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 σ .
Figure 3: Comparison of kernel functions. (a) Single-sided kernel [ 8 ] : non-zero only inside the triangle ( x≤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(−σ) is flexible with respect to σ , 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 σ . However, only our elastic kernel produces seamless rendering with low σ (hard boundary) thanks to its flexible value on the boundary.
λσ
PSNR ↑
SSIM ↑
LPIPS ↓
CD( ×10−3 ) ↓
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 λσ 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 ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
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
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
✓
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
λe
λn
λlap
0.1×
0.891
0.648
0.694
1×
0.607
0.607
0.607
10×
2.097
0.666
1.935
Table 5: Regularization sensitivity on a FAMOUS subset. Values are Chamfer Distance ( ×10−3 ) ↓ after changing one regularization weight at a time.
training time ↓
# triangles
FPS ↑
rendering time ↓
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
λσ
-
-
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( ×10−3 ) ↓ 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 ↑ 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 ↑ 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 ↓ on Mip-NeRF 360 [ 1 ] .
Train
Truck
Drjohnson
Playroom
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
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 ] .
Triangle- and mesh-based neural rendering seeks to bridge neural representations and graphics engines through triangle primitives compatible with standard rasterization. Yet existing methods were developed and evaluated under inconsistent settings, with few direct comparisons and little evaluation in graphics engines. We introduce MeshSuite, the first benchmark for systematic evaluation of engine deployment and Non-Engine Render in triangle- and mesh-based neural rendering. MeshSuite evaluates two deployment settings: Standard Deployment, using an opaque mesh pipeline with vertex colors and hardware Z-buffering; and Dedicated Deployment, using method-specific engine implementations to preserve appearance and compositing features. For mesh splatting, we also audit exported surfaces for topological and geometric integrity as downstream graphics assets. Our results show that engine deployment reduces image quality in both settings; among evaluated methods, MeshSplatting degrades least under Standard Deployment. Dedicated Deployment preserves most fidelity at roughly 5-33x the runtime of Standard Deployment across two deployment datasets. For mesh splatting, explicit connectivity and shared indexing remain limited: shared vertex indexing alone guarantees neither manifoldness nor global connectivity. These findings show that rasterizability is only part of graphics readiness and underscore the need to assess engine deployment. Project page: https://surrey-up-lab.github.io/MeshSuite.
Kaixuan Zhang, Minxian Li, Mingwu Ren +1
Nanjing University of Science and Technology · University of Surrey
Splatting-based radiance-field representations have recently gained popularity in robotic perception, enabling high-quality novel-view synthesis (NVS) through explicit scene primitives. However, existing splatting methods typically rely on iterative nonlinear optimization that jointly optimizes millions of coupled primitive parameters, resulting in substantial computation time, peak GPU memory consumption, and susceptibility to local minima. In this paper, we propose CF-Splat, a novel closed-form triangle-splatting framework for efficient scene appearance estimation over a known mesh either pre-existing or provided by upstream mapping or reconstruction pipelines. Our key insight is that, by properly formulating the problem over the given mesh geometry and calibrated cameras, the rendered pixel colors can be expressed linearly with respect to the per-vertex spherical harmonics (SH) coefficients, allowing multi-view scene appearance estimation to be formulated as a global linear least-squares (LLS) problem without the need of iterative nonlinear optimization. To guarantee a well-posed solution under uneven multi-view observation coverage, we introduce adaptive SH regularization that constrains insufficiently supported high-order SH coefficients and yields a unique global minimizer. To ensure the scalability to large-scale scenes, we develop a triangle-structured matrix-free preconditioned conjugate-gradient solver that avoids explicitly constructing the large sparse normal system while producing the same LLS solution. Experiments on diverse real-world datasets demonstrate that CF-Splat achieves competitive or superior NVS quality compared with state-of-the-art splatting methods. While, CF-Splat requires approximately 27.5-378.0 times less computation time and 1.8 times lower peak GPU memory on average than 3DGS across scenes of different scales on a consumer-grade laptop.
Recent radiance field methods represent scenes with 2D primitives that offer surface alignment and efficient rasterization, from Gaussian disks to triangles, yet all rely on convex boundaries: curved and concave structures demand excessive primitives. We introduce Deformable Triangle Splatting, which augments each triangle with K control points per edge, each parameterized by a single learnable scalar displacement that shifts the boundary inward or outward, enabling non-convex shape representation while preserving the three base vertices that define the 3D plane. To render these non-convex primitives differentiably, we design a rasterization pipeline in the triangle's barycentric coordinate space, ensuring view-consistent rendering. A winding number test determines whether each pixel lies inside the deformed primitive, and a window function controlled by two learnable parameters, sharpness and corner smoothness, together with a per-primitive scalar opacity, produces the smooth opacity transition from interior to boundary. Validation is done in a variety of real-world scenes, outperforming recent works based on non-volumetric primitives in terms of visual quality and versatility while still achieving competitive rendering efficiency.
Oriol Jiménez-Ayguadé, Antonio Agudo
Institut de Robòtica i Informàtica Industrial, CSIC-UPC, Barcelona, Spain