Objects as volumes: A stochastic geometry view of opaque solids

TL;DR

This work develops a rigorous stochastic-geometry framework for representing opaque solids as volumes, deriving conditions under which exponential volumetric transport holds and expressing the attenuation coefficient as a function of the occupancy/vacancy distributions. It extends to anisotropic transport and stochastic implicit surfaces, ensuring reciprocity and reversibility in the volume rendering equation. By showing NeuS and VolSDF as special cases of their theory, the authors diagnose reciprocity violations in prior models and propose principled, physically plausible alternatives. Experiments on standard 3D reconstruction benchmarks demonstrate that the proposed reciprocal, anisotropy-aware formulations yield improved accuracy and more meaningful scalar fields (mean implicit function, vacancy, and density). This framework provides a practical toolbox for designing volumetric representations that better capture solid geometry in neural rendering pipelines and 3D reconstruction tasks.

Abstract

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.

Figures (10)

• Figure 1: Volumetric representations replace deterministic (left) with stochastic (right) ray casting: rather than find the first intersection with deterministic geometry, they use the free-flight distribution along a ray to represent the probability of first intersection with stochastic geometry. Classical volumetric representations describe stochastic microparticle geometry (top). We derive volumetric representations for stochastic solid geometry (bottom).
• Figure 2: Overview of our theory, presented in \ref{['thm:coefficient', 'def:anisotropy', 'pro:implicit']}.
• Figure 3: The attenuation coefficient optimized for the bear scene in BlendedMVS behaves as anticipated by our theory: isotropically in the object interior, and anisotropically near its surface.
• Figure 4: When optimizing for the clock scene in BlendedMVS using NeuS, the $\mathrm{ReLU}$ term leads to attenuation coefficient (top) and transmittance (bottom) values that violate reciprocity. By contrast, using our representation leads to reciprocal results.
• Figure 5: Visualization of shape and key quantities of our volumetric representation learned for scenes in the BlendedMVS dataset.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• proof
• proof