Research map / 3D vision / Face and shape
Objects as Volumes: A Stochastic Geometry View of Opaque Solids
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Derives a stochastic-geometry theory showing when opaque solids can be modeled with exponential volumetric transport, giving closed-form attenuation coefficients and a reciprocal, anisotropic representation that improves neural 3D reconstruction over NeuS and VolSDF.
Problem
Neural volume rendering uses volumetric representations for opaque solids without a mathematical justification for why such solids can be treated as volumes, and prior representations are heuristic and sometimes physically implausible (e.g., violating reciprocity).
Approach
Model opaque solids as random indicator functions; show the free-flight distribution is exponential iff the indicator restricted to a ray is a continuous-time Markov process, derive the attenuation coefficient as density times a projected area over a distribution of normals, and apply it to stochastic implicit surfaces (Gaussian/logistic/Laplace sigmoids) with spatially varying anisotropy.
Key results
- On DTU, the proposed representation achieves mean Chamfer distance 1.57 versus 2.17 for NeuS and 1.84 for VolSDF.
- On NeRF Realistic Synthetic, mean Chamfer distance is 0.113 versus 0.201 (NeuS) and 0.252 (VolSDF).
- Ablation on DTU shows Gaussian implicit distribution (mean 1.78) beats logistic (1.98) and Laplace (1.96), and spatially varying mixture of normals (1.75) beats constant mixture (1.97).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace VolSDF and NeuS representations with the proposed representation | 3D reconstruction · DTU | mean Chamfer distance: 2.17 -> 1.57 | Helped |
| Replace VolSDF and NeuS representations with the proposed representation | 3D reconstruction · DTU | median Chamfer distance: 1.99 -> 1.56 | Helped |
| Replace VolSDF and NeuS representations with the proposed representation | 3D reconstruction · NeRF RS | mean Chamfer distance: 0.201 -> 0.113 | Helped |
| Replace VolSDF and NeuS representations with the proposed representation | 3D reconstruction · NeRF RS | median Chamfer distance: 0.085 -> 0.057 | Helped |
| Use Gaussian Ψ instead of logistic Ψ (ablation) | Proposed volumetric representation · DTU | mean Chamfer distance: 1.98 -> 1.78 | Helped |
| Use Gaussian Ψ instead of Laplace Ψ (ablation) | Proposed volumetric representation · DTU | mean Chamfer distance: 1.96 -> 1.78 | Helped |
| Use Gaussian Ψ instead of logistic Ψ (ablation) | Proposed volumetric representation · DTU | median Chamfer distance: 1.86 -> 1.74 | Helped |
| Use Gaussian Ψ instead of Laplace Ψ (ablation) | Proposed volumetric representation · DTU | median Chamfer distance: 1.92 -> 1.74 | Helped |
| Remove the reciprocity-violating ReLU term from delta-normal projected area (ablation) | Delta-normal representation · DTU | mean Chamfer distance: 2.17 -> 1.98 | Helped |
| Remove the reciprocity-violating ReLU term from delta-normal projected area (ablation) | Delta-normal representation · DTU | median Chamfer distance: 1.99 -> 1.86 | Helped |
| Use a spatially varying mixture of normals instead of a delta distribution (ablation) | Proposed volumetric representation · DTU | mean Chamfer distance: 2.17 -> 1.75 | Helped |
| Use a spatially varying mixture of normals instead of a spatially constant mixture (ablation) | Proposed volumetric representation · DTU | mean Chamfer distance: 1.97 -> 1.75 | Helped |
Surprising findings
- Including the reciprocity-violating ReLU term from NeuS underperforms omitting it, showing enforcing reciprocity improves reconstruction.
- The free-flight distribution of opaque solids is generally non-exponential for Gaussian excursion sets, so exponential transport is an approximation outside the derived conditions.
Limitations
- Excludes semi-transparent solids and global illumination coupling, which are only briefly treated in appendices.
- Assumes exponential transport, which may not hold for opaque solids (e.g., Gaussian excursion sets give non-exponential first-passage times).
- Experiments are a simplified NeuS variant aimed at design exploration rather than state-of-the-art performance.
Experimental setup
Datasets: DTU BlendedMVS NeRF Realistic Synthetic
Benchmarks: DTU BlendedMVS NeRF Realistic Synthetic
Metrics: Chamfer distance
Baselines: NeuS VolSDF
Builds on: NeuS VolSDF NeRF microflake models
Closest papers by meaning
- Geometry Field Splatting with Gaussian Surfels (CVPR 2025 · similarity 0.88)
- Subsurface Scattering for Gaussian Splatting (NeurIPS 2024 · similarity 0.84)
- Volumetric Surfaces: Representing Fuzzy Geometries with Layered Meshes (CVPR 2025 · similarity 0.84)
- MaterialRefGS: Reflective Gaussian Splatting with Multi-view Consistent Material Inference (NeurIPS 2025 · similarity 0.84)
- Ghost on the Shell: An Expressive Representation of General 3D Shapes (ICLR 2024 · similarity 0.83)
- Light Transport-aware Diffusion Posterior Sampling for Single-View Reconstruction of 3D Volumes (CVPR 2025 · similarity 0.83)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.