CF-Splat: Closed-Form Triangle Splatting with Mesh Priors

Yixin Fang; Liang Zhao

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

\times

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 ]	$\checkmark$	surgical	tiny	$540\times 675$	colonoscopic video / phantom
OmniObject3D [ 42 ]	$\checkmark$	object-centric	small	$800\times 800$	structured-light scanner
BundleFusion [ 25 ]	$\checkmark$	indoor	medium	$480\times 640$	RGB-D camera
Oxford Spires [ 7 ]	$\checkmark$	outdoor	large	$1080\times 1440$	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\geq 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 $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
$0/0$	3DGS	28.08	0.852	0.253
$0/0$	CF-Splat	29.16	0.854	0.232
$0.1/0.5$	3DGS	26.56	0.815	0.293
$0.1/0.5$	CF-Splat	28.07	0.823	0.267
$0.3/2$	3DGS	23.00	0.751	0.378
$0.3/2$	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 $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	Error (cm) $\downarrow$
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 $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	Time (s) $\downarrow$	GPU (GB) $\downarrow$	FPS $\uparrow$
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 $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	Time (s) $\downarrow$	GPU (GB) $\downarrow$	FPS $\uparrow$
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.

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