Spherical Voronoi: Directional Appearance as a Differentiable Partition of the Sphere

🏆 Accepted at CVPR 2026

📄 Project Website | 📝 arXiv Paper

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Authors

Francesco Di Sario¹², Daniel Rebain³, Dor Verbin⁵, Marco Grangetto¹, Andrea Tagliasacchi²⁴

¹ University of Torino · ² Simon Fraser University · ³ University of British Columbia · ⁴ University of Toronto · ⁵ Google DeepMind

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Overview

Spherical functions like the shown environment maps have a multitude of applications in Computer Graphics and 3D Computer Vision. Classical representations like Spherical Harmonics optimize well but struggle in representing high-frequency functions. Explicit representations like Spherical Gaussians are capable of representing localized functions, but they are difficult to optimize due to the locality of the Gaussian kernel. We propose Spherical Voronoi as a new explicit representation that is capable of modeling high frequencies effectively, provides an adaptive decomposition of the spherical domain, and is easier to optimize.

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🆕 Update — May 2026

This release brings significant improvements to Spherical Voronoi:

• Native CUDA implementation for the Spherical Voronoi kernel, delivering improved speed and stability.
• Unified learning rates across all models and datasets
• Improved results across all benchmarks (see tables below).
• All checkpoints and evaluation renders are publicly available: 📦 Google Drive
• Release of Spherical Voronoi for 3DGS-MCMC and 2DGS.

⚠️ Important — MipNeRF-360 evaluation: For correct and fair evaluation, use the images_2 folder for indoor scenes and images_4 for outdoor scenes. Do not use manual downsampling via the -r flag.

Results — Beta Splatting + Spherical Voronoi

Dataset	PSNR	SSIM	LPIPS
MipNeRF-360	28.71	0.837	0.228
Tanks & Temples	25.00	0.874	0.170
Deep Blending	30.63	0.917	0.298
Blender	34.58	0.973	0.032

Results — 3DGS-MCMC + Spherical Voronoi

Dataset	PSNR	SSIM	LPIPS
MipNeRF-360	28.51	0.837	0.220
Tanks & Temples	24.67	0.867	0.186
Deep Blending	30.04	0.905	0.315
Blender	34.35	0.973	0.034

Results — 2DGS + Spherical Voronoi

Dataset	PSNR	SSIM	LPIPS
Blender	33.56	0.969	0.039
MipNeRF-360	27.70	0.807	0.279

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Repository Structure

This repository contains two main components:

• radiance/ - Uses Spherical Voronoi instead of Spherical Harmonics for directional appearance. Backbone: Beta Splatting.
• reflection/ - Uses Spherical Voronoi–parameterized light probes for reflections. Backbone: 2D Gaussian Splatting with deferred rendering.

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Installation

Clone the repository

git clone https://github.com/sphericalvoronoi/sphericalvoronoi.git

Install PyTorch

Install PyTorch according to your CUDA version following the official instructions.

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Modeling Radiance

Setup

cd radiance
pip install -e .
pip install ./gsplat
pip install ./spherical-voronoi

Training and Evaluation

Ready-to-run scripts are provided in the scripts/ folder. For example:

# Edit $DATA_ROOT inside the script to point to your dataset
bash scripts/train_mipnerf360.sh
bash scripts/eval_mipnerf360.sh

A typical manual command is:

python train.py --eval \
  --source_path /DATASET_DIR/bonsai \
  --model_path ./output/voronoi/nerf_real_360/bonsai \
  --color_rep voronoi \
  --scene bonsai \
  --config ./config/indoor.json

The training script:

• Trains and evaluates the model
• Saves rendered images
• Reports PSNR, SSIM, LPIPS

Datasets

• Mip-NeRF 360
• Tanks and Temples / Deep Blending
• NeRF Synthetic

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Reflection

Setup

cd reflection
pip install -e submodules/diff-surfel-rasterization
pip install -e submodules/diff-surfel-rasterization-real
pip install -e submodules/diff-surfel-2dgs
pip install -e submodules/sv-probes

Note: You will also need PyTorch3D and nvdiffrast.

Training and Evaluation

Ready-to-run scripts are provided in the scripts/ folder. A typical manual command is:

python train.py --eval --m /OUTPUT_DIR -s /DATASET_DIR --rand_bg

The training script:

• Trains and evaluates the model
• Saves rendered images
• Reports PSNR, SSIM, LPIPS, and FPS

Datasets

• Ref-NeRF
• Ref-Real
• Glossy Synthetic (use nero2blender.py to convert to Blender format)

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Citation

If you use this work in your research, please cite:

@InProceedings{Di_Sario_2026_CVPR,
    author = {Di Sario, Francesco and Rebain, Daniel and Verbin, Dor and Grangetto, Marco and Tagliasacchi, Andrea},
    title = {Spherical Voronoi: Directional Appearance as a Differentiable Partition of the Sphere},
    booktitle = {Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)},
    month = {June},
    year = {2026},
    pages = {22529-22538}
}

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Acknowledgments

This work builds upon several excellent open-source projects. We are grateful to the authors for making their code publicly available:

• Beta Splatting
• 2D Gaussian Splatting
• Ref-GS
• gsplat

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Contact

For questions or inquiries, please contact Francesco Di Sario.

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Full Results Tables

Beta Splatting + Spherical Voronoi — per scene

Dataset	Scene	PSNR	SSIM	LPIPS
MipNeRF-360	bonsai	34.87	0.957	0.231
	kitchen	32.98	0.937	0.147
	room	33.26	0.937	0.257
	counter	31.04	0.930	0.225
	bicycle	25.75	0.795	0.196
	garden	27.92	0.876	0.114
	flowers	22.18	0.639	0.337
	stump	27.32	0.806	0.211
	treehill	23.08	0.656	0.333
	avg	28.71	0.837	0.228
Tanks & Temples	train	23.25	0.845	0.214
	truck	26.76	0.904	0.125
	avg	25.00	0.874	0.170
Deep Blending	drjohnson	29.96	0.914	0.304
	playroom	31.30	0.920	0.292
	avg	30.63	0.917	0.298
Blender	lego	37.15	0.986	0.015
	mic	37.39	0.994	0.006
	ship	31.49	0.911	0.117
	chair	37.18	0.990	0.012
	ficus	36.93	0.991	0.009
	hotdog	38.49	0.988	0.020
	materials	30.98	0.967	0.036
	drums	27.05	0.959	0.037
	avg	34.58	0.973	0.032

3DGS-MCMC + Spherical Voronoi — per scene

Dataset	Scene	PSNR	SSIM	LPIPS
MipNeRF-360	bonsai	34.15	0.952	0.244
	kitchen	32.79	0.934	0.156
	room	32.66	0.932	0.269
	counter	31.06	0.938	0.135
	bicycle	25.72	0.796	0.202
	garden	27.87	0.878	0.110
	flowers	22.01	0.638	0.327
	stump	27.41	0.810	0.207
	treehill	22.95	0.654	0.331
	avg	28.51	0.837	0.220
Tanks & Temples	train	22.92	0.833	0.238
	truck	26.42	0.901	0.134
	avg	24.67	0.867	0.186
Deep Blending	drjohnson	29.45	0.894	0.323
	playroom	30.63	0.917	0.306
	avg	30.04	0.905	0.315
Blender	lego	36.45	0.985	0.018
	mic	37.71	0.995	0.006
	ship	31.36	0.912	0.119
	chair	36.96	0.989	0.015
	ficus	35.87	0.989	0.011
	hotdog	38.39	0.988	0.023
	materials	31.06	0.967	0.038
	drums	27.03	0.959	0.038
	avg	34.35	0.973	0.034

2DGS + Spherical Voronoi — per scene

Dataset	Scene	PSNR	SSIM	LPIPS
MipNeRF-360	bonsai	33.37	0.943	0.267
	room	31.81	0.919	0.304
	counter	29.91	0.909	0.274
	kitchen	32.05	0.926	0.169
	bicycle	24.87	0.741	0.282
	garden	27.26	0.853	0.149
	flowers	21.21	0.589	0.386
	stump	26.35	0.760	0.278
	treehill	22.43	0.621	0.406
	avg	27.70	0.807	0.279
Blender	lego	35.62	0.981	0.023
	mic	35.46	0.991	0.008
	ship	30.92	0.906	0.130
	chair	35.75	0.986	0.018
	hotdog	37.80	0.985	0.028
	ficus	36.25	0.989	0.013
	materials	29.98	0.960	0.046
	drums	26.69	0.956	0.044
	avg	33.56	0.969	0.039