DALES [49]: DALES is a large-scale aerial point cloud dataset collected in Surrey, Canada. Data were acquired using a Riegl Q1560 dual-channel system at an altitude of approximately 1300 m, a speed of 72 m/s, and an average point density of 50 points/m². All buildings are provided with instance-level annotations.
• Vaihingen [50]: Vaihingen was released by the German Society for Photogrammetry and Remote Sensing (ISPRS) and collected in Vaihingen, Germany. The test area in this paper belongs to the benchmark data of the ISPRS “Urban Classification and 3D Building Reconstruction Test Project.” Data was acquired using a Leica ALS50 system with a 45° field of view, at an altitude of approximately 500 m, and an average point density of 4 points/m². Building point clouds can be directly obtained through the provided classification labels.

4.1.2. Evaluation Indicators

To quantitatively evaluate the proposed method in this paper, we examine it from two aspects: plane segmentation and model reconstruction. According to [51], the evaluation metrics for plane segmentation include: Completeness ($C_m$), Correctness ($C_r$), and Quality ($Q_l$), defined as follows:

$$\begin{array}{l}{C}_m={\displaystyle \frac{\left|TP\right|}{\left|TP\right|+\left|FN\right|}}\\ C_r={\displaystyle \frac{\left|TP\right|}{\left|TP\right|+\left|FP\right|}}\\ Q_l={\displaystyle \frac{\left|TP\right|}{\left|TP\right|+\left|FP\right|+\left|FN\right|}}\end{array}$$

(10)

where $TP$ (True Positive) refers to correctly segmented roof surfaces; $FN$ (False Negative) refers to missed roof surfaces; $FP$ (False Positive) refers to incorrectly segmented roof surfaces.

During the model reconstruction process, since no ground-truth reconstruction results are available for the buildings in the three datasets, the commonly used accuracy metric, Root Mean Square Error (RMSE), is employed to evaluate the quality of each reconstructed model. In the context of surface reconstruction, given a point cloud set $\mathcal{P}={\left\{p_i\right\}}_{i=1}^N$, RMSE is defined as the square root of the mean squared Euclidean distance from the points to the reconstructed model:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum }_{i=1}^N\parallel p_i-{\widehat{p}}_i{\parallel }^2}$$

(11)

where $p_i$ is the $i$-th point in the point cloud; ${\widehat{p}}_i$ is the projection point of point $p_i$ onto the surface of the reconstruction model; and $\parallel p_i-{\widehat{p}}_i\parallel$ is the Euclidean distance.

4.1.3. Parameter Settings

The method described in this paper involves several parameters whose values are determined based on the point cloud density of the dataset and experimental experience. During the planar segmentation stage, the mean squared error threshold $T_m$ for hypervoxel clustering and merging is set to 0.25. In the vertical plane inference stage, the height difference threshold ${\epsilon }_h$ for height change detection is set to 0.5 m. The horizontal neighborhood radius ${\epsilon }_d$ is adaptively adjusted based on the point cloud density of different datasets, set to twice the average inter-point distance. During the building model reconstruction stage, the feature line matching overlap rate ${\tau }_o$ is set to 0.9, and the weight ${\lambda }_l$ of the feature line constraint term is set to 0.08.

4.2. Planar Segmentation Results

To validate the effectiveness and robustness of the proposed planar segmentation algorithm, comparative experiments were conducted against three classical plane detection methods: the RANSAC algorithm [19], the Region Growing (RG) algorithm [21], and the Global Optimization (GO) algorithm [52]. Partial segmentation results of the four methods on different datasets are shown in Figure 6. The figure displays six representative building instances: the top two rows originate from the AHN3 dataset, the middle two rows from the DALES dataset, and the bottom two rows from the Vaihingen dataset. From left to right: (a) Input building point cloud, (b) Manually annotated reference segmentation result, (c) Segmentation result from the proposed method, (d) Segmentation result from the RANSAC method, (e) Segmentation result from the RG method, (f) Segmentation result from the GO method.

Figure 6. Roof plane segmentation results. (a) Input point cloud; (b) Manually annotated reference segmentation result; (c) Segmentation result from the proposed method; (d) Segmentation result from the RANSAC method; (e) Segmentation result from the RG method; (f) Segmentation result from the GO method.

Overall, the proposed method demonstrates significant advantages in segmentation completeness, boundary accuracy, and adaptability to complex roof structures. For instance, as observed in Figure 6d, while the RANSAC method generates relatively clear planar boundaries, it tends to produce oversegmentation in sparse point cloud regions. In Figure 6e, the RG method struggles to form regular and clear planar boundaries at different plane intersections. Also, it tends to exhibit oversegmentation or undersegmentation in complex roof structure areas. In Figure 6f, although the GO method achieves high overall segmentation accuracy with relatively smooth boundaries, it is prone to undersegmentation in regions with indistinct boundary features; simultaneously, oversegmentation still occurs in low-density point cloud areas.

In contrast, the proposed method maintains both segmentation accuracy and planar integrity under uneven point cloud density. This advantage stems from converting discrete point clouds into structurally stable hypervoxels, effectively mitigating risks of oversegmentation and undersegmentation. Concurrently, incorporating hierarchical clustering and global graph cut optimization progressively enhances planar integrity and boundary precision.

Table 1 presents quantitative evaluation results from three datasets, with 10 buildings selected from each dataset for statistical analysis. A comprehensive review of all evaluation metrics reveals that the proposed method outperforms other comparison algorithms across different datasets in terms of segmentation completeness, accuracy, and overall quality. This further validates the effectiveness and generalization capability of the proposed planar segmentation method in diverse architectural scenarios.

Table 1. Quantitative evaluation results of different plane segmentation methods. $C_m$, $C_r$ and $Q_l$ represent Completeness, Correctness and Quality defined in Equation (10).

Method	AHN3			DALES			Vaihingen
	${\mathit{C}}_{\mathit{m}}$	${\mathit{C}}_{\mathit{r}}$	${\mathit{Q}}_{\mathit{l}}$	${\mathit{C}}_{\mathit{m}}$	${\mathit{C}}_{\mathit{r}}$	${\mathit{Q}}_{\mathit{l}}$	${\mathit{C}}_{\mathit{m}}$	${\mathit{C}}_{\mathit{r}}$	${\mathit{Q}}_{\mathit{l}}$
Proposed method	96.7	95.1	94.3	94.3	94.9	90.7	93.6	91.3	89.9
RANSAC	92.6	89.9	89.0	89.4	87.6	86.7	85.2	84.3	74.7
RG	85.3	87.6	86.7	83.6	84.9	82.5	74.3	81.4	67.8
GO	94.2	92.5	91.4	93.1	92.2	89.8	92.7	90.7	89.1

To further analyze the impact of planar segmentation results on subsequent building reconstruction, the segmentation outputs from different algorithms were input into the reconstruction framework proposed in this paper to obtain corresponding reconstruction results, as shown in Figure 7.

Figure 7. Model reconstruction results using different planar segmentation methods. (a) Segmentation and reconstruction results of the proposed method; (b) Segmentation and reconstruction results of the RANSAC method; (c) Segmentation and reconstruction results of the RG method; (d) Segmentation and reconstruction results of the GO method.

As shown in Figure 7a, the reconstructed roof structure generated from the planar segmentation results based on the proposed method exhibits regular and reasonable features, indicating that this segmentation approach can provide accurate and reliable topological planar primitives for subsequent reconstruction processes. Within the red-boxed area in Figure 7b, RANSAC incorrectly classifies distinct roof surfaces with significant height differences as belonging to the same plane, resulting in topological errors in the reconstructed model. In Figure 7c, the RG method struggles to generate clear plane boundaries, resulting in numerous small, fragmented surfaces at the intersections of multiple roof planes in the reconstruction. Within the red-boxed area of Figure 7d, the GO method exhibits a small number of oversegmented and undersegmented plane primitives, causing partial roof structures to be reconstructed incorrectly.

In summary, the proposed planar segmentation algorithm effectively enhances segmentation accuracy while preserving planar integrity, providing stable and reliable topological planar primitives for subsequent model reconstruction.

4.3. Model Reconstruction Results

Figure 8a shows a large test area within the AHN3 dataset, where red points represent building point clouds. After manually cropping the building point clouds in this area into individual buildings, they were input into the reconstruction process proposed in this paper, yielding the corresponding reconstruction results shown in Figure 8b. It can be observed that for most buildings, the proposed method generates structurally complete and compact 3D building models. Within this test area, a total of 273 buildings were included, with an overall reconstruction time of approximately 17 min. The average reconstruction time per building was about 3.77 s. These results demonstrate that the method proposed in this paper exhibits good processing efficiency and scalability in large-scale scenarios.

Figure 8. Large-scale building model reconstruction results from the AHN3 dataset. (a) Input point cloud; (b) Building model.

To better illustrate model details, representative buildings from the three test datasets are selected and magnified in Figure 9. Despite issues such as uneven point cloud density and data gaps in the input, as well as significant variations in roof structures across different buildings, the proposed method successfully reconstructs visually complete and plausible 3D building models. This demonstrates the effectiveness of the method in extracting roof planes and inferring vertical planes from airborne LiDAR point clouds, as well as its ability to efficiently utilize these planar primitives during model reconstruction.

Figure 9. Reconstructed building models from different datasets. (1–6) from the AHN3 dataset; (7–12) from the DALES dataset; (13–16) from the Vaihingen dataset.

The quantitative evaluation results for the reconstruction models of each building in Figure 9 are shown in Table 2. The results show that the proposed method achieves high reconstruction accuracy across all test buildings, with RMSE values ranging from 0.06 m to 0.25 m. This improvement can be attributed to the proposed feature line constraint term, which effectively guides the selection of model planes, thereby enhancing reconstruction accuracy and enabling finer model details to better fit the input point clouds. In terms of reconstruction efficiency, the proposed method also performs well. Except for a few buildings with extremely complex roof structures, the reconstruction time for most building models is kept under 1 min. Furthermore, compared to traditional dense triangular mesh reconstruction methods [53], the proposed method generates simplified polygonal models with fewer faces and more compact topology, making it more suitable for large-scale urban modeling applications.

Table 2. Quantitative Evaluation Results of Reconstruction Models for Buildings in Figure 9. The report includes the number of input point clouds for each building, the number of faces in the reconstructed model, the fitting error, and the running time.

Dataset	Model	Points	Faces	RMSE (m)	Time (s)
AHN3	(1)	1713	56	0.07	14
	(2)	9286	112	0.11	63
	(3)	15,309	764	0.19	316
	(4)	1455	58	0.06	9
	(5)	2980	94	0.14	19
	(6)	2300	130	0.08	26
DALES	(7)	6548	134	0.06	13
	(8)	15,126	218	0.14	49
	(9)	18,319	237	0.22	121
	(10)	8851	107	0.12	33
	(11)	6749	235	0.21	78
	(12)	13,528	196	0.13	38
Vaihingen	(13)	7759	139	0.22	22
	(14)	11,927	176	0.19	47
	(15)	9479	86	0.25	56
	(16)	1019	98	0.19	22
	(17)	1742	82	0.17	38
	(18)	755	117	0.21	13

To further validate the effectiveness of the proposed reconstruction method, comparative experiments were conducted on three datasets—AHN3, DALES, and Vaihingen—against two state-of-the-art building reconstruction methods: City3D [28] and Geoflow [29]. Visual comparison results of some reconstructed models are shown in Figure 10. The comparison reveals that the model generated by our method demonstrates the highest structural integrity and regularity. In contrast, City3D’s reconstruction results exhibit deficiencies in detail recovery and are prone to topological errors in areas where multiple roof surfaces intersect. Geoflow’s reconstruction outcomes are significantly influenced by the quality of the input point cloud, struggling to reconstruct regular roof structures when the point cloud is sparse.

Figure 10. Comparison of Reconstruction Results from Different Methods: (a) Input point cloud; (b) Reconstruction result from the proposed method; (c) Reconstruction result from the City3D method; (d) Reconstruction result from the Geoflow method; (e) Reference reconstruction result.

Table 3 presents quantitative comparisons of the three methods across metrics, including model polygon count, reconstruction time, and RMSE. It is evident that while Geoflow demonstrates certain advantages in terms of polygon count and runtime, its reconstruction accuracy is significantly lower than that of the other two methods. The proposed method exhibits comparable computational efficiency to City3D, yet generates models with fewer polygons and more compact structures, achieving the highest reconstruction accuracy among the three approaches.

Table 3. Quantitative Evaluation Statistics for Reconstruction Comparison Methods. The report includes the total number of faces, average error, and running time.

Dataset	Method	Faces	RMSE (m)	Time (s)
AHN3	Ours	311	0.13	91
	City3D	486	0.16	84
	Geoflow	209	0.22	24
DALES	Ours	419	0.16	133
	City3D	1563	0.21	129
	Geoflow	312	0.25	30
Vaihingen	Ours	186	0.15	73
	City3D	497	0.24	109
	Geoflow	147	0.29	27

Overall, both cross-dataset experimental results and multi-method comparative analyses demonstrate that the proposed method exhibits comprehensive advantages in terms of structural integrity, topological accuracy, and model compactness. It provides robust and reliable technical support for 3D reconstruction of urban buildings based on airborne LiDAR point clouds.

4.4. Discussion

As demonstrated by the aforementioned experimental results, when the roof point cloud is relatively complete, the proposed method can generate structurally sound and topologically consistent building models. However, in airborne LiDAR data, roof point clouds often exhibit various forms of missing data due to factors such as tree occlusion, shielding by adjacent high-rise buildings, or limited observation angles—a common occurrence in complex urban environments.

As illustrated in Figure 11a, missing data in the roof point cloud primarily occurs along building boundaries, resulting in incomplete roof outline information. Since our method relies on the roof outline to infer the overall building structure, damage to the outline makes it difficult to accurately reconstruct the building’s entire boundary, leading to missing data in localized areas of the reconstructed model.

Figure 11. Reconstruction Results and Topological Error Examples in Different Roof-Missing Scenarios. (a) Missing roof boundary points; (b) Cavities inside the roof; (c) Ring-shaped building.

For cases with cavities inside the roof (Figure 11b), since the outer contour remains intact and the roof plane extends and intersects in space within the “assume-select” framework, such cavities can be partially filled. However, the filled results may deviate significantly from the actual structure. Furthermore, when the input point cloud represents a ring-shaped building (Figure 11c), the extended roof plane may fill what should be an internal void, introducing topological errors.

The reconstruction framework presented in this paper utilizes only planar primitives, which suffices for the reconstruction needs of most urban buildings. However, in real-world scenarios, buildings with curved surfaces still exist, such as those indicated by the red rectangular box in Figure 12. Under this framework, such structures can only be approximated as segmented planar geometries.

Figure 12. Example of Segmented Planar Approximation Reconstruction Results for Curved Surfaces. (a) Input point cloud; (b) Reconstructed model.

Finally, it should be noted that the method described in this paper focuses primarily on reconstructing roof structures. Vertical facades are inferred from the three-dimensional points of the roof, and their positions may not perfectly align with the actual walls. As shown in Figure 13, for buildings with eaves, the reconstructed exterior walls of the model exhibit a certain degree of deviation from their true locations.

Figure 13. Example of exterior wall position deviation caused by eave structure. (a) Input point cloud; (b) Reconstructed model.

5. Conclusions

This paper proposes a method for generating closed, two-dimensional manifold-based three-dimensional polygonal building surface models from airborne LiDAR point clouds of individual structures. Overall, the main contributions of this work are threefold: (1) A planar segmentation method combining adaptive hypervoxels with graph cut optimization is introduced, which maintains both segmentation accuracy and planar integrity even under point cloud density variations; (2) It constructs a vertical plane inference strategy based on point cloud projection contours and height change detection, effectively completing the vertically oriented wall structures commonly missing in airborne LiDAR data; (3) It introduces feature line constraints within a “hypothesis-selection” reconstruction framework, enabling structured prior guidance for candidate plane selection. Experimental results on three publicly available airborne LiDAR point cloud datasets demonstrate that the proposed method outperforms existing approaches in model integrity, structural regularity, and reconstruction accuracy.

However, the proposed method still has room for improvement: when significant gaps exist in the roof point cloud, the completeness and topological correctness of the reconstruction results remain compromised. Additionally, the current reconstruction framework primarily relies on planar primitives, limiting its expressive power for surfaces or free-form structures. Future work will explore utilizing deep learning methods to complete missing roof point clouds and further extend the reconstruction primitives to include curved or hybrid primitive representations. This will enhance the method’s applicability and expressiveness in complex urban environments.