BG-Triangle: Bézier Gaussian Triangle for 3D Vectorization and Rendering

TL;DR

BG-Triangle addresses the challenge of preserving sharp geometry in differentiable rendering by introducing a hybrid 3D representation that couples Bézier triangle primitives with Gaussian rendering. It bridges explicit, edge-preserving geometry and probabilistic rendering by using Bezier surfaces for shape and pixel-aligned Gaussians for appearance, together with discontinuity-aware blending and adaptive subdivision. The approach achieves comparable rendering quality to existing probabilistic methods like $3$DGS while using significantly fewer primitives, enabling faster training and simpler scene representations. This vectorized, boundary-aware framework promises a practical path toward high-fidelity, editable 3D reconstructions and efficient view synthesis.

Abstract

Differentiable rendering enables efficient optimization by allowing gradients to be computed through the rendering process, facilitating 3D reconstruction, inverse rendering and neural scene representation learning. To ensure differentiability, existing solutions approximate or re-formulate traditional rendering operations using smooth, probabilistic proxies such as volumes or Gaussian primitives. Consequently, they struggle to preserve sharp edges due to the lack of explicit boundary definitions. We present a novel hybrid representation, Bézier Gaussian Triangle (BG-Triangle), that combines Bézier triangle-based vector graphics primitives with Gaussian-based probabilistic models, to maintain accurate shape modeling while conducting resolution-independent differentiable rendering. We present a robust and effective discontinuity-aware rendering technique to reduce uncertainties at object boundaries. We also employ an adaptive densification and pruning scheme for efficient training while reliably handling level-of-detail (LoD) variations. Experiments show that BG-Triangle achieves comparable rendering quality as 3DGS but with superior boundary preservation. More importantly, BG-Triangle uses a much smaller number of primitives than its alternatives, showcasing the benefits of vectorized graphics primitives and the potential to bridge the gap between classic and emerging representations.

Figures (11)

• Figure 1: BG-Triangle can transform multi-view images into vectorized 3D scenes, enabling the rendering of sharp novel views.
• Figure 2: Rendering pipeline of BG-Triangles. The BG-Triangle rendering pipeline has three modules. The Primitive Rasterization module tessellates the Bézier triangle, producing coordinate and index maps, as well as boundary points. These maps are then used in the sub-primitive generation module to create pixel-aligned sub-primitives for differentiable rendering. The discontinuity-aware alpha blending module utilizes the boundary points to render images with sharp edges. Finally, the fully differentiable pipeline allows gradient backpropagation along the blue arrows to optimize the control points of BG-Triangles.
• Figure 3: Sub-Primitive Generation. For a differentiable rendering pipeline, we generate 3D Gaussian sub-primitives on a degree-2 Bézier triangle (left). Coordinates (purple) are computed from control points (yellow) via barycentric interpolation, while other attributes are interpolated barycentrically or via a multi-resolution attribute map.
• Figure 4: Discontinuity-Aware Alpha Blending. Primitives $\mathbf{R}_{a}$ and $\mathbf{R}_{b}$ share a boundary. For pixel $p_{1}$, which lies within both the boundary region and $\mathbf{R}_{a}$, the blending coefficient from Gaussian $\mathcal{G}_{a}$ (associated with $\mathbf{R}_{a}$) is larger than that from $\mathcal{G}_{b}$ (associated with $\mathbf{R}_{b}$). In contrast, pixel $p_{2}$, located well inside $\mathbf{R}_{b}$ and away from the boundary, is unaffected by Gaussians $\mathcal{G}_{a}$ and $\mathcal{G}_{c}$ (not associated with $\mathbf{R}_{a}$ nor $\mathbf{R}_{b}$).
• Figure 5: Initial validation. Our method is capable of fitting the object's surface, silhouette, and boundaries by using Bézier triangles, maintaining sharp edges even when zoomed in, and sometimes appearing clearer than the ground truth. In contrast, 3DGS shows blurring under magnification.