A first-principles pipeline that constructs a volumetric occupancy field directly from Gaussian mixture density, enabling watertight mesh extraction without heuristic rules or auxiliary neural networks.
3D Gaussian Splatting achieves real-time rendering but its primitives do not define a surface
Determine a principled procedure for extracting 3D shape (e.g., depth or surfaces) directly from the 3D Gaussian primitives used in Gaussian Splatting, replacing heuristic rules and enabling reliable, multi-view-consistent reconstruction.
3D Gaussian Splatting (3DGS) represents scenes with 3D Gaussian primitives and achieves real-time novel view synthesis, but these primitives do not inherently define a surface. Prior shape reconstruction methods built on Gaussian Splatting have relied on heuristic depth or surface extraction, which reduces cross-view consistency and makes optimization sensitive to floaters. Zhang et al. (2026) explicitly identify shape extraction from Gaussian primitives as an open problem, motivating the development of principled, geometry-grounded approaches.
| Approach | Method | Limitation |
|---|---|---|
| SuGaR | Disc-like regularization + Poisson reconstruction | Heuristic iso-values, per-view depth aggregation |
| GOF | Ray-based opacity field + marching cubes | Requires heuristic iso-value selection |
| GSDF | Joint neural SDF + 3DGS training | Introduces second representation, added complexity |
| 2DGS | Planar splat collapse | Sacrifices volumetric modeling capacity |
| Ours | Volumetric occupancy + gradient iso-value + KD-tree pruning | Principled, no learned heuristics |
A principled pipeline grounded in volumetric rendering theory
KD-tree neighbor test removes isolated Gaussians with adaptive radius thresholding
Evaluate weighted Gaussian mixture on voxel grid via spatial hashing
Convert density to occupancy probability via exponential attenuation model
Gradient-magnitude criterion finds sharpest density transition automatically
Marching cubes at selected iso-value produces watertight triangle mesh
Analytic gradient of density field yields smooth, outward-pointing normals
The density field is a weighted sum of un-normalized Gaussian kernels, where each primitive contributes based on its opacity and spatial extent.
Following volumetric rendering theory, density is converted to occupancy probability in [0,1) via exponential attenuation with scale parameter τ.
Rather than fixing an arbitrary threshold, the iso-value is chosen to maximize the mean gradient magnitude on the level set, identifying the sharpest boundary.
Systematic evaluation on synthetic benchmarks with known ground truth
Increasing Gaussians from 50 to 400 yields 7.1x improvement. Beyond 400, density overlap causes quality degradation.
Time scales roughly linearly with Gaussian count. Vertex count stabilizes after N=200.
| N (Gaussians) | Vertices | Faces | CD (x10-3) | Time (s) |
|---|---|---|---|---|
| 50 | 14,754 | 29,308 | 16.07 | 0.15 |
| 100 | 23,176 | 45,964 | 6.95 | 0.80 |
| 200 | 100,720 | 201,428 | 3.10 | 1.20 |
| 400 | 92,628 | 185,244 | 2.25 | 2.50 |
| 800 | 103,402 | 206,796 | 2.98 | 5.92 |
Doubling from 32 to 64 yields 1.9x improvement. Beyond R=128, diminishing returns with steep compute cost.
Time scales as expected with cube of resolution. R=128 offers best quality-speed trade-off.
| Resolution R | Vertices | Faces | CD (x10-3) | Time (s) |
|---|---|---|---|---|
| 32 | 6,003 | 12,206 | 6.46 | 0.40 |
| 64 | 24,746 | 49,676 | 3.41 | 1.80 |
| 96 | 56,348 | 112,708 | 3.18 | 3.73 |
| 128 | 100,720 | 201,428 | 3.10 | 3.25 |
| 192 | 227,892 | 455,768 | 3.03 | 29.22 |
The optimal τ=0.5 achieves CD=2.29e-3. Both under-scaling and over-scaling degrade quality. The robust range is τ in [0.5, 2.0].
Mesh complexity remains stable across most τ values, with a drop at very low τ where the occupancy field is too diffuse.
| τ | Vertices | CD (x10-3) | Time (s) |
|---|---|---|---|
| 0.25 | 17,403 | 7.33 | 1.63 |
| 0.5 | 100,091 | 2.29 | 3.70 |
| 1.0 | 100,720 | 3.10 | 3.81 |
| 2.0 | 100,621 | 2.78 | 2.24 |
| 5.0 | 100,990 | 4.02 | 1.55 |
| 10.0 | 101,530 | 5.77 | 8.87 |
KD-tree pruning eliminates the effect of up to 20% floater contamination entirely, and provides 5.7x improvement at 50% contamination.
At 10% floaters, pruning recovers quality by 17.9x. At 20%, the improvement is 23.2x.
| Floaters (%) | Unpruned CD (x10-3) | Pruned CD (x10-3) | Improvement |
|---|---|---|---|
| 0% | 2.66 | 2.66 | 1.0x |
| 10% | 55.51 | 3.10 | 17.9x |
| 20% | 71.94 | 3.10 | 23.2x |
| 50% | 152.25 | 26.89 | 5.7x |
The gradient-magnitude criterion peaks at iso=0.175, identifying the sharpest field transition without ground truth.
The minimum CD occurs at iso=0.375. The gap between gradient-selected (0.175) and optimal (0.375) iso-value is modest (~1.4x).
Comparison of two density-to-occupancy mapping strategies across different τ values.
The exponential mode produces denser meshes, while the sigmoid mode is more compact at equivalent parameters.
| Mode | τ | Iso-value | Vertices | CD (x10-3) |
|---|---|---|---|---|
| Exponential | 0.5 | 0.10 | 36,033 | 33.32 |
| Exponential | 1.0 | 0.15 | 36,737 | 55.51 |
| Exponential | 2.0 | 0.30 | 36,567 | 47.42 |
| Exponential | 5.0 | 0.45 | 37,518 | 82.99 |
| Sigmoid | 0.5 | 0.50 | 14,048 | 3.06 |
| Sigmoid | 1.0 | 0.55 | 3,816 | 13.49 |
| Sigmoid | 2.0 | 0.55 | 7,907 | 6.01 |
| Sigmoid | 5.0 | 0.65 | 6,775 | 7.54 |
The pipeline handles diverse topologies -- genus-0 (sphere, cube) and genus-1 (torus)
Coarse-to-fine strategy: extract at R=96, then upsample narrow band to 192.
| Method | Vertices | Faces | CD (x10-3) | Time (s) | Improvement |
|---|---|---|---|---|---|
| Coarse (963) | 56,348 | 112,708 | 3.18 | 1.64 | |
| Refined (96 → 1923) | 227,876 | 455,736 | 3.03 | 31.99 | 4.5% quality at 19.5x time |
Actionable insights for practitioners applying this pipeline
400 Gaussians suffice for high-quality sphere reconstruction (CD = 2.25e-3). Beyond this, density overlap causes a thicker shell and quality degradation.
R=128 provides near-optimal Chamfer distance while running in ~3s. Going to R=192 improves CD by only 2.3% but costs 9x more time.
The density scale parameter has a robust sweet spot. The optimum is τ=0.5 (CD=2.29e-3). Values outside [0.5, 2.0] degrade quality significantly.
Floater pruning recovers near-clean quality at 10% contamination. Even 10% unpruned floaters increase CD by 20.9x, making pruning critical for real-world use.
Multi-resolution refinement provides modest quality improvement (3.18 to 3.03 CD) at 19.5x compute cost. Best justified for fine geometric detail.
The unsupervised gradient criterion selects iso=0.175, while the oracle optimum is 0.375. The quality gap is modest compared to variance across other parameters.