Curve-Aware Gaussian Splatting for 3D Parametric Curve Reconstruction¶
Conference: ICCV 2025 arXiv: 2506.21401 Code: Project Page Area: 3D Vision Keywords: Parametric curve reconstruction, 3DGS, Bézier curves, end-to-end optimization, edge reconstruction
TL;DR¶
This paper proposes CurveGaussian, a single-stage method that establishes a bidirectional coupling mechanism between parametric curves and edge-oriented Gaussian primitives, enabling direct end-to-end optimization of 3D parametric curves from multi-view edge maps. By eliminating the error accumulation inherent in two-stage pipelines, the method achieves comprehensive improvements over prior approaches in accuracy, efficiency, and compactness.
Background & Motivation¶
Parametric curves are indispensable geometric primitives in CAD and industrial applications, and accurately reconstructing them from multi-view images is an important open problem.
Fundamental limitations of two-stage methods: 1. Existing methods (NEF, EMAP, EdgeGaussians) all follow a two-stage pipeline of "3D edge point cloud reconstruction → parametric curve fitting"
Error accumulation: Noise from 2D edge detection propagates to the 3D point cloud and further to parametric curve fitting, resulting in spurious branches and broken curves.
Greedy fitting and local optima: Iterative fitting methods such as RANSAC tend to produce redundant curves, with complexity growing exponentially with scene scale.
Low efficiency: NEF requires 1.5 hours and EMAP requires 2.5 hours of training.
Core challenge: Parametric curves are inherently incompatible with rendering-based multi-view optimization (lacking differentiable rendering capability), while neural rendering frameworks (NeRF/3DGS) cannot preserve the geometric continuity of curves.
Proposed solution: Establish a bidirectional coupling between parametric curves and Gaussian primitives, letting Gaussians serve as "renderable proxies" for curves, and directly optimize curve control points via backpropagation through rendering losses.
Method¶
Overall Architecture¶
Input: Multi-view 2D edge maps + camera poses Output: A set of 3D parametric curves (cubic Bézier curves + first-order Bézier line segments) Pipeline: Random curve initialization → curve-aware Gaussian splatting rendering → backpropagation of rendering loss to optimize control points → adaptive topology optimization
Key Designs¶
-
Bidirectional curve–Gaussian coupling:
- Each curve \(c_j\) is uniformly sampled to generate \(N=12\) Gaussian primitives.
- Gaussian attributes are fully determined by curve geometry:
- Position: anchored at curve sample points \(\mathbf{p}_j(t_i)\), \(t_i = \frac{i+0.5}{N}\)
- Orientation: principal axis \(\mathbf{v}_0\) aligned with curve tangent \(\mathbf{T}_j(t_i)\); remaining axes orthogonalized
- Scale: \(\mathbf{s}^{j,i} = [\|\Delta\mathbf{p}_j^i\|, d_j, d_j]^\top\), with the principal axis far exceeding the others, forming stick-shaped edge-oriented Gaussians
- Opacity: inherited from the parent curve attribute \(o_j\)
- A learnable importance mask \(m^{j,i} \in [0,1]\) is introduced to automatically identify redundant curve segments.
- Bidirectionality: Gaussian attributes are constrained by curve parameters, and rendering gradients are backpropagated to update curve control points.
-
Parametric curve representation:
- Cubic Bézier curve: 4 control points, \(\mathbf{c}_j(t) = \sum_{k=0}^{3} B_k^3(t)\mathbf{P}_j^k\)
- First-order Bézier (line segment): 2 endpoints, \(\mathbf{c}_j(t) = (1-t)\mathbf{P}_j^0 + t\mathbf{P}_j^1\)
- Each curve carries additional opacity \(o_j\) and thickness \(d_j\)
-
Adaptive topology optimization (four strategies):
- Linearization: If a cubic Bézier approximates a straight line (mean squared error from sampled points to the fitted line \(< \tau_l\)), it is replaced by a first-order Bézier.
- Merging: Adjacent line segments are merged when the directional angle \(\theta < \tau_{la}\) and endpoint distance \(d < \tau_{ld}\); adjacent Bézier curves are merged if the post-merge error \(< \tau_b\).
- Splitting: When a geometric discontinuity is detected (principal axis angle between adjacent Gaussians \(> \theta_s\)), the curve is split at the discontinuity; segments with low mask values (\(m_j^i < \tau_m\)) are removed while preserving both sides.
- Pruning: An entire curve is removed when opacity \(< \tau_d\) or all Gaussian masks fall below the threshold.
- Scheduling: Linearization begins at 3k iterations, merging at 7k, and both are executed once every 1k iterations.
Loss & Training¶
Edge-aware rendering loss — addressing gradient collapse caused by sparse edge pixels: $\(\mathcal{L}_{edge} = \frac{|M_I|}{|E_I|}\sum_{i \in N_I}\|I_i - \hat{I}_i\|_2^2 + \frac{|N_I|}{|E_I|}\sum_{i \in M_I}\|I_i - \hat{I}_i\|_2^2\)$ Complementary weights balance the contributions of edge and non-edge pixels.
Smooth connectivity regularization: \(\mathcal{L}_{conn}\) — minimizes the distance between endpoints of adjacent curves. Curve smoothness regularization: \(\mathcal{L}_{smo}\) — minimizes orientation differences between adjacent Gaussians. Compactness regularization: \(\mathcal{L}_{reg} = \sum \log(1 + o_j^2/0.5)\) — encourages low-opacity curves to be pruned. Mask loss: \(\mathcal{L}_m\) — eliminates redundant Gaussians.
Total loss: \(\mathcal{L}_{all} = \mathcal{L}_{edge} + \lambda_1\mathcal{L}_{conn} + \lambda_2\mathcal{L}_{smo} + \lambda_3\mathcal{L}_{reg} + \lambda_4\mathcal{L}_m\)
Training runs for 10k iterations; after 7k, opacity is frozen and the mask loss is activated.
Key Experimental Results¶
Main Results¶
ABC-NEF dataset (82 ABC models, DexiNed edge detector):
| Method | Acc.↓ | Comp.↓ | F5↑ | F10↑ | F20↑ | Time↓ |
|---|---|---|---|---|---|---|
| NEF | 21.9 | 15.7 | 10.8 | 42.1 | 76.8 | 1.5 h |
| EMAP | 8.8 | 8.9 | 59.1 | 88.9 | 94.9 | 2.5 h |
| EdgeGaussians | 9.6 | 8.4 | 45.2 | 93.7 | 95.7 | 4 min |
| Ours | 8.2 | 7.5 | 73.7 | 94.0 | 96.2 | 3 min |
Efficiency and compactness comparison:
| Method | Training Time↓ | # Bézier↓ | # Segments↓ | # Total Curves↓ |
|---|---|---|---|---|
| NEF | 1.5 h | 22.9 | 0 | 22.9 |
| EMAP | 2.5 h | 9.2 | 36.6 | 45.8 |
| EdgeGaussians | 4 min | 13.0 | 84.9 | 97.9 |
| Ours | 3 min | 6.9 | 22.0 | 28.9 |
Ablation Study¶
Ablation of key components:
| Configuration | Acc.↓ | Comp.↓ | F5↑ | Note |
|---|---|---|---|---|
| w/o edge-aware loss | degraded | degraded | degraded | gradient collapse |
| w/o topology optimization | increased | increased | decreased | cannot eliminate redundancy |
| w/o smoothness regularization | increased | increased | decreased | non-smooth curves |
| Full model | 8.2 | 7.5 | 73.7 | — |
Key Findings¶
- 14.5% accuracy improvement: F5 increases from 45.2% (EdgeGaussians) to 73.7% (DexiNed).
- 70.5% reduction in curve count: Total curves reduced from 97.9 to 28.9, yielding more compact reconstructions.
- 33% training speedup: 3 minutes vs. 4 minutes for EdgeGaussians.
- On the Replica real-world dataset, the proposed method produces parametric curves that are more complete and less redundant.
Highlights & Insights¶
- Paradigm shift: Moving from two-stage to single-stage optimization directly in parametric space fundamentally eliminates error accumulation.
- Elegant coupling design: Stick-shaped Gaussians naturally conform to curve geometry (principal axis = tangent, principal scale = segment length, minor scale = line width), allowing rendering gradients to flow naturally to control points.
- Adaptive topology: During training, a large initial set of curves is progressively refined into a small, accurate set — the reverse of the conventional 3DGS "grow from few" strategy.
- Extreme parametric compactness: Each curve requires only 4 control points and 2 scalar values, far fewer parameters than independent Gaussians.
Limitations & Future Work¶
- Reconstruction quality depends on the quality of the 2D edge detector (different detectors affect F5 by ~15%).
- Random initialization may require more iterations to converge to the optimal topology.
- Only cubic and first-order Bézier curves are supported; more general parametric forms such as NURBS are not covered.
- Curve grouping with semantic information has not been explored.
Related Work & Insights¶
- NEF/EMAP: Two-stage pioneers that reconstruct edge fields with NeRF, but suffer from error accumulation in the fitting stage.
- EdgeGaussians: Accelerates edge reconstruction with 3DGS but remains constrained by the two-stage paradigm.
- DiffVG: Differentiable 2D vector graphics rendering; this work extends analogous ideas to 3D parametric curves.
- 3DGS: Provides an efficient differentiable rendering foundation.
Rating¶
- Novelty: ⭐⭐⭐⭐⭐ (single-stage parametric curve optimization + curve–Gaussian coupling)
- Technical Depth: ⭐⭐⭐⭐ (comprehensive adaptive topology strategy)
- Experimental Thoroughness: ⭐⭐⭐⭐ (multi-dataset + efficiency comparison + ablation)
- Value: ⭐⭐⭐⭐ (3-minute training, compact output, well-suited for CAD pipelines)