Elastic Triangle Splatting

Tian Shi^1,2* Shenhan Qian^1,2* Daniel Cremers^1,2

¹ TU Munich ² MCML

^* Equal contribution.

arXiv 2026

page arxiv

Elastic Triangle Splatting explores kernel properties by two complementary tasks. Shape reconstruction (top) deforms a spherical triangle mesh into an accurate surface. Novel-view synthesis (bottom) optimizes an unstructured triangle soup for photorealistic free-viewpoint rendering. Triangles are splatted onto the image plane with a soft kernel and supervised by posed RGB images. Each triangle is parameterized by vertex positions $\bm{v}$, face color $\bm{c}$, face opacity $o$, and edge softness $\sigma$.

Abstract

While neural rendering methods such as 3D Gaussian Splatting achieve remarkable visual fidelity, traditional polygonal meshes remain the backbone of established graphics pipelines. Triangle splatting bridges this gap by optimizing triangle primitives as differentiable splats, producing representations that are closer to mesh-based workflows. Central to these methods is the kernel function that softens triangle boundaries to propagate gradients to vertex positions. Existing triangle splatting methods make inconsistent choices of kernel functions, and analysis on these kernels’ optimization behavior has been limited to unstructured triangle soups for novel-view synthesis. In this work, we consider triangle splatting as a generic tool for photometric optimization, comparing kernel properties through two complementary tasks: mesh optimization for shape reconstruction and triangle soup optimization for novel-view synthesis. Along with the analysis, we introduce an elastic kernel function that features bilateral gradient support across the boundary and flexible values on the edge, which are shown to be essential to robust optimization. Under isolated comparison, our elastic kernel outperforms existing kernels on shape reconstruction and in the majority of novel-view synthesis benchmarks, demonstrating the importance of kernel design in the effectiveness and versatility of triangle splatting.

Kernel Functions

We identify two key properties of kernel functions that are essential for robust optimization:

1. Bilateral gradient support across the triangle boundary ensures that gradients are propagated to vertex positions even when the triangle is small or far away.

2. Flexible value on the edge maintains constant visual size of a softened triangle and prevents edge artifacts between hard boundaries (see referenced figure).

Comparison of kernel functions. (a) Single-sided kernel: non-zero only inside the triangle ($x \leq 1$), providing no exterior gradient and shrinking when softened. (b) Bilateral kernel: extends beyond the triangle boundary, but the boundary weight at $x=1$ is fixed, creating visual expansion when softened and edge artifacts when sharpened (see referenced figure). (c) Our elastic kernel: the boundary weight $G(1) = \exp(-\sigma)$ is flexible with respect to $\sigma$, enabling wide gradient support while keeping stable visual size.

Edge artifacts for connected triangles. Transparent seams are unavoidable for soft kernels with high $\sigma$. However, only our elastic kernel produces seamless rendering with low $\sigma$ (hard boundary) thanks to its flexible value on the boundary.

Shape Reconstruction

Convergence speed comparison for mesh optimization. Our elastic kernel converges faster and better than the single-sided (TS) and bilateral (2DTS) kernels on shapes from the FAMOUS dataset. Results are averaged over all objects reported in the paper.

1 / 4

Mesh optimization processes on Armadillo, Demosthenes, Goathead, and Hand Closed from the FAMOUS dataset. Our elastic kernel recovers finer geometric details in a shorter time than Triangle Splatting and 2DTS.

Armadillo

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

Nvdiffrast-F

Nvdiffrast-V

GT

Triangle Splatting

2DTS

Ours

1 / 14

Interactive viewers for optimized meshes. Our elastic kernel recovers more overall structure and finer geometric details than Triangle Splatting and 2DTS.

Novel-View Synthesis

Method	Mip-NeRF 360			Tanks&Temples			Deep Blending
	PSNR↑	SSIM↑	LPIPS↓	PSNR↑	SSIM↑	LPIPS↓	PSNR↑	SSIM↑	LPIPS↓
3DGS	27.58	0.813	0.220	23.83	0.853	0.169	29.79	0.910	0.238
2DGS	26.82	0.794	0.260	23.19	0.833	0.212	29.53	0.900	0.256
TS	27.02	0.804	0.200	23.10	0.855	0.144	29.02	0.891	0.242
2DTS	27.16	0.808	0.194	23.00	0.851	0.143	29.20	0.895	0.229
Ours	27.33	0.813	0.189	23.08	0.858	0.134	29.27	0.898	0.224

Quantitative results of novel-view synthesis on Mip-NeRF 360, Tanks&Temples, and Deep Blending. Among triangle-based methods, our elastic kernel achieves the best perceptual quality (SSIM, LPIPS) across all three benchmarks.

Colored Triangle Soup

1 / 3

Interactive viewers for optimized triangle soups. The triangle soups are optimized with our elastic kernel and are directly compatible with modern 3D graphics frameworks. We show colored faces, colored wireframes, normal map, and depth map in the preview image.

Acknowledgments

This project follows Triangle Splatting and 2DTS. We sincerely thank their authors for pioneering the research and sharing their code.

This work was supported by the ERC Advanced Grant “SIMULACRON” (agreement #884679), the GNI Project “AI4Twinning”, and the DFG project CR 250/26-1 “4DYoutube”.