Delaunay Canopy: Building Wireframe Reconstruction from Airborne LiDAR Point Clouds via Delaunay Graph¶

Conference: ECCV 2026
Paper: ECCV Official
Area: Autonomous Driving
Keywords: Airborne LiDAR / Building Wireframe Reconstruction / Delaunay Graph / Geometric Prior / Curvature Guidance

TL;DR¶

Delaunay Canopy constructs a Delaunay graph over airborne LiDAR point clouds to compute face normals, edge dihedral angles, and vertex corner scores as explicit curvature signatures, establishing an adaptive 3D search space for precise corner and wire selection while overcoming the search explosion of 3D baselines and the loss of internal corners in 2D heightmap projections.

Background & Motivation¶

Acquiring 3D building models from airborne LiDAR point clouds provides fundamental structural assets for autonomous driving HD maps, smart cities, metaverse platforms, and robotics simulation. Compared with bulky dense meshes or unstructured raw point clouds, wireframes offer a compact, lightweight, and topology-centric representation that directly captures structural boundaries and roof configurations. However, airborne LiDAR scans suffer from inherent sparsity, irregular spatial distribution, and measurement noise. Because sharp geometric corners and roof-ridge topological connections are acutely sensitive to missing observations and noise, robustly recovering structural wireframes from raw scans remains an open challenge.

Prior learning-based wireframe reconstruction methods, such as Point2Roof and PC2WF, decompose the task into a sequential two-stage pipeline: first detecting 3D corners and then predicting connecting wires. However, conducting an unconstrained search across large-scale building point clouds that are dominated by featureless planar points creates an excessively large search space, inevitably producing spurious corner detections and topological false positives. To constrain this search space, BWFormer (CVPR 2025) proposed projecting point clouds onto a 2D Bird's-Eye View (BEV) heightmap. While this dimensional compression reduces computation, quantizing continuous 3D points onto a discrete 2D grid forfeits crucial 3D geometric context; worse, the 2D projection severely obscures internal corners that lie inside roof junctions rather than on exterior boundaries.

Consequently, building wireframe reconstruction requires an adaptive search space that eliminates the detrimental noise of planar regions while preserving full 3D spatial fidelity and interior roof topologies. This paper recognizes that airborne roof scans naturally exhibit open surface topologies without self-occlusions, making Delaunay triangulation an ideal prior to approximate the underlying surface manifold. Core idea: build a Delaunay graph over the point cloud to extract curvature signatures via face normals, edge dihedral angles, and vertex-wise corner scores, using these geometric priors to adaptively isolate high-curvature corner candidates and dynamically scale wire queries via graph shortest-path dihedral averages for fully 3D wireframe reconstruction.

Method¶

Overall Architecture¶

The input to Delaunay Canopy is an airborne LiDAR building point cloud $C \in \mathbb{R}^{N \times 3}$, and the output is a clean building wireframe composed of 3D corner vertices and connecting wires. The pipeline comprises two core stages: (i) Delaunay Graph Scoring, which computes hierarchical curvature metrics across faces, edges, and vertices on the triangulated mesh-graph $G = (V, E, F)$; and (ii) a geometry-guided Corner and Wire Selection pipeline, which isolates the top-$K$ high-scoring corner candidates for Transformer-based corner refinement, followed by graph shortest-path path scoring that dynamically scales wire queries during Transformer-based edge classification.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Input Airborne LiDAR Point Cloud C"] --> B["Delaunay Triangulation<br/>Construct Graph G = (V, E, F)"]
    B --> C["Delaunay Graph Scoring<br/>Face Normals → Edge Dihedral Angles → Corner Scores"]
    C --> D["Corner Score Sampling<br/>Top-K High-Curvature Pruning"]
    D --> E["Corner Selection Module<br/>Transformer Decoder Refinement"]
    E --> F["Form Candidate Wires wij between Corners"]
    C -.->|Shortest Graph Path| G["Wire-Wise Path Score & Query Scaling<br/>Mean Dihedral Angle Prior Modulation"]
    F --> G
    G --> H["Wire Selection Module<br/>Transformer Decoder Edge Classification"]

Key Designs¶

1. Delaunay Graph Scoring: Explicit Curvature Signatures from Surface Topology To address the lack of local geometric constraints in raw point sets, this design capitalizes on the single-view, non-self-occluding nature of aerial roof point clouds to construct a Delaunay graph $G = (V, E, F)$. Curvature metrics are calculated hierarchically. First, for each triangular face $f = (v_a, v_b, v_c)$, the face normal is computed as $n_f = (v_b - v_a) \times (v_c - v_a)$ and oriented such that its $z$-component is non-negative. Second, for every edge $e$ shared by adjacent faces $f_1$ and $f_2$, the dihedral angle $\theta_e$ is derived via the inner product of their normal vectors: $\theta_e = \arccos(n_1 \cdot n_2)$, while boundary edges touching only a single face are assigned $\theta_e = \pi$. Third, at the vertex level, the corner score $S_c(v)$ is computed by averaging the dihedral angles of all incident edges $E(v)$: $$S_c(v) = \frac{1}{|E(v)|} \sum_{e \in E(v)} \theta_e$$ Planar regions yield dihedral angles near zero, whereas ridge lines, roof eaves, and sharp corners produce significant normal divergence and elevated scores. This training-free geometric scoring reliably isolates surface curvature even in sparse or noisy conditions.

2. Corner Score Sampling & Selection: Adaptive Search Space Pruning Conventional farthest point sampling (FPS) scatters candidates uniformly across flat roof surfaces, overburdening the downstream detector with uninformative points and false-positive risks. In contrast, this design sorts all input points $C \in \mathbb{R}^{N \times 3}$ by their vertex corner scores $S_c(v)$ and selects the top-$K$ highest-scoring vertices to form a compact candidate subset $C' \in \mathbb{R}^{K \times 3}$ ($K=150$). This focus effectively strips away redundant planar points and concentrates the candidate pool around structural boundaries and junctions. The refined subset $C'$ is then processed by a 6-layer Transformer-based corner selection module $\tilde{C} = \mathcal{F}_{\text{corner}}(C')$ that models spatial relationships and regresses the final set of $M$ 3D corners $\tilde{C}$.

3. Wire-Wise Path Score & Prior-Based Query Scaling: Shortest-Path Modulation of Topology Queries Pairing all $M$ predicted corners yields $O(M^2)$ candidate wires $W = \{w_{ij} = (v_i, v_j)\}$. Treating every candidate equally inside a Transformer decoder wastes model capacity on implausible connections that cross open planar spaces. To suppress these false candidates, this design maps each candidate $w_{ij}$ onto the Delaunay graph $G$ and finds the shortest path $P^G_{i \rightarrow j}$ between its endpoints. Because physical building edges follow continuous ridge lines with elevated dihedral angles, the path score $S_P(v_i, v_j)$ is computed as the average dihedral angle across all edges $L(P^G_{i \rightarrow j})$ along this path: $$S_P(v_i, v_j) = \frac{1}{|L(P^G_{i \rightarrow j})|} \sum_{e \in L(P^G_{i \rightarrow j})} \theta_e$$ Within the initial 3 layers of the 6-layer Transformer wire decoder, each wire query $Q_w^l$ is element-wise scaled using a sigmoid-activated gating factor: $Q_w^{\prime l} = Q_w^l \odot \sigma(S_P^w)$. This modulation attenuates queries for geometrically unaligned wires while amplifying promising ridge candidates, enabling the attention layers to focus on valid structural edges.

Loss & Training¶

The framework is trained end-to-end across two stages for 300 epochs using AdamW with an initial learning rate of $10^{-4}$. All LiDAR point coordinates are normalized into $[0, 256]$. Both the corner selection module and wire selection module employ 6-layer Transformer decoder backbones. Corner selection is optimized using Hungarian bipartite matching with smooth L1/Chamfer coordinate regression and classification objectives. Wire selection utilizes edge-level self-attention and cross-edge feature pooling, trained under binary cross-entropy loss against ground-truth wireframe adjacency.

Key Experimental Results¶

Main Results¶

Evaluations were conducted on the Building3D benchmark across both the Tallinn City dataset (32,618 pairs: 30,000 train / 2,618 test) and the Entry-Level dataset (5,698 pairs: 5,000 train / 698 test). Metrics include Wireframe Edit Distance (WED ↓), Average Corner Offset (ACO ↓), Corner Precision/Recall/F1 (CP, CR, CF1 ↑), and Edge Precision/Recall/F1 (EP, ER, EF1 ↑).

Table 1: Main quantitative comparison on Building3D Tallinn City dataset (Paper Table 1)

Method	WED ↓	ACO ↓	CP (%) ↑	CR (%) ↑	CF1 (%) ↑	EP (%) ↑	ER (%) ↑	EF1 (%) ↑
Building3D (PointNet)	0.264	0.241	88.2	77.6	82.6	80.4	71.9	75.9
PC2WF (CVPR 2021)	0.554	0.508	21.4	50.5	30.1	2.8	16.7	4.8
Point2Roof (ISPRS 2022)	0.260	0.236	89.3	78.5	83.6	81.2	72.2	76.4
PointTransformer* (ICCV 2021)	0.257	0.238	89.7	79.1	84.1	81.3	72.4	76.6
PointMLP* (2022)	0.255	0.233	90.6	79.0	84.4	81.6	72.8	76.9
PointNeXt* (2022)	0.256	0.229	90.4	79.4	84.5	82.0	73.9	78.0
PointMeta* (CVPR 2023)	0.251	0.225	90.6	79.8	84.9	82.4	73.3	77.6
BWFormer (CVPR 2025)	0.245	0.213	91.4	80.1	85.4	84.2	74.6	79.1
Delaunay Canopy (Ours)	0.232	0.194	94.5	82.8	88.3	87.1	74.0	80.0

On the Entry-Level dataset, Delaunay Canopy similarly achieves state-of-the-art performance across all metrics: WED reaches 0.230 (vs. 0.242 for BWFormer), ACO reaches 0.190 (vs. 0.201 for BWFormer), CF1 improves to 90.2% (+2.9%), and EF1 reaches 82.4% (+1.3%).

Ablation Study¶

The components were ablated on the Tallinn City dataset by evaluating Corner Score Sampling against standard FPS ($K=150$) and Prior-Based Query Scaling against an unscaled standard Transformer decoder.

Table 2: Ablation study on core components (Paper Table 2)

Config	Score Sampling	Query Scaling	WED ↓	ACO ↓	CP (%) ↑	CR (%) ↑	CF1 (%) ↑	EP (%) ↑	ER (%) ↑	EF1 (%) ↑	Note
Baseline	-	-	0.257	0.234	88.9	78.2	83.2	81.1	72.1	76.3	Standard FPS + Transformer
+ Score Sampling	✓	-	0.239	0.210	93.1	81.1	86.7	85.6	73.7	79.2	Curvature sampling (+3.5% CF1)
+ Query Scaling	-	✓	0.242	0.223	92.4	80.3	85.9	84.9	72.5	78.2	Shortest-path dihedral scaling
Full Model	✓	✓	0.232	0.194	94.5	82.8	88.3	87.1	74.0	80.0	Full model achieves best results

Key Findings¶

Sampling budget $K$ trade-off (Paper Table 3): Performance peaks sharply at $K=150$ points (WED 0.232 / CF1 88.3%). When $K$ drops to 100, the search space becomes overly restricted and misses valid corners (CF1 drops to 84.5%); conversely, expanding $K$ to 200 admits excess planar noise, diluting the geometric prior (WED degrades to 0.260).
Early-layer modulation boundary (Paper Table 4): Applying prior-based query scaling to the first 3 layers produces the best EF1 (80.0%) and WED (0.232). Extending query scaling across all 6 layers degrades EF1 to 78.3%, demonstrating that strong geometric priors should serve as an early hypothesis filter without overriding high-level learned contextual features in deeper layers.
Geometric prior vs. architectural depth (Paper Table 5): Replacing the Point2Roof wire head with a Transformer without priors yields marginal improvement (CF1 86.0% $\rightarrow$ 86.7%, EF1 78.8% $\rightarrow$ 79.2%). Injecting the Delaunay geometric prior drives substantial gains (CF1 88.3%, EF1 80.0%), confirming that explicit geometric guidance is the primary driver of performance.

Highlights & Insights¶

Physics-grounded discrete curvature quantification: By modeling single-view aerial scans as open topological surfaces, Delaunay Canopy computes face normals, dihedral angles, and vertex curvature scores in a parameter-free manner, extracting rich continuous curvature signatures at minimal computational cost.
Adaptive 3D search space avoids projection pitfalls: Unlike 2D BEV heightmap methods that distort geometries through rasterization and discard internal corners, Delaunay Canopy operates natively in continuous 3D space, capturing subtle interior folds while maintaining a compact, bounded candidate pool.
Synergistic graph shortest-path query gating: Leveraging shortest-path dihedral line integrals on the Delaunay graph as dynamic query gates bridges discrete computational geometry with differentiable Transformer attention, providing an effective blueprint for structured 3D topological inference.

Limitations & Future Work¶

Reliance on open 2.5D surface topology: The formulation assumes an open surface scanned from above without self-occluding layers or vertical undercuts. Extending this formulation to multi-story facades, cantilevered architecture, or dense mobile street scans would cause 2D/2.5D Delaunay triangulations to produce degenerate overlapping faces.
Sensitivity to severe LiDAR data dropouts: In regions with extreme laser absorption or shadowing, the shortest path on the Delaunay graph may bridge across missing gaps, generating erroneous edge curvature scores.
Future directions: Adapting the formulation to 3D volumetric Delaunay tetrahedralization or coupling the structural prior with implicit surface representations (e.g., 3D Gaussians or Neural SDFs) could generalize wireframe reconstruction to full-building 3D envelopes.

vs Point2Roof / PC2WF: Early deep wireframe baselines rely on unconstrained 3D point searches that are overwhelmed by planar redundancy and sensor noise; Delaunay Canopy leverages curvature-scored Delaunay graphs to construct an adaptive search space focused exclusively on structural candidates.
vs BWFormer (CVPR 2025): BWFormer compresses point clouds into 2D BEV heightmaps to constrain search space, but suffers from rasterization quantization errors and fails to reconstruct internal roof corners; Delaunay Canopy retains full 3D contextual reasoning and reliably detects subtle interior foldings.
vs Kinetic Shape Reconstruction / City3D: Mesh-fitting approaches impose global planar and watertightness priors that over-regularize noisy real-world scans; Delaunay Canopy focuses on lightweight, flexible wireframes that can subsequently be extruded into clean, high-fidelity meshes (Paper Fig. 9).

Rating¶

Novelty: ⭐⭐⭐⭐☆ [Uses Delaunay dihedral angles and shortest-path priors to resolve the tension between 3D search space explosion and 2D projection distortion]
Experimental Thoroughness: ⭐⭐⭐⭐⭐ [Extensive evaluations on Building3D benchmarks with thorough ablations on candidate budgets, scaling layers, baseline backbones, and qualitative edge cases]
Writing Quality: ⭐⭐⭐⭐⭐ [Clear structural exposition, insightful comparative figures, and well-motivated mathematical formulation]
Value: ⭐⭐⭐⭐☆ [Provides an accurate, lightweight representation pipeline for large-scale urban 3D modeling and autonomous driving HD map generation]