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Alpha Invariance: On Inverse Scaling Between Distance and Volume Density in Neural Radiance Fields
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Analyzes how volume density scales inversely with scene size in NeRFs (alpha invariance) and proposes log-space distance/density parameterization (GumbelCDF activation) with a closed-form high-transmittance initialization for robustness across scene scalings.
Problem
NeRF volume densities have magnitude ambiguity under arbitrary scene scaling: density must scale inversely with distance, causing optimization failures when scenes are scaled down (densities too large to form solid geometry) or up (initial cloudiness traps optimization).
Approach
Define alpha invariance and parameterize density as exp(x) combined with log-distance so log density and log distance cancel (GumbelCDF form), plus an additive offset log(log(1/T0)) - log L - tau^2/2 that sets initial ray transmittance to T0 independent of discretization or scene scale.
Key results
- Vanilla NeRF with the proposed recipe maintains consistent PSNR across scene scalings k from 0.1 to 10 on Blender scenes, while baseline NeRF-Pytorch shows random failures (e.g., chair PSNR 14.04 at k=10 baseline vs 31.36 ours).
- TensoRF baseline fails on several Blender scenes at k=0.1 (fail), while the recipe converges (e.g., chair 35.41, drums 26.01).
- Nerfacto baseline collapses at k=10 on Mip-NeRF 360 (bicycle 12.42 vs 22.40 with the high-transmittance offset).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use log-space density parameterization and high-transmittance initialization | NeRF-PyTorch · Blender synthetic scenes across scene scales | PSNR and convergence consistency: 27.97 / 28.25 | Helped |
| Add high-transmittance initialization | Nerfacto · Mip-NeRF 360, k=10, bicycle | PSNR: 12.42 / 22.40 | Helped |
| Add high-transmittance initialization | TensoRF with exp activation · Blender scenes across distance scales | percentage of samples with alpha < 0.01 | Helped |
| Set Lout equal to Lin for the outer scene (ablation) | Nerfacto · Mip-NeRF 360, k=10 | PSNR: 24.71 -> 26.27 | Helped |
| Set Lout to the metric ray distance (10000) (ablation) | Nerfacto · Mip-NeRF 360, k=10 | PSNR: 24.71 | Hurt |
| Set Lout=0, applying no transmittance offset to the outer scene (ablation) | Nerfacto · Mip-NeRF 360, k=10 | PSNR: 25.21 | Hurt |
| Reduce the initial learning rate from 30.0 to 0.3 (ablation) | Plenoxels · Blender | PSNR: 31.66 -> 29.51 | Hurt |
| Reduce the initial learning rate from 30.0 to 0.003 (ablation) | Plenoxels · Blender | PSNR: 31.66 -> 23.51 | Hurt |
| Reduce the initial learning rate from 30.0 to 0.3 (ablation) | Plenoxels · LLFF | PSNR: 24.51 -> 23.22 | Hurt |
| Reduce the initial learning rate from 30.0 to 0.003 (ablation) | Plenoxels · LLFF | PSNR: 24.51 -> 21.09 | Hurt |
| Use exp activation without the high-transmittance initialization offset (ablation) | TensoRF · Blender scenes at large scene scales | percentage of samples with alpha < 0.01 | Hurt |
| Use the high-transmittance initialization offset with exp activation (ablation) | TensoRF · Blender scenes across scene scales | convergence | Helped |
Surprising findings
- Vanilla NeRF with ReLU is surprisingly capable of producing large densities, but optimization randomly converges to poor local minima.
- Empirically, density changes by a factor close to 1/k with scene scale k but the factor deviates from 1/k in practice.
Limitations
- Closed-form initialization derived only for exp activation; ReLU and softplus require numerical Monte Carlo tuning of offsets.
- Assumes i.i.d. initial density samples, which is imperfect due to feature/voxel sharing.
- Random failure modes remain for vanilla NeRF even with Kaiming initialization.
Experimental setup
Datasets: NeRF Blender synthetic Mip-NeRF 360 LLFF Tanks and Temples
Benchmarks: NeRF Blender synthetic Mip-NeRF 360 LLFF Tanks and Temples
Metrics: PSNR CDF of volume density percentage of empty space
Baselines: Vanilla NeRF TensoRF DVGO Plenoxels Nerfacto NeRF-Pytorch
Builds on: NeRF Instant-NGP Mip-NeRF 360 TensoRF DVGO
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- I2-NeRF: Learning Neural Radiance Fields Under Physically-Grounded Media Interactions (NeurIPS 2025 · similarity 0.85)
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