DiffSoup: Direct Differentiable Rasterization of Triangle Soup for Extreme Radiance Field Simplification

Kenji Tojo; Bernd Bickel; Nobuyuki Umetani

Figure 1: (a) Given multi-view RGB images, (b) we reconstruct a simplified radiance field as a textured triangle soup using differentiable rasterization. (c) The resulting scene can be rendered within a standard depth-tested rasterization pipeline, enabling seamless visualization across devices—including a high-end desktop with a dedicated GPU, a consumer-grade laptop (MacBook Pro with an M3 Pro chip), and even a smartphone (iPhone 15).

Figure 2: Overview of our algorithm. (a) We represent radiance fields as a textured triangle soup, where barycentric coordinates map to learnable color and alpha textures. (b) At run time, this representation is rendered with binary opacity, enabling efficient rasterization via standard depth testing. (c) Rasterization produces a high-dimensional color-feature image, which is then mapped to RGB values using a small shared MLP trained to minimize the error between the rendering and the ground-truth image. (d) During training, we initialize triangles around structure-from-motion (SfM) keypoints. (e) We optimize opacity through stochastic opacity masking, while vertex positions are updated using edge gradients for opaque triangles.

Figure 3: (a) Alpha blending computes cumulative transmittance over depth-sorted samples and produces the final color as a weighted average. (b) Our stochastic opacity masking, in contrast, simulates a discrete stochastic process over the surface, allowing the renderer to output the exact color of the selected fragment.

Figure 4: Multi-resolution triangle texture. Top: We parameterize the texture using multiple grid resolutions, corresponding to different triangle subdivision levels. Bottom: (a) At run time, the renderer queries only the highest-resolution triangle texture. (b) At training time, this highest-resolution texture is over-parameterized with lower-resolution grids to improve optimization stability.

Figure 5: Qualitative results on the MipNeRF360 [ 3 ] dataset. (Left to right) Ground-truth test images; views synthesized by 3DGS [ 24 ] , Triangle Splatting [ 17 ] , TexGS [ 5 ] , our method; and a visualization of our reconstructed triangle geometry. Rows correspond to different scenes, and all models use 15K primitives.

Method	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	# Prims
3DGS [ 24 ]	23.72	0.664	0.420	15K
TS [ 17 ]	22.81	0.634	0.430	15K
TexGS [ 5 ]	24.80	0.697	0.270	15K
Ours	24.76	0.748	0.204	15K

Table 1: Quantitative evaluation on the MipNeRF360 [ 3 ] dataset. For PSNR, SSIM, and LPIPS, we report scene-level averages. The number of primitives is strictly enforced for all methods.

Figure 6: Qualitative results on synthetic scenes. (Left to right) Ground-truth views; view-synthesis results from MobileNeRF; a mesh decimated from the MobileNeRF output using QEM [ 12 ] ; our results optimized from random initialization; and our results initialized from the MobileNeRF-decimated mesh.

Method	NeRF-Synthetic [ 36 ] (PSNR $\uparrow$ )					Shelly [ 50 ] (PSNR $\uparrow$ )
	ship	chair	mic	drums	# Faces $\downarrow$	khady	kitten	pug	# Faces $\downarrow$
MobileNeRF [ 7 ]	26.06	31.02	29.18	23.82	159K	26.22	30.05	29.98	275K
MobileNeRF + QEM [ 12 ]	8.44	16.34	17.35	13.35	15K	11.21	14.42	11.25	15K
Ours w/ QEM initialization	26.68	32.11	30.18	23.94	15K	26.55	31.23	30.51	15K
Ours w/ random initialization	25.29	31.71	29.64	23.47	15K	26.67	30.68	30.16	15K

Table 2: Comparison with MobileNeRF [ 7 ] on synthetic scenes from the NeRF-Synthetic [ 36 ] and Shelly [ 50 ] datasets. We report PSNR for each scene. The face count for MobileNeRF is averaged across scenes, whereas all other methods use exactly 15K faces.

	FPS $\uparrow$ across resolutions
Method	Full	1/2	1/4	# Params
3DGS [ 24 ]	115	482	1.32K	885K
TS [ 17 ]	88.8	370	1.00K	885K
TexGS [ 5 ]	16.8	49.1	94.8	151M
Ours (CUDA)	1.96K	6.11K	13.7K	67.5M
Ours (laptop)	146	447	879	67.5M

Table 3: Rendering time in the MipNeRF360 [ 3 ] experiment. “Full” denotes the original image resolution. The number of trainable parameters is included for reference.

Figure 7: Ablation studies. (a) Without opacity learning, triangles are unable to represent fine sub-triangle boundaries. (b) Without our multi-resolution texture scheme, the instability of color optimization leads to noisier geometry (pink arrows). For a clearer comparison, we disable coarse-to-fine optimization in setting (b).

Figure A1 : (Left) Starting from a randomly initialized triangle soup, training gradually drives irrelevant triangles to become transparent, and these are regularly removed by our adaptive resampling step. This process yields a clean geometry that captures even thin structures and challenging topology. (Right) The corresponding ground-truth image is shown for reference.

Figure A2 : Additional qualitative results on the MipNeRF360 [ 3 ] dataset. (Left to right) Ground-truth test images; views synthesized by 3DGS [ 24 ] , Triangle Splatting [ 17 ] , TexGS [ 5 ] , our method; and a visualization of our reconstructed triangle geometry. Rows correspond to different scenes, and all models use 15K primitives.

Method	NeRF-Synthetic [ 36 ] (SSIM $\uparrow$ )					Shelly [ 50 ] (SSIM $\uparrow$ )
	ship	chair	mic	drums	# Faces $\downarrow$	khady	kitten	pug	# Faces $\downarrow$
MobileNeRF [ 7 ]	0.817	0.958	0.961	0.910	159K	0.810	0.931	0.857	275K
MobileNeRF + QEM [ 12 ]	0.554	0.793	0.857	0.739	15K	0.686	0.847	0.729	15K
Ours w/ QEM init.	0.848	0.975	0.975	0.931	15K	0.838	0.951	0.903	15K
Ours w/ random init.	0.828	0.972	0.972	0.927	15K	0.838	0.950	0.899	15K

Table A1 : Additional quantitative comparisons with MobileNeRF [ 7 ] on synthetic scenes from the NeRF-Synthetic [ 36 ] and Shelly [ 50 ] datasets. In addition to the PSNR values reported in the main paper, we report SSIM (top table) and LPIPS (bottom table) scores for each scene. The face count for MobileNeRF is averaged across scenes, whereas all other methods use the exact face count specified. See the main paper for further discussion.

Method	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
w/o opacity	24.65	0.744	0.205
$R_{\mathrm{min}}=R_{\mathrm{max}}=5$	23.54	0.719	0.226
$R_{\mathrm{min}}=2,\,R_{\mathrm{max}}=5$	24.48	0.747	0.203
Ours full	24.76	0.748	0.204

Table A2 : Quantitative results for the ablation study on the MipNeRF360 [ 3 ] dataset. From top to bottom: (1) our method with opacity fixed to 1, (2) a variant that uses only the highest-resolution feature grid with

R_{\mathrm{max}}=5

(i.e., without multi-resolution features and without the coarse-to-fine strategy), (3) a variant that uses multi-resolution features but does not use the coarse-to-fine strategy, and (4) our full method combining both multi-resolution features and the coarse-to-fine strategy. PSNR, SSIM, and LPIPS scores are reported as scene-level averages. The number of triangles is fixed at

\lvert\mathcal{T}\rvert=15\mathrm{K}

for all methods.

Figure A3 : We represent RGB color and opacity within each triangle by linearly interpolating values stored at subdivision points. (a) Each subdivision point stores eight 32-bit floating-point values: a 7-dimensional color feature vector and one opacity value. (b) The color feature vector is decoded into RGB using a small neural network, while the final channel is used directly as opacity.

Figure A4 : (Top row, left to right) Ground-truth images for the Bicycle scene from the MipNeRF360 [ 3 ] dataset. This scene exhibits an extremely large spatial extent, as indicated by the distant objects highlighted with orange arrows. (a) Triangle-soup rendering and reconstructed geometry using our method with

\lvert\mathcal{T}\rvert=15\mathrm{K}

triangles and a texture resolution of

R_{\mathrm{max}}=5

. (b) Rendering and geometry using

\lvert\mathcal{T}\rvert=60\mathrm{K}

triangles and

R_{\mathrm{max}}=4

. (c) Rendering and geometry using

\lvert\mathcal{T}\rvert=100\mathrm{K}

triangles and

R_{\mathrm{max}}=4

. (Bottom row) Test error measured as mean absolute error (MAE) over training iterations for the three configurations, demonstrating convergence behavior and clarifying the representational capacity.

Figure A5 : Effect of varying the triangle budget on reconstruction quality for NeRF-Synthetic [ 36 ] scenes. Each row shows (left to right) a ground-truth view followed by our results with

|\mathcal{T}|=15\mathrm{K}

,

5\mathrm{K}

, and

1\mathrm{K}

triangles, respectively.

Method	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
3DGS [ 24 ]	23.72	0.664	0.420
3DGS-MCMC [ 25 ]	24.22	0.692	0.389
TexGS [ 5 ]	24.80	0.697	0.270
Ours	24.76	0.748	0.204

Table A3 : Reconstruction quality comparison with 3DGS-MCMC [ 25 ] on the MipNeRF360 [ 3 ] dataset. All methods use 15K primitives.

	FPS $\uparrow$ across resolutions
Method	Full	1/2	1/4
3DGS [ 24 ]	115	482	1.32K
3DGS-MCMC [ 25 ]	214	598	943
TexGS [ 5 ]	16.8	49.1	94.8
Ours (CUDA)	1.96K	6.11K	13.7K

Table A4 : Rendering speed comparison with 3DGS-MCMC [ 25 ] on the MipNeRF360 [ 3 ] dataset. All methods use 15K primitives.

Method	Memory $\downarrow$	Storage $\downarrow$	# Prims
3DGS [ 24 ]	3.38MB	3.38MB	15K
TexGS [ 5 ]	576MB	576MB	15K
MobileNeRF [ 7 ]	329MB	118MB	209K
Ours	65.9MB	44.3MB	15K

Table A5 : Average GPU memory footprint and storage cost of the models used in our experiments. The average primitive count is also reported.

	PSNR $\uparrow$
# Triangles	Ship	Mic	Lego	Hotdog	Ficus
$\|\mathcal{T}\|=15\mathrm{K}$	26.68	30.18	29.78	33.53	27.51
$\|\mathcal{T}\|=5\mathrm{K}$	26.12	30.03	28.21	33.45	27.39
$\|\mathcal{T}\|=1\mathrm{K}$	24.34	28.26	24.36	31.40	24.27

Table A6 : Reconstruction quality (PSNR) across triangle budgets on NeRF-Synthetic [ 36 ] scenes. Quality degrades gracefully from

|\mathcal{T}|{=}15\mathrm{K}

to

5\mathrm{K}

, with a more noticeable drop at

1\mathrm{K}

.

DiffSoup: Direct Differentiable Rasterization of Triangle Soup for Extreme Radiance Field Simplification

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