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Steepest Descent Density Control for Compact 3D Gaussian Splatting
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Proposes Steepest Density Control (SDC), a theoretically grounded densification rule for 3DGS that splits only Gaussians with indefinite splitting matrices into two offspring displaced along the least-eigenvector, halving opacity, yielding SteepGS with ~50% fewer Gaussians.
Problem
Adaptive density control in 3D Gaussian Splatting produces redundant Gaussians, increasing memory, rendering cost, and storage, while existing densification rules are heuristic and lack optimization-theoretic justification.
Approach
Analyzes the loss change after splitting via a second-order expansion with a per-Gaussian splitting matrix built from the Hessian of each Gaussian's position; splitting helps only when this matrix has a negative eigenvalue, and the optimal split is two offspring displaced along ±the smallest eigenvector with opacity weights 0.5.
Key results
- On Mip-NeRF 360, SteepGS reaches 28.734 PSNR with 1.606M points versus 3DGS 29.037 PSNR with 3.339M points, roughly a 50% reduction.
- On Tanks & Temples, SteepGS achieves 23.684 PSNR with 0.958M points versus 3DGS-MCMC 22.545 at a similar count.
- On Deep Blending, SteepGS achieves 29.963 PSNR with 1.296M points, exceeding 3DGS (29.690 PSNR, 2.818M points).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SteepGS density control instead of standard 3DGS ADC | 3DGS · Mip-NeRF 360 | # Points (million): 3.339 -> 1.606 | Helped |
| Use SteepGS density control instead of standard 3DGS ADC | 3DGS · Mip-NeRF 360 | PSNR: 29.037 -> 28.734 | Hurt |
| Use SteepGS density control instead of standard 3DGS ADC | 3DGS · Tank & Temple | # Points (million): 1.830 -> 0.958 | Helped |
| Use SteepGS density control instead of standard 3DGS ADC | 3DGS · Tank & Temple | PSNR: 23.743 -> 23.684 | Hurt |
| Use SteepGS density control instead of standard 3DGS ADC | 3DGS · Deep Blending | # Points (million): 2.818 -> 1.296 | Helped |
| Use SteepGS density control instead of standard 3DGS ADC | 3DGS · Deep Blending | PSNR: 29.690 -> 29.963 | Helped |
| Split a Gaussian into two offspring and halve each offspring's opacity | SteepGS · 3D Gaussian Splatting density control | loss | Helped |
| Displace the two offspring along positive and negative directions of the least-eigenvalue eigenvector | SteepGS · 3D Gaussian Splatting density control | loss | Helped |
| Do not split when the splitting matrix is positive semidefinite | SteepGS · 3D Gaussian Splatting density control | loss | No clear effect |
Surprising findings
- Densifying a random point can increase the loss, so the original ADC has no guarantee of loss decrease after splitting.
- On Mip-NeRF 360, SteepGS is slightly below 3DGS in PSNR (28.734 vs 29.037) despite using about half the Gaussians.
Limitations
- Gains over 3DGS on Mip-NeRF 360 PSNR are not present; rendering quality is slightly lower than the original ADC on that dataset.
- Theory relies on a second-order approximation and bounded third-order derivatives, with only position parameters used for the splitting matrix.
Experimental setup
Datasets: Mip-NeRF 360 Tanks & Temples Deep Blending
Benchmarks: Mip-NeRF 360 Tanks & Temples Deep Blending
Metrics: PSNR SSIM LPIPS number of points
Baselines: 3DGS 3DGS + Thres. 3DGS-MCMC Revising 3DGS
Builds on: 3D Gaussian Splatting Adaptive Density Control S2D neuron splitting Firefly
Closest papers by meaning
- FlashGS: Efficient 3D Gaussian Splatting for Large-scale and High-resolution Rendering (CVPR 2025 · similarity 0.92)
- DashGaussian: Optimizing 3D Gaussian Splatting in 200 Seconds (CVPR 2025 · similarity 0.92)
- Improving Gaussian Splatting with Localized Points Management (CVPR 2025 · similarity 0.92)
- Gaussian Herding across Pens: An Optimal Transport Perspective on Global Gaussian Reduction for 3DGS (NeurIPS 2025 · similarity 0.92)
- GaussianPro: 3D Gaussian Splatting with Progressive Propagation (ICML 2024 · similarity 0.92)
- Speedy-Splat: Fast 3D Gaussian Splatting with Sparse Pixels and Sparse Primitives (CVPR 2025 · similarity 0.92)
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