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.
Figures & tables
Fig. 1: Evaluation of CF-Splat against 3DGS [ 5 ] on the college scene from the large-scale Oxford Spires dataset [ 7 ] . The input consists of 760 views at 1080 × 1440 resolution and a mesh with 1.55M vertices reconstructed using the SDF produced by VDBFusion [ 8 ] and the LiDAR poses estimated by HBA [ 9 ] . Given the same input geometry source, 3DGS jointly optimizes millions of coupled Gaussian parameters through 30K iterations to minimize the photometric residual ( L1 loss ), which improves image-space fitting at the cost of increased geometric error ( Chamfer distance ). In contrast, CF-Splat directly admits a closed-form solution without any iteration , while achieving 2.84 dB higher PSNR on the visualized test view, requiring only 1.15 min of computation time and 4.18 GB of peak GPU memory , compared with 48.93 min and 7.18 GB for 3DGS.
Algorithm 1 Closed-Form Scene Appearance Estimation
Algorithm 2 Triangle-Structured Matrix-Free PCG
Dataset
Real-world
Type
Scene Scale
RGB Resolution
Data Source
C3VD [ 41 ]
✓
surgical
tiny
540×675
colonoscopic video / phantom
OmniObject3D [ 42 ]
✓
object-centric
small
800×800
structured-light scanner
BundleFusion [ 25 ]
✓
indoor
medium
480×640
RGB-D camera
Oxford Spires [ 7 ]
✓
outdoor
large
1080×1440
RGB camera / LiDAR
TABLE I: The Datasets Used for Evaluation.
Method
c_t1_b
c_t2_b
c_t3_a
s_t1_a
s_t2_a
s_t3_a
t_t1_b
t_t2_c
t_t4_a
t_t4_b
Average
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
3DGS [ 5 ]
31.95
0.902
0.356
34.31
0.918
0.267
33.49
0.912
0.235
37.17
0.914
0.394
35.65
0.893
0.263
36.17
0.895
0.325
37.67
0.924
0.306
32.11
0.909
0.416
32.64
0.888
0.246
32.24
0.847
0.332
34.34
0.900
0.314
2DGS [ 11 ]
28.01
0.870
0.450
31.77
0.903
0.308
30.43
0.886
0.304
36.01
0.911
0.404
34.37
0.876
0.305
33.86
0.883
0.367
37.01
0.920
0.317
30.71
0.906
0.417
29.81
0.860
0.302
30.90
0.823
0.373
32.29
0.884
0.355
PGSR [ 12 ]
30.72
0.892
0.369
32.57
0.904
0.284
31.81
0.889
0.274
36.57
0.913
0.374
35.10
0.887
0.262
35.40
0.892
0.314
37.24
0.920
0.289
31.61
0.906
0.415
31.98
0.874
0.259
32.13
0.836
0.337
33.51
0.891
0.318
MVGS [ 10 ]
32.63
0.907
0.326
34.90
0.923
0.249
33.95
0.915
0.222
37.67
0.918
0.377
36.83
0.913
0.208
36.67
0.903
0.298
38.48
0.930
0.278
30.88
0.903
0.399
33.35
0.895
0.218
32.92
0.855
0.306
34.83
0.906
0.288
GaMeS [ 13 ]
30.79
0.896
0.335
29.50
0.888
0.327
31.39
0.887
0.271
36.41
0.913
0.384
32.70
0.845
0.344
34.22
0.873
0.360
35.88
0.909
0.316
33.65
0.920
0.358
30.36
0.849
0.336
20.71
0.752
0.476
31.56
0.873
0.351
TABLE II: Quantitative NVS Comparison Results on C3VD Dataset [ 41 ] . First , Second , and Third Indicate Ranking Results.
Fig. 2: Qualitative NVS comparison results on c_t1_b , s_t2_a and t_t2_c scenes from C3VD dataset [ 41 ] . The red boxes highlight the differences.
Method
apple
backpack
ball
bed
banana
battery
book
bottle
bowl
box
Average
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
3DGS [ 5 ]
24.91
0.577
0.119
38.03
0.982
0.054
34.22
0.961
0.090
37.62
0.982
0.048
33.74
0.926
0.033
33.82
0.981
0.019
40.64
0.996
0.020
25.57
0.679
0.127
38.45
0.990
0.058
23.98
0.859
0.138
33.10
0.893
0.070
2DGS [ 11 ]
41.46
0.985
0.035
39.45
0.988
0.038
38.05
0.986
0.038
38.15
0.989
0.027
38.38
0.982
0.024
37.63
0.992
0.011
40.77
0.997
0.016
43.52
0.998
0.007
39.57
0.994
0.035
37.08
0.984
0.026
39.41
0.989
0.026
PGSR [ 12 ]
40.36
0.986
0.033
38.99
0.986
0.038
35.94
0.979
0.046
39.31
0.987
0.029
42.19
0.991
0.009
43.23
0.996
0.005
40.50
0.995
0.016
42.82
0.996
0.008
39.49
0.991
0.040
39.10
0.988
0.020
40.19
0.990
0.024
MVGS [ 10 ]
46.04
0.994
0.013
40.08
0.989
0.024
39.36
0.989
0.021
40.94
0.988
0.019
46.30
0.996
0.005
46.65
0.996
0.002
42.40
0.995
0.012
43.81
0.995
0.008
40.99
0.993
0.024
42.50
0.994
0.009
42.91
0.993
0.014
GaMeS [ 13 ]
34.51
0.981
0.037
37.75
0.981
0.053
34.47
0.943
0.117
36.54
0.977
0.056
43.68
0.996
0.008
42.39
0.997
0.006
40.54
0.996
0.018
43.76
0.998
0.008
38.26
0.991
0.042
37.95
0.989
0.029
38.98
0.985
0.037
TABLE III: Quantitative NVS Comparison Results on OmniObject3D Dataset [ 42 ] . First , Second , and Third Indicate Ranking Results.
Fig. 3: Qualitative NVS comparison results on backpack , book and box from OmniObject3D Dataset [ 42 ] . The red boxes highlight the rendering differences.
Method
apt0
apt1
apt2
copyroom
office0
office1
office2
office3
Average
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
3DGS [ 5 ]
27.49
0.838
0.231
27.32
0.815
0.269
22.28
0.709
0.347
28.29
0.903
0.194
27.09
0.853
0.258
25.36
0.791
0.278
24.87
0.742
0.314
25.00
0.823
0.265
25.96
0.809
0.270
2DGS [ 11 ]
27.26
0.846
0.271
27.34
0.831
0.308
24.01
0.766
0.347
28.38
0.910
0.219
27.27
0.857
0.298
25.56
0.804
0.317
25.34
0.774
0.365
25.69
0.844
0.288
26.36
0.829
0.302
PGSR [ 12 ]
27.37
0.841
0.261
26.69
0.815
0.302
23.62
0.753
0.339
28.25
0.908
0.209
26.89
0.853
0.280
25.38
0.799
0.301
24.51
0.756
0.348
25.42
0.835
0.285
26.02
0.820
0.291
MVGS [ 10 ]
28.45
0.852
0.236
28.03
0.828
0.275
23.98
0.746
0.335
29.67
0.916
0.185
25.14
0.848
0.308
24.69
0.796
0.325
24.47
0.769
0.381
24.97
0.842
0.293
26.18
0.825
0.292
GaMeS [ 13 ]
26.71
0.838
0.259
26.29
0.820
0.299
23.50
0.753
0.353
25.76
0.890
0.232
25.21
0.848
0.292
24.50
0.789
0.317
24.09
0.756
0.357
25.03
0.840
0.289
25.14
0.817
0.300
TABLE IV: Quantitative NVS Comparison Results on BundleFusion Dataset [ 25 ] . First , Second , and Third Indicate Ranking Results.
Fig. 4: Qualitative NVS comparison results on apt1 , copyroom and office0 from BundleFusion Dataset [ 25 ] . The red boxes highlight the differences.
Method
college
quarter
library
church
palace
Average
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
3DGS [ 5 ]
24.55
0.754
0.351
18.86
0.685
0.402
17.72
0.689
0.434
17.26
0.680
0.432
19.71
0.743
0.404
19.62
0.710
0.405
2DGS [ 11 ]
22.82
0.731
0.430
18.55
0.710
0.456
17.09
0.696
0.492
17.11
0.708
0.548
20.09
0.773
0.512
19.13
0.724
0.488
PGSR [ 12 ]
22.77
0.728
0.442
19.56
0.722
0.490
19.49
0.730
0.513
18.04
0.713
0.548
20.22
0.777
0.537
20.02
0.734
0.506
MVGS [ 10 ]
25.08
0.767
0.376
19.88
0.720
0.442
18.96
0.726
0.490
18.04
0.712
0.526
19.67
0.758
0.512
20.33
0.737
0.469
GaMeS [ 13 ] ∗
22.39
0.730
0.485
18.46
0.681
0.492
–
–
–
19.31
0.686
0.558
20.21
0.734
0.612
20.09
0.708
0.537
TABLE V: Quantitative NVS Comparison Results on Oxford Spires Dataset [ 7 ] . First , Second , and Third Indicate Ranking Results.
Fig. 5: Qualitative NVS comparison results on college , quarter and palace from Oxford Spires Dataset [ 7 ] . The red boxes highlight the rendering differences.
Fig. 6: Surface geometry and appearance evaluation on the apple , banana , and bottle scenes from OmniObject3D [ 42 ] . For each method, we show the reconstructed surface, appearance-error map and geometric-error map from left to right. CIEDE2000 reference is attached.
Fig. 7: Surface geometry and appearance evaluation on the office1 scene from BundleFusion [ 25 ] and the palace scene from Oxford Spires [ 7 ] . The top pictures are the estimated scene of our CF-Splat.
Fig. 8: Efficiency comparison on the college scene from Oxford Spires [ 7 ] . The curves show PSNR as a function of computation time for the iterative baselines, while the marker denotes the result obtained by CF-Splat. The inset reports the corresponding peak GPU memory consumption.
Method
C3VD [ 41 ]
OmniObject3D [ 42 ]
BundleFusion [ 25 ]
Oxford Spires [ 7 ]
Time (min)
GPU (GB)
Time (min)
GPU (GB)
Time (min)
GPU (GB)
Time (min)
GPU (GB)
3DGS [ 5 ]
11.52
2.18
11.34
1.47
25.53
5.93
48.90
7.14
2DGS [ 11 ]
14.74
2.47
14.94
1.83
41.42
6.96
42.50
6.43
PGSR [ 12 ]
51.32
3.06
43.17
2.24
85.19
7.07
63.58
7.26
MVGS [ 10 ]
43.99
4.28
36.69
3.07
77.69
7.43
74.00
7.76
GaMeS [ 13 ]
45.67
2.54
30.90
1.74
87.05
5.08
119.40
7.47
TABLE VI: Efficiency comparison across the evaluated datasets. First , Second , and Third indicate the ranking results.
Fig. 9: Numerical accuracy and efficiency of matrix-free PCG under different stopping tolerances on the backpack scene from OmniObject3D [ 42 ] , with the explicitly assembled solver used as the numerical reference. Explicit means solving the same global LLS formulation through a factorization-based solver. (a) Runtime and PCG iteration count. (b) GPU memory and mean RGB residual MAE.
Fig. 10: Scalability of CF-Splat (w/o densification) with respect to the number of input views on the college scene from Oxford Spires [ 7 ] , with 1.55M input mesh vertices. (a–c) Reconstruction quality, runtime, and peak GPU memory compared with baseline methods. (d) Runtime breakdown of CF-Splat. (e) PCG iteration counts and final relative residuals, with the dashed line indicating the convergence tolerance.
Fig. 11: Scalability of CF-Splat (w/o densification) with respect to the number of input mesh vertices on the college scene from Oxford Spires [ 7 ] , with 760 input views. (a) Reconstruction quality. (b) Peak GPU memory. (c) Runtime breakdown of CF-Splat. (d) PCG iteration counts and final relative residuals, with the dashed line indicating the convergence tolerance.
Fig. 12: Effect of adaptive SH regularization under uneven observations, on the c_t1_b scene from C3VD [ 41 ] , using K=7 input views and maximum SH degree L=3 . (a) PSNR on input and test views. (b) PCG solving records, with solve time, iteration count, and final residual MSE. (c) Per-vertex SH magnitudes over the number of observing input views. (d) SH magnitude distributions over observed vertices. Dashed and solid lines in (c) and (d) denote the DC ( l=0 ) and non-DC ( l≥1 ) SH components, respectively.
Fig. 13: Qualitative effect of adaptive SH regularization in Fig. 12 . The regions highlighted by the purple and blue boxes are observed in input view 1 but not in input view 2, resulting in insufficient directional support for higher-order SH components according to ( 25 ). Without adaptive SH regularization, the weakly constrained SH components produce noisy colors in the test view.
Rot./Trans. ( ∘ /cm)
Method
PSNR ↑
SSIM ↑
LPIPS ↓
0/0
3DGS
28.08
0.852
0.253
CF-Splat
29.16
0.854
0.232
0.1/0.5
3DGS
26.56
0.815
0.293
CF-Splat
28.07
0.823
0.267
0.3/2
3DGS
23.00
0.751
0.378
CF-Splat
25.08
0.759
0.333
TABLE VII: Effect of pose noise on the Apt0 scene from BundleFusion [ 25 ] . Rotation and translation noise denote per-axis standard deviations.
Fig. 14: Visualized comparison of 3DGS and CF-Splat under increasing camera pose perturbations, on the apt0 scene from BundleFusion [ 25 ] . PSNR of visualized test view is reported for each noise setting.
Voxel size
Vertices (M)
PSNR ↑
SSIM ↑
LPIPS ↓
Error (cm) ↓
0.5 cm
11.92
29.17
0.872
0.212
8.29
1 cm
2.39
29.10
0.869
0.219
8.98
2 cm
0.48
29.07
0.867
0.222
9.86
4 cm
0.10
28.51
0.863
0.223
11.81
TABLE VIII: Effect of Mesh Reconstruction Errors Induced by Different TSDF voxel sizes on the apt0 scene from BundleFusion [ 25 ] . Best Results Are Highlighted in Bold.
Fig. 15: Render (top) and mesh (bottom) results of CF-Splat using input meshes reconstructed with different TSDF voxel sizes. PSNR is reported for each setting.
SH Degree
PSNR ↑
SSIM ↑
LPIPS ↓
Time (s) ↓
GPU (GB) ↓
FPS ↑
0
35.39
0.968
0.062
0.96
1.153
3027.6
1
38.46
0.979
0.054
0.98
1.154
3018.0
2
38.63
0.982
0.043
1.19
1.157
2874.4
3
39.61
0.989
0.025
1.53
1.160
2853.3
TABLE IX: Ablation of the maximum SH degree L on the backpack scene from OmniObject3D [ 42 ] .
Rounds
Vertices (K)
PSNR ↑
SSIM ↑
LPIPS ↓
Time (s) ↓
GPU (GB) ↓
FPS ↑
0
22.4
39.31
0.986
0.027
0.81
1.146
3959.9
10
51.4
39.61
0.989
0.025
1.53
1.160
2853.3
20
77.2
39.66
0.989
0.023
2.43
1.174
2159.6
40
125.9
39.67
0.990
0.023
4.28
1.192
1581.8
TABLE X: Ablation of Gradient-Guided Triangle Densification on The backpack Scene from OmniObject3D [ 42 ] .
Fig. 16: Qualitative effect of gradient-guided adaptive densification on the backpack scene from OmniObject3D [ 42 ] . The highlighted regions show the appearance details.
Fig. 17: Preconditioner ablation on the apt0 scene from BundleFusion [ 25 ] , using a PCG tolerance of 10−6 and a maximum of 300 iterations. No precond. means the proposed PCG solver without any preconditioner. (a) PCG solve time and iteration count. (b) Peak GPU memory and mean RGB residual MAE. OOM exceeds the available GPU memory.
Fig. 18: Comparison between CF-Splat and CF-Splat (Adam) on the college scene from Oxford Spires [ 7 ] . (a) PSNR vs. computation time. (b) Final SSIM and LPIPS.
Explicit primitive-based radiance fields such as 3D Gaussian Splatting typically model view-dependent appearance using low-order spherical harmonics (SH). Although efficient to evaluate, SH coefficients dominate per-primitive storage and memory traffic, while their band-limited basis restricts angular detail. We present a thorough, end-to-end comparison of SH and recent spherical appearance models and introduce an implicit alternative that decodes compact per-primitive latent codes using a tiny shared MLP. We integrate all models into the same optimized pipeline, fusing their forward and backward passes into a differentiable CUDA rasterizer and provide a portable WebGL viewer for laptop and mobile GPUs. Our evaluation across reconstruction quality, memory use, and optimization and rendering performance shows that recent spherical models offer the strongest overall quality-efficiency trade-off. Our neural representation is the most compact model evaluated and, compared to third-degree SH, reduces the per-primitive appearance footprint from 192 to 28 bytes, accelerates optimization by 1.3×, while improving reconstruction quality. We further analyze how appearance parametrization shapes optimization, identifying differences in recovered geometry and the tendency of expressive models to absorb non-static scene content. Together, our framework and analysis provide practical guidance for replacing SH beyond what image metrics alone can capture.
Florian Hahlbohm, Jorge Condor, Linus Franke +2
TU Braunschweig · Universit`a della Svizzera italiana · University of Würzburg +1
While 3D Vision Foundation Models (3DVFMs) have demonstrated remarkable zero-shot capabilities in visual geometry estimation, their direct application to generalizable novel view synthesis (NVS) remains challenging. In this paper, we propose AirSplat, a novel training framework that effectively adapts the robust geometric priors of 3DVFMs into high-fidelity, pose-free NVS. Our approach introduces two key technical contributions: (1) Self-Consistent Pose Alignment (SCPA), a training-time feedback loop that ensures pixel-aligned supervision to resolve pose-geometry discrepancy; and (2) Rating-based Opacity Matching (ROM), which leverages the local 3D geometry consistency knowledge from a sparse-view NVS teacher model to filter out degraded primitives. Experimental results on large-scale benchmarks demonstrate that our method significantly outperforms state-of-the-art pose-free NVS approaches in reconstruction quality. Our AirSplat highlights the potential of adapting 3DVFMs to enable simultaneous visual geometry estimation and high-quality view synthesis.
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.