DFG-PCN: Degree-Flexible Point Graph Completion

• The paper introduces a novel degree-flexible method that adaptively assigns node degrees based on feature variation and curvature to enhance point cloud completion.
• It employs geometry-aware graph integration using Manhattan distance to capture fine details and complex local structures in 3D shapes.
• Experimental results on PCN, ShapeNet, and KITTI benchmarks demonstrate lower Chamfer Distances and improved reconstruction accuracy compared to fixed-degree models.

The Degree-Flexible Point Graph Completion Network (DFG-PCN) is a point cloud completion framework designed to reconstruct fully dense 3D shapes from partial observations, addressing the fundamental limitation of fixed-degree graph models in representing complex, non-uniform geometric structures. This approach introduces detail-aware, adaptive node degree assignment along with geometry-aware graph integration to achieve improved accuracy in reconstructing fine-grained details and handling regions with complex local structure.

1. Motivation and Problem Context

In point cloud completion, the central problem is to infer complete 3D shapes from incomplete data—often resulting from occlusion or limited sensor resolution. Traditional deep learning models typically rely on uniform node degrees (e.g., k-nearest neighbor graphs with fixed $k$), which fail to capture the highly non-uniform distribution of geometric complexity across a shape. Regions with high curvature, sharp edges, or structural discontinuities are under-represented when all points are treated equally. DFG-PCN addresses this by adaptively focusing computational and representational capacity on informative, detail-rich regions, allocating higher node degree where complexity warrants denser local connectivity.

2. Architectural Framework and Main Components

DFG-PCN is organized as a two-stage framework. Its critical modules are:

• Feature Extractor and Seed Generator: Hierarchical aggregation (PointNet, Point Transformer) generates global and local per-point features. The Upsample Transformer produces an initial coarse shape, which is fused with the input (using FPS downsampling) into a seed cloud for subsequent refinement.
• Degree-Flexible Point Graph Module (Core Innovation): Instead of a uniform $k$-NN connection scheme, this module assigns adaptive node degrees using a detail-aware metric:

$$D_Q = \sum_c |Q - Q_{\downarrow\uparrow}| + |Q - H_{i-1}|$$

where $Q$ are per-point features, $Q_{\downarrow\uparrow}$ are recovered features after down-up sampling, and $H_{i-1}$ is the prior-stage feature. The degree $d_i$ for point $i$ is then:

$$d_i = \text{round}\left(B \frac{D_i}{\sum_{i=1}^n D_i}\right), \qquad B = \alpha \sum_{i=1}^n (D_i + \kappa_i)$$

with $\kappa_i$ denoting local curvature and $\alpha$ a scaling parameter. Points in high-variation regions receive more neighbors and thus richer local features.

• Geometry-Aware Graph Integration: Edge aggregation leverages Manhattan distance rather than Euclidean:

$$M^i_{j,k} = \gamma (|x^i_j - x^i_k| + |y^i_j - y^i_k| + |z^i_j - z^i_k|)$$

with $\gamma$ the scaling constant, making the model sensitive to axis-aligned variation (helpful for boundaries and discontinuities).

The edge-conditioned aggregation is expressed as:

$$\begin{aligned} \hat{a}^i_{j,k} &= \beta_i \left[ (f^i_j + \delta) \ominus (f^i_k + \delta) + C \cdot (\neg G_i) + M^i_{j,k} \right] \ a^i_{j,k} &= \exp(\hat{a}^i_{j,k}) / \sum_{k\in\mathcal{N}(j)} \exp(\hat{a}^i_{j,k}) \ h^i_j &= \sum_{k\in\mathcal{N}(j)} a^i_{j,k} \odot (h^{i-1}_k + \delta) \end{aligned}$$

where $\beta_i$ is a learnable MLP, $\delta$ is positional encoding, $C$ a negative constant penalizing non-connected nodes, $\ominus$ a parameterized similarity measure, and $\odot$ denotes Hadamard product.

• Graph Fusion Module (GFM): Combines local and global features using self-attention, with queries, keys, and values computed over enriched feature sets. Local detail richness is incorporated into the fusion, using attention for weighted integration:

$$\begin{aligned} \hat{H}_l &= h^i_j + D^i_j;~ Q = \hat{H}_l W_Q;~ K = \hat{H}_l W_K;~ V = \hat{H}_l W_V \ H_l &= \text{softmax}(Q K^T / \sqrt{d_k}) V \end{aligned}$$

This mechanism enables effective information propagation between regions of varying complexity.

• Upsampling: Progressive coordinate refinement uses deconvolutions and MLPs to upscale the point cloud guided by graph features.

3. Technical Details and Mathematical Formulation

The core technical contributions can be summarized as follows:

• Detail-Richness Metric: Adaptive node degree assignment based on both per-point feature variation across scales and local curvature:

$$D_Q = \sum_c |Q - Q_{\downarrow\uparrow}| + |Q - H_{i-1}|$$

$$d_i = \text{round}(B \frac{D_i}{\sum_{i=1}^n D_i}),\quad B = \alpha \sum_{i=1}^n (D_i + \kappa_i)$$

• Manhattan Distance Integration: Increased sensitivity to nuanced variations via axis-aligned metrics:

$$M^i_{j,k} = \gamma (|x^i_j - x^i_k| + |y^i_j - y^i_k| + |z^i_j - z^i_k|)$$

• Edge-Conditioned Feature Aggregation:

$$a^i_{j,k} = \exp(\hat{a}^i_{j,k}) / \sum_{k\in\mathcal{N}(j)} \exp(\hat{a}^i_{j,k})$$

• Self-Attention Fusion of Local and Global Graph Features: Weighted propagation based on detail-aware metric.

4. Experimental Protocols and Quantitative Results

Extensive experiments were performed on a set of benchmarks:

• PCN Dataset: 28,974 train / 800 val / 1,200 test, point clouds with 2,048 points.
• ShapeNet-55 and ShapeNet-34: 55 and 34 categories for within- and cross-category evaluation.
• KITTI: Real-world LiDAR scans with isolated car instances.

Metrics include Chamfer Distance ($\mathrm{CD}_{\ell_1}, \mathrm{CD}_{\ell_2}$), Minimal Matching Distance (MMD), and Fidelity Distance (FD, KITTI).

DFG-PCN achieves consistently lower $\mathrm{CD}$ (e.g., $\mathrm{CD}_{\ell_1} = 6.42$ on PCN, outperforming SeedFormer, FBNet, PointAttN) and preserves smooth surfaces and fine structural details, as confirmed in visual comparisons. Ablation studies confirm the contribution of adaptive degree assignment, local/global graph sampling, and the feature fusion module.

5. Comparative Perspective and Related Methods

DFG-PCN advances over prior models that rely on fixed-degree graphs (e.g., $\text{k}$-NN), which exhibit inefficiency and local blindness in regions of high curvature or geometric complexity. Compared to SoftPoolNet (Wang et al., 2020), which reorganizes features by soft pooling and regional convolutions, and approaches such as Graph-Guided Deformation Networks (Shi et al., 2021) that deform mesh-like structures via Laplacian coordinates, DFG-PCN uniquely combines dynamic node degree assignment with explicit geometric sensitivity (via the Manhattan metric) and multi-scale fusion. The method demonstrates superior robustness over standard PCN (Yuan et al., 2018), which employs a fixed permutation-invariant encoder and a hybrid coarse-to-fine decoder, by dynamically focusing modeling capacity where it is needed most.

6. Implications and Future Directions

The degree-flexible graph construction paradigm enables more precise reconstruction of fine details, smoother surfaces, and suppression of completion noise. Ongoing work aims to further optimize flexible graph construction, refine connection allocation mechanisms, and enhance cross-dataset generalization. Extensions to related tasks such as reconstruction and upsampling are foreseen, leveraging the strengths of adaptive local-global graph integration.

A plausible implication is that DFG-PCN’s innovations in node degree assignment and geometry-aware representation could inform broader classes of geometric learning tasks where local structural complexity varies widely, including mesh reconstruction, scene parsing, and semantic segmentation.

7. Summary Statement

DFG-PCN represents a significant development in point cloud completion by introducing a degree-flexible, geometry-aware graph integration mechanism. Through adaptive node degree assignment based on feature variation and curvature, along with edge-conditioned aggregation using Manhattan distance, the framework achieves enhanced representation of complex geometric structures. Experimental validation across multiple benchmarks consistently demonstrates improved completion accuracy and detail preservation compared to leading contemporary methods (Shu et al., 28 Sep 2025).