Bézier Splatting for Fast and Differentiable Vector Graphics Rendering¶

Conference: NeurIPS 2025 arXiv: 2503.16424 Code: https://xiliu8006.github.io/Bezier_splatting_project Area: Vector Graphics Rendering Keywords: Differentiable Rendering, Bézier Curves, Gaussian Splatting, Vectorization, Image Optimization

TL;DR¶

Bézier Splatting integrates the Gaussian Splatting framework with Bézier curves by uniformly sampling 2D Gaussian points along each curve and rendering via α-blending to achieve differentiable vector graphics. The method achieves 30× forward and 150× backward speedups over DiffVG while maintaining or surpassing the image quality of methods such as LIVE.

Background & Motivation¶

Background: Differentiable vector graphics renderers (e.g., DiffVG, LIVE) optimize Bézier curve parameters for image vectorization. DiffVG performs per-pixel color accumulation, while LIVE adds curves layer by layer; both require hours of optimization for high-resolution images.

Limitations of Prior Work: DiffVG's backward pass requires sampling at every boundary pixel, making computation proportional to the number of curves and image resolution—extremely slow. LIVE's layer-wise optimization takes 2.6–5.1 hours for 512 curves. No existing method achieves high-quality vectorization within minutes.

Key Challenge: Traditional differentiable rendering must precisely handle anti-aliased gradients at curve boundaries, which creates a computational bottleneck. A rendering approach that bypasses per-pixel boundary sampling is needed.

Goal: Substantially accelerate differentiable vector graphics rendering while preserving output quality, reducing optimization time from hours to minutes.

Key Insight: Gaussian Splatting has demonstrated efficient differentiable rendering in 3D scenes. By parameterizing Bézier curves as a set of 2D Gaussian points distributed along each curve, the efficient Gaussian Splatting rendering pipeline can be directly reused.

Core Idea: Uniformly sample 2D Gaussian points along each Bézier curve, derive rotation from the local tangent direction and scale from inter-point distances, and achieve fast differentiable vectorization via α-blending rendering combined with adaptive pruning and densification.

Method¶

Overall Architecture¶

Input image → Initialize a set of Bézier curves → Uniformly sample points along each curve → Compute 2D Gaussian parameters per sample (position = sampled point, rotation = tangent angle, scale = distance to neighboring points) → α-blending rendering → Optimize curve control points and colors with L2 + Xing loss → Adaptively prune low-opacity curves and densify new curves in high-error regions.

Key Designs¶

Parameterization of Bézier Curves as 2D Gaussians:
Function: Convert each Bézier curve into a sequence of 2D Gaussian points.
Mechanism: Position = uniformly sampled points on the curve; rotation \(\theta\) = local tangent angle; \(\sigma_x\) (along the curve) = distance to neighboring sample points; \(\sigma_y\) (perpendicular to the curve) = learnable width parameter. For closed curves, a "paired Bézier curve" structure linearly interpolates control points between two boundary curves to fill the interior region.
Design Motivation: 2D Gaussians are the native representation in Gaussian Splatting, enabling direct reuse of its efficient rendering pipeline. Automatically deriving parameters from tangent direction and inter-point distances avoids additional learning overhead.
α-blending Rendering (Non-accumulative):
Function: Correctly handle occlusion relationships between curves.
Mechanism: Curves are sorted by depth and rendered front-to-back as \(C = \sum_i T_i \alpha_i c_i\), where \(T_i = \prod_{j<i}(1-\alpha_j)\). Unlike DiffVG's per-pixel color accumulation, α-blending naturally supports occlusion.
Design Motivation: DiffVG's accumulative rendering leads to color aliasing, whereas α-blending better reflects the physical occlusion model.
Adaptive Pruning and Densification:
Function: Dynamically adjust the number and distribution of curves.
Mechanism: Pruning—remove curves whose opacity falls below a threshold or whose area is too small; Densification—randomly initialize new curves in regions with high residual error; executed periodically.
Design Motivation: Initially random curves may be unevenly distributed; the adaptive strategy concentrates curves in regions requiring fine detail.

Loss & Training¶

\(\mathcal{L} = \lambda_1 \|\hat{I} - I\|_2^2 + \lambda_2 \mathcal{L}_{Xing}\)
The Xing loss constrains Bézier curves to be self-intersection-free (preserving convexity).
Optimization completes in 3.2–8.6 minutes (vs. 2.6–5.1 hours for LIVE).

Key Experimental Results¶

Main Results¶

Method	SSIM↑	PSNR↑	LPIPS↓	Forward Speedup	Backward Speedup	Optimization Time
DiffVG	—	—	—	1×	1×	Hours
LIVE	0.601	21.70	0.521	—	—	2.6–5.1 h
Bézier Splatting	0.607	22.11	0.528	30×	150×	3.2–8.6 min

Ablation Study¶

Configuration	SSIM	PSNR
w/o pruning/densification	0.590	21.10
w/ pruning/densification	0.607	22.11
Layer-wise strategy	0.613	— (2× training time)

Key Findings¶

The backward pass speedup is most pronounced (150×), as it entirely bypasses DiffVG's per-boundary sampling.
The paired-curve structure for closed curves enables gradient optimization over filled regions.
The method generalizes across diverse image types including photographs, artistic paintings, and cartoons.
Adaptive densification contributes substantially to quality (SSIM +0.017, PSNR +1.01).

Highlights & Insights¶

Creative reuse of the Gaussian Splatting framework: Mapping the vector graphics problem onto the Gaussian Splatting pipeline cleverly leverages existing CUDA-accelerated implementations.
Natural parameterization design: Encoding tangent direction as rotation and inter-point distance as scale transfers curve geometry to Gaussian parameters with minimal information loss.
150× backward speedup opens the door to interactive vectorization: Reducing optimization from hours to minutes enables real-time application scenarios.

Limitations & Future Work¶

LPIPS scores are comparable to or slightly worse than LIVE, leaving room for perceptual quality improvement.
The paired-curve structure for closed curves limits shape expressiveness to strip-like regions.
No comparison is made against recent neural vectorization methods such as VectorFusion.
The Xing loss enforcing curve convexity may restrict the representation of complex shapes.

vs. DiffVG: DiffVG samples boundary gradients per pixel, which is prohibitively slow; the proposed method substitutes Gaussians to bypass boundary computation entirely.
vs. LIVE: LIVE's layer-wise curve addition yields high quality but is slow; this work's global optimization with adaptive strategies achieves comparable quality at over 100× higher speed.
vs. 3DGS: 3DGS targets 3D scene reconstruction; this work combines 2D Gaussians with Bézier curves for vector graphics rendering.

Rating¶

Novelty: ⭐⭐⭐⭐ The combination of Gaussian Splatting and Bézier curves is a novel contribution.
Experimental Thoroughness: ⭐⭐⭐⭐ Multi-dataset evaluation with ablations and speed comparisons.
Writing Quality: ⭐⭐⭐⭐ Method description is clear and well-structured.
Value: ⭐⭐⭐⭐ Substantially advances the efficiency of differentiable vector graphics rendering.