Elastic Triangle Splatting

Tian Shi; Shenhan Qian; Daniel Cremers

cs.CVAug 29, 2026

Elastic Triangle Splatting

Authors: Tian Shi, Shenhan Qian, Daniel Cremers

Organizations: Technical University of Munich · MCML

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 of 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 an adaptive boundary value, which are shown to be essential for 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.

Figures & tables

Figure 2: A triangle with a softened boundary.

Kernel	Support	$G(1)$	Exterior grad.	Main limitation
Triangle Splatting [ 8 ]	inside	0	no	dead zones
2DTS [ 26 ]	bilateral	fixed	yes	boundary bias
Ours	bilateral	adaptive	yes	–

Table 1: Kernel design axes. Existing triangle kernels cover either interior-only support or bilateral support with fixed boundary weight. Our kernel keeps bilateral gradients while making the boundary value adaptive through

\sigma

Figure 3: Comparison of kernel functions. (a) Single-sided kernel [ 8 ] : non-zero only inside the triangle ( $x\leq 1$ ), providing no exterior gradient and shrinking when softened. (b) Bilateral kernel [ 26 ] : extends beyond the triangle boundary, but the boundary weight at $x=1$ is fixed, creating edge artifacts when sharpened and visual expansion when softened. (c) Our elastic kernel: the boundary weight $G(1)=\exp(-\sigma)$ is flexible with respect to $\sigma$ , enabling wide gradient support while maintaining a stable visual size.

Figure 4: 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.

	$\lambda_{\sigma}$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	CD( $\times 10^{-3}$ ) $\downarrow$
Nvdiffrast-F [ 14 ]	-	35.99	0.983	0.031	1.775
Nvdiffrast-V [ 14 ]	-	40.95	0.995	0.008	0.469
Triangle Splatting [ 8 ]	$10^{-3}$	37.15	0.983	0.040	2.989
2DTS [ 26 ]	$10^{-3}$	38.43	0.987	0.038	0.926
Ours	$10^{-3}$	40.67	0.991	0.023	0.717
Triangle Splatting [ 8 ]	$10^{-4}$	29.06	0.958	0.068	5.369

Table 2: Quantitative results of shape reconstruction on the FAMOUS dataset [ 27 ] . Our elastic kernel consistently outperforms splatting-based baselines under both

\lambda_{\sigma}

settings, approaching the accuracy of Nvdiffrast-V, which additionally relies on triangle connectivity to interpolate vertex colors.

Figure 5: Convergence speed comparison for mesh optimization. We compare our elastic kernel against the single-sided (TS) and bilateral (2DTS) kernels on shapes from the FAMOUS dataset, averaged over all objects reported in the paper.

Figure 6: Qualitative mesh reconstruction on the FAMOUS dataset [ 27 ] . Our elastic kernel recovers sharper surfaces and finer geometric details than Triangle Splatting [ 8 ] and 2DTS [ 26 ] .

Figure 7: Qualitative results of novel-view synthesis on the Mip-NeRF 360 [ 1 ] and Tanks&Temples [ 13 ] datasets. Our kernel function results in better visual details and fewer floaters and artifacts.

	Mip-NeRF 360			Tanks&Temples			Deep Blending
	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
3DGS [ 11 ]	27.58	0.813	0.220	23.83	0.853	0.169	29.79	0.910	0.238
2DGS [ 9 ]	26.82	0.794	0.260	23.19	0.833	0.212	29.53	0.900	0.256
TS [ 8 ]	27.02	0.804	0.200	23.10	0.855	0.144	29.02	0.891	0.242
2DTS [ 26 ]	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

Table 3: Quantitative results of novel-view synthesis on Mip-NeRF 360 [ 1 ] , Tanks&Temples [ 13 ] , and Deep Blending [ 6 ] . Our method is the most competitive among all triangle-based methods.

Figure 8: Sharp and opaque rendering of optimized triangle soups on the Garden, Bicycle, and Kitchen scenes from Mip-NeRF 360 [ 1 ] .

cull small tri.	Mip-NeRF 360			Tanks&Temples			Deep Blending
cull small tri.	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
✓	27.33	0.813	0.189	23.08	0.858	0.134	29.27	0.898	0.224
	27.32	0.813	0.190	23.12	0.859	0.133	29.28	0.900	0.224

Table 4: Ablation study on small triangle filtering . Not culling small triangles has negligible effect on novel-view synthesis performance.

Scale	$\lambda_{e}$	$\lambda_{n}$	$\lambda_{\text{lap}}$
$0.1\times$	0.891	0.648	0.694
$1\times$	0.607	0.607	0.607
$10\times$	2.097	0.666	1.935

Table 5: Regularization sensitivity on a FAMOUS subset. Values are Chamfer Distance (

\times 10^{-3}

)

\downarrow

after changing one regularization weight at a time.

	training time $\downarrow$	# triangles	FPS $\uparrow$	rendering time $\downarrow$
Triangle Splatting	58.54 min	3.71M	61.9	4.3 ms
2DTS	64.23 min	3.58M	58.4	4.8 ms
Ours	68.19 min	3.72M	57.5	4.7 ms

Table 6: Efficiency comparison on Mip-NeRF 360. Rendering time is measured per 1 million triangles.

	Nvdiffrast-F	Nvdiffrast-V	TS	2DTS	Ours	TS	2DTS	Ours
$\lambda_{\sigma}$	-	-	$10^{-3}$	$10^{-3}$	$10^{-3}$	$10^{-4}$	$10^{-4}$	$10^{-4}$
armadillo	0.14	0.061	1.649	0.084	0.079	7.789	0.079	0.077
boot	1.515	0.082	0.601	0.631	0.192	2.373	0.174	0.129
brucewick	3.463	2.146	11.177	2.393	2.613	7.474	2.381	2.453
bunny_hr	0.782	0.047	2.362	0.15	0.066	4.548	0.072	0.064
cat-low-resolution	0.571	0.05	0.713	0.436	0.059	1.123	0.121	0.056

Table 7: CD(

\times 10^{-3}

)

\downarrow

on FAMOUS [ 27 ] .

	bicycle	bonsai	counter	flowers	garden	kitchen	room	stump	treehill
3DGS	25.15	32.48	29.18	21.41	27.37	31.56	31.90	26.69	22.50
2DGS	24.61	31.38	28.15	20.81	26.66	30.28	30.83	26.19	22.44
TS	24.54	32.13	28.90	20.47	27.00	31.42	31.10	26.00	21.61
2DTS	24.70	32.63	28.99	20.56	27.19	31.50	31.44	25.91	21.54
Ours	24.92	32.79	29.24	20.79	27.42	31.41	31.77	26.10	21.53

Table 8: PSNR

\uparrow

on Mip-NeRF 360 [ 1 ] .

	bicycle	bonsai	counter	flowers	garden	kitchen	room	stump	treehill
3DGS	0.748	0.948	0.916	0.588	0.857	0.933	0.928	0.768	0.635
2DGS	0.713	0.936	0.900	0.554	0.834	0.921	0.915	0.753	0.619
TS	0.738	0.949	0.914	0.585	0.849	0.932	0.930	0.744	0.593
2DTS	0.743	0.952	0.917	0.590	0.852	0.933	0.933	0.751	0.603
Ours	0.754	0.954	0.920	0.597	0.858	0.935	0.936	0.758	0.608

Table 9: SSIM

\uparrow

on Mip-NeRF 360 [ 1 ] .

	bicycle	bonsai	counter	flowers	garden	kitchen	room	stump	treehill
3DGS	0.241	0.180	0.182	0.359	0.122	0.116	0.195	0.243	0.346
2DGS	0.306	0.204	0.214	0.404	0.163	0.138	0.222	0.290	0.398
TS	0.224	0.143	0.157	0.316	0.122	0.107	0.163	0.242	0.322
2DTS	0.220	0.137	0.150	0.307	0.117	0.106	0.161	0.235	0.317
Ours	0.208	0.138	0.148	0.299	0.111	0.105	0.160	0.226	0.311

Table 10: LPIPS

\downarrow

on Mip-NeRF 360 [ 1 ] .

	Train			Truck			Drjohnson			Playroom
	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
3DGS	22.21	0.82	0.195	25.44	0.89	0.142	29.49	0.91	0.236	30.08	0.91	0.240
2DGS	21.26	0.79	0.250	25.12	0.87	0.173	28.90	0.90	0.257	30.15	0.91	0.260
TS	21.22	0.82	0.182	24.99	0.89	0.105	28.87	0.89	0.250	29.18	0.89	0.234
2DTS	21.24	0.82	0.180	24.77	0.88	0.106	28.81	0.89	0.240	29.58	0.90	0.217
Ours	21.47	0.83	0.162	24.68	0.89	0.105	28.91	0.90	0.235	29.64	0.90	0.214

Table 11: Per-scene results of novel-view synthesis on Tanks&Temples [ 13 ] and Deep Blending [ 6 ] .

Differentiable Rendering

Differentiable rendering makes image formation and its gradients computable, enabling visual scenes and graphics to be optimized from observations.

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