We develop a theory for the representation of opaque solids as volumetric models. 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 derive expressions for the volumetric attenuation coefficient as a functional of the probability distributions underlying the stochastic model, and we generalize our theory to account for isotropic and anisotropic scattering, as well as stochastic implicit surfaces. Our representation is derived from first principles, which ensures that it satisfies physical constraints such as reciprocity and reversibility. We use our theory to explain and improve upon previous volumetric representations used in 3D reconstruction.

Since the introduction of NeRF, volumetric representations have proliferated in neural rendering techniques. Yet their application to opaque objects in transparent media differs fundamentally from classical uses in translucent materials such as fog, smoke, and biological tissue. The authors pose three central questions: "Why can we use volumetric light transport to simulate images of scenes with only light-surface interactions? What is the mathematical underpinning for modeling an opaque object as a volume?" and what optical properties such volumes should possess.

Rather than adopting heuristic approaches used in prior work, the authors derive volumetric representations "from first principles, which ensures that it satisfies physical constraints such as reciprocity and reversibility." They frame volumetric representations as a formalism for querying stochastic geometry, where transmittance answers visibility queries and free-flight distributions address ray-casting queries. This perspective unifies the treatment of opaque solids within the established framework of radiative transfer theory.

The paper's contributions are threefold. First, the authors explain previous volumetric representations as special cases within their unified framework, highlighting critical defects such as lack of reciprocity and reversibility. Second, they propose principled extensions that systematically address these limitations. Third, the theory is evaluated experimentally, demonstrating improved 3D reconstruction performance on common datasets when incorporated into existing neural rendering pipelines. The authors provide open-source code and interactive visualizations to enable reproducibility.

Volumetric light transport models scenes with stochastic geometry. The framework handles both microscopic particles in translucent materials and macroscopic opaque objects. A key insight is that "deterministic light transport algorithms interact with scene geometry only through two geometric queries." The first query addresses visibility V(x,y) between two points x and y, taking values in {0,1}. The second involves computing the free-flight distance t* along a ray — the distance before first intersection with scene geometry.

Transmittance represents "the probability of visibility from the ray origin — equivalently, the tail distribution of the free-flight distance," defined as T(x,w)(t) = Pr{V(x,w)(t) = 1}. The free-flight distribution p^ff provides the probability density function of when intersection occurs: p^ff(x,w)(t) = -dT/dt. The attenuation coefficient sigma(x,w) equals "the probability density of zero free-flight distance — equivalently, probability density of ray termination." Transmittance satisfies three crucial properties: it is reciprocal, monotonically non-increasing, and equals 1 at distance zero.

Most volumetric approaches assume the free-flight distance follows an exponential distribution — termed "exponential transport." This assumption yields the classical transmittance formula: T(x,w)(t) = exp(-integral from 0 to t of sigma(r(s),w) ds), and the free-flight distribution: p^ff(x,w)(t) = sigma(r(t),w) * T(x,w)(t). The reciprocity condition requires that sigma(x,w) = sigma(x,-w), meaning the attenuation coefficient must be symmetric with respect to ray direction. Isotropic transport further assumes sigma(x,w) = sigma(x), making the coefficient direction-independent, while anisotropic transport allows directional variation.

The authors introduce Definition 2, characterizing an opaque solid through an indicator function I: R^3 -> {0,1}, defining a solid O as the set where I(x)=1. A solid is opaque when "for every point x in O and direction w, the visibility satisfies V(x,w)(t) = delta(t)," meaning light cannot penetrate into the solid's interior. Definition 3 then introduces stochastic solids by treating the indicator function as a random field, enabling two derived quantities: occupancy o(x) representing the probability that point x is occupied, and vacancy v(x) = 1 - o(x) representing the probability of being unoccupied.

Theorem 4 provides the central theoretical result. It establishes that "the free-flight distribution is exponential if and only if the restriction of the indicator function on this ray is a continuous-time discrete-space Markov process." The theorem further proves that reciprocity and reversibility hold precisely when the attenuation coefficient equals: sigma_delta(x,w) = |w * grad(log(v(x)))| = |w * grad(v(x))| / v(x). This expression provides a unique, physically principled formula connecting the volumetric attenuation to the spatial gradient of the vacancy field.

The authors decompose the attenuation coefficient into two physically meaningful components: sigma_delta(x,w) = sigma_parallel(x) * sigma_perpendicular(x,w), where sigma_parallel(x) = ||grad(v(x))|| / v(x) represents an isotropic "density" term, and sigma_perpendicular(x,w) = |w * n(x)| represents an anisotropic "projected area" term. Here n(x) is the unit normal to the level set of vacancy at point x. This decomposition reveals that the attenuation of an opaque solid is inherently anisotropic — it depends on the angle between the ray direction and the local surface normal.

Definition 5 generalizes the framework through a distribution of normals D_x: S^2 -> R. The projected area becomes: sigma_perpendicular_D(x,w) = integral over S^2 of |w * m| * D_x(m) dm. This formulation allows modeling directionally-dependent foreshortening effects that mimic deterministic surfaces near the boundary while remaining isotropic in the interior. In the stochastic microparticle geometry setting, isotropic transport is equivalent to a model where microparticles are rotationally-symmetric scatterers — a condition that generally does not hold for opaque macroscopic objects.

This section extends the theory to implicit surface representations, common in neural rendering. Rather than working directly with occupancy fields, the authors derive volumetric parameters from implicit surfaces represented as zero level sets of signed distance functions or other scalar fields. The key insight is that the vacancy field v(x) can be expressed as a function of the implicit surface's level-set value, typically through a sigmoid or cumulative distribution function transformation. This connects the theoretical framework to practical representations used by methods like NeuS and VolSDF.

The extension maintains the fundamental theoretical properties — reciprocity, reversibility, and physical plausibility — while adapting the attenuation coefficient expressions to work with implicit geometry representations that practitioners already employ in neural rendering pipelines. Importantly, the authors show that previous methods such as NeuS and VolSDF correspond to specific choices within this framework, but these choices do not always satisfy the physical constraints derived in Theorem 4. The stochastic implicit surface formulation thus provides principled alternatives that preserve reciprocity and reversibility by construction.

The paper positions its theoretical framework as explaining previous volumetric representations as special cases. The authors demonstrate that their rigorous, first-principles derivation unifies disparate heuristic approaches used in neural rendering. The theory clarifies how prior methods by Mildenhall et al. (NeRF), Yariv et al. (VolSDF), and others correspond to different stochastic modeling assumptions about opaque geometry. Importantly, the authors identify "critical defects of previous volumetric representations (e.g., lack of reciprocity and reversibility)" and show how their approach systematically addresses these limitations.

The framework reveals that existing methods made arbitrary choices about which geometric properties to preserve when converting deterministic representations into volumetric ones, whereas this work's approach follows directly from volumetric transport axioms. For instance, NeRF's isotropic density formulation discards the anisotropic projected-area term that naturally arises from the theory, while NeuS's weight function does not correspond to a valid free-flight distribution in all cases. These insights provide concrete, actionable guidelines for designing improved volumetric representations.

Though primarily theoretical, the paper demonstrates practical utility through experimental validation. The authors note that their improved representation can be "readily incorporated within existing volumetric neural rendering pipelines." Testing on standard 3D reconstruction datasets shows that "replacing previous volumetric representations with ours leads to significantly better (qualitatively and quantitatively) 3D reconstructions." The work provides open-source code and interactive visualizations, enabling reproducibility and practical adoption.

The experimental validation confirms that the theoretically-motivated anisotropic treatment of surface versus interior regions produces measurable performance gains in 3D reconstruction tasks. Specifically, incorporating the projected-area term into the attenuation coefficient leads to sharper surface reconstructions and more accurate geometry recovery, particularly in regions with grazing angles and thin structures where the isotropic assumption of standard NeRF is most severely violated. These results validate the practical significance of the theoretical distinction between isotropic and anisotropic volumetric transport for opaque solids.

Comparisons with NeRF, NeuS, and VolSDF demonstrate that the theoretically-principled representation achieves superior or competitive surface reconstruction quality across all tested scenarios. The improvements are most pronounced in challenging geometries — objects with sharp edges, concavities, and fine surface detail — where heuristic approaches tend to produce smoothed or inaccurate geometry. The authors emphasize that these improvements come "at no additional computational cost" beyond replacing the attenuation coefficient formula, making the proposed representation a drop-in replacement for existing volumetric pipelines.