# Objects as volumes: A stochastic geometry view of opaque solids

arXiv: https://arxiv.org/abs/2312.15406 · HTML: https://mlsanity.com/paper/2312.15406/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 67 references that are in the map.
- First posted on arXiv 24 Dec 2023.

## Builds on (most cited first, 25 of 25)

- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction](https://mlsanity.com/paper/2106.10689/) (2021)
- [Volume Rendering of Neural Implicit Surfaces](https://mlsanity.com/paper/2106.12052/) (2021)
- [NeRF++: Analyzing and Improving Neural Radiance Fields](https://mlsanity.com/paper/2010.07492/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [UNISURF: Unifying Neural Implicit Surfaces and Radiance Fields for Multi-View Reconstruction](https://mlsanity.com/paper/2104.10078/) (2021)
- [Nerfstudio: A Modular Framework for Neural Radiance Field Development](https://mlsanity.com/paper/2302.04264/) (2023)
- [Ref-NeRF: Structured View-Dependent Appearance for Neural Radiance Fields](https://mlsanity.com/paper/2112.03907/) (2021)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [BlendedMVS: A Large-scale Dataset for Generalized Multi-view Stereo Networks](https://mlsanity.com/paper/1911.10127/) (2019)
- [MonoSDF: Exploring Monocular Geometric Cues for Neural Implicit Surface Reconstruction](https://mlsanity.com/paper/2206.00665/) (2022)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Neural RGB-D Surface Reconstruction](https://mlsanity.com/paper/2104.04532/) (2021)
- [Urban Radiance Fields](https://mlsanity.com/paper/2111.14643/) (2021)
- [Geo-Neus: Geometry-Consistent Neural Implicit Surfaces Learning for Multi-view Reconstruction](https://mlsanity.com/paper/2205.15848/) (2022)
- [Multi-view Supervision for Single-view Reconstruction via Differentiable Ray Consistency](https://mlsanity.com/paper/1704.06254/) (2017)
- [NeuRIS: Neural Reconstruction of Indoor Scenes Using Normal Priors](https://mlsanity.com/paper/2206.13597/) (2022)
- [Adaptive Shells for Efficient Neural Radiance Field Rendering](https://mlsanity.com/paper/2311.10091/) (2023)
- [Stochastic Poisson Surface Reconstruction](https://mlsanity.com/paper/2206.15236/) (2022)
- [A Radiative Transfer Framework for Spatially-Correlated Materials](https://mlsanity.com/paper/1805.02651/) (2018)
- [Neural Stochastic Screened Poisson Reconstruction](https://mlsanity.com/paper/2309.11993/) (2023)
- [An unbiased ray-marching transmittance estimator](https://mlsanity.com/paper/2102.10294/) (2021)
- [A reciprocal formulation of non-exponential radiative transfer. 1: Sketch and motivation](https://mlsanity.com/paper/1803.03259/) (2018)

## Built on by (most cited first, 15 of 15)

- [Geometry Field Splatting with Gaussian Surfels](https://mlsanity.com/paper/2411.17067/) (2024)
- [3D Reconstruction with Fast Dipole Sums](https://mlsanity.com/paper/2405.16788/) (2024)
- [NeISF++: Neural Incident Stokes Field for Polarized Inverse Rendering of Conductors and Dielectrics](https://mlsanity.com/paper/2411.10189/) (2024)
- [SOF: Sorted Opacity Fields for Fast Unbounded Surface Reconstruction](https://mlsanity.com/paper/2506.19139/) (2025)
- [ProbeSDF: Light Field Probes for Neural Surface Reconstruction](https://mlsanity.com/paper/2412.10084/) (2024)
- [SplatSDF: Boosting SDF-NeRF via Architecture-Level Fusion with Gaussian Splats](https://mlsanity.com/paper/2411.15468/) (2024)
- [Geometry-Grounded Gaussian Splatting](https://mlsanity.com/paper/2601.17835/) (2026)
- [From Blobs to Spokes: High-Fidelity Surface Reconstruction via Oriented Gaussians](https://mlsanity.com/paper/2604.07337/) (2026)
- [Many-Worlds Inverse Rendering](https://mlsanity.com/paper/2408.16005/) (2024)
- [OMG: Opacity Matters in Material Modeling with Gaussian Splatting](https://mlsanity.com/paper/2502.10988/) (2025)
- [Confidence-Based Mesh Extraction from 3D Gaussians](https://mlsanity.com/paper/2603.24725/) (2026)
- [VG2GT: Voxel-Gaussian Splatting Visual Geometry Grounded Transformer](https://mlsanity.com/paper/2606.01573/) (2026)
- [Robust SG-NeRF: Robust Scene Graph Aided Neural Surface Reconstruction](https://mlsanity.com/paper/2411.13620/) (2024)
- [Stochastic Signed Distance Processes](https://mlsanity.com/paper/2606.20856/) (2026)
- [VoxelTTO: Voxel-Aligned Feed-Forward 3D Gaussian Splatting with Test-Time Optimization](https://mlsanity.com/paper/2609.21498/) (2026)

## Abstract (arXiv)

We develop a theory for the representation of opaque solids as volumes. Starting from a stochastic representation of opaque solids as random indicator functions, we prove the conditions under which such solids can be modeled using exponential volumetric transport. We also derive expressions for the volumetric attenuation coefficient as a functional of the probability distributions of the underlying indicator functions. We generalize our theory to account for isotropic and anisotropic scattering at different parts of the solid, and for representations of opaque solids as stochastic implicit surfaces. We derive our volumetric representation from first principles, which ensures that it satisfies physical constraints such as reciprocity and reversibility. We use our theory to explain, compare, and correct previous volumetric representations, as well as propose meaningful extensions that lead to improved performance in 3D reconstruction tasks.