Solving partial differential equations in participating media

Bailey Miller; Ioannis Gkioulekas; Keenan Crane; Rohan Sawhney

REVIEW 3 major objections 5 minor 7 references

Solving partial differential equations in participating media

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proposes volumetric walk on spheres and volumetric walk on stars, Monte Carlo estimators that solve Laplace boundary value problems directly in stochastic microparticle geometry described by the Poisson Boolean model, without…

desk verdict Genuinely new problem and useful estimators; the walk-memory conditioning claim is asserted, not proven, and Equation 32 in Appendix A is wrong as printed. read the letter →

arxiv 2506.08237 v1 pith:XFHOVGMB submitted 2025-06-09 cs.GR cs.NAmath.NA

Bailey Miller , Rohan Sawhney , Keenan Crane , Ioannis Gkioulekas This is my paper

classification cs.GRcs.NAmath.NA MSC 65C0560D0535J05

keywords participatingmediawalkonspheresstarsPoissonBooleanmodelMonteCarloPDEsimulationstochasticgeometryhomogenizationensembleaveraging

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Solving PDEs in domains perforated by huge numbers of microscopic particles is usually done either by averaging many explicit random particle configurations (ensemble averaging, expensive) or by homogenizing, which is biased unless particles are infinitesimal. This paper proposes treating the particle geometry as a participating medium governed by the Poisson Boolean model, and asks for the mean solution, the expected value of the PDE solution over random configurations. It derives a boundary integral equation for this mean solution and two recursive single-sample Monte Carlo estimators, volumetric walk on spheres (VWoS) and volumetric walk on stars (VWoSt), that estimate it directly. The estimators replace the deterministic closest-point queries of walk on spheres with closest-point sampling conditioned on the history of the walk. Experiments show that VWoS and VWoSt match ensemble averaging in expectation while running faster, and avoid the bias of homogenization at finite particle sizes and densities.

What carries the argument

The load-bearing object is the walk memory M_k, summarized by the empty-ball memory E(M_k), the union of all walk spheres that must contain no particle centers, and the sampled-particle memory C(M_k), the particles discovered during the walk, treated as deterministic geometry. Conditional closest-point sampling then reduces to thinning the Poisson Boolean model density to zero inside the dilation E(M_k) ⊕ R, and taking the closer of the sampled stochastic boundary point and the deterministic closest point on ∂(V \ C(M_k)). The other essential ingredient is the polar representation of the Poisson point process of particle centers: the cubed distance from a point to its closest center is an exponential random variable, which makes closest-center sampling straightforward, including in heterogeneous media via acceptance-rejection thinning.

What would settle it

Compare VWoS against a high-sample-count ensemble average on a dense, large-radius medium engineered so that walk spheres overlap heavily; if the conditional sampling for conjunctions of empty-ball events is biased, the VWoS mean will drift from the ensemble average by more than Monte Carlo error, and the drift should grow with average walk length.

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Extended reading notes

Core claim

The paper claims that the ensemble-averaged Laplace solution in a Poisson Boolean medium satisfies a boundary integral equation in which each step is conditioned on the walk's memory, and that recursive single-sample Monte Carlo estimation of this equation gives an unbiased estimator of the mean solution. The VWoS estimator is structurally identical to walk on spheres except that sphere radii come from conditional closest-point sampling rather than deterministic queries, and every step updates a memory of empty balls and sampled particles. The same construction yields VWoSt for mixed Dirichlet-Neumann problems, with the particle boundary carrying Neumann conditions, via closest silhouette point sampling. The paper therefore claims that no particle configuration ever needs to be explicitly sampled or solved: the mean solution is estimated directly, with per-walk cost proportional to the local medium rather than to the full configuration.

Load-bearing premise

The argument assumes that the walk's memory M_k, empty balls plus sampled particles, captures everything about past closest-point events, so conditioning on many such events amounts only to zeroing the medium density inside the dilated empty balls and treating sampled particles as fixed geometry; this sufficiency is proven for a single conditioning set but assumed, not proven, for the conjunction of overlapping events that a walk accumulates.

Editorial extensions

If this is right

If correct, VWoS and VWoSt give unbiased estimates of the mean solution for finite, not infinitesimal, particle size and density, removing homogenization's limiting assumption.
Because each walk discovers only local particles, the methods are output-sensitive: cost does not scale with the number of particles in the whole volume, giving large speedups over ensemble averaging, especially for sparse evaluation points or dense media.
Memory is not optional: a memoryless variant is both biased and slower, so any implementation must store and update empty balls and sampled particles.
The framework extends to screened Poisson and Poisson equations and, through VWoSt, to Neumann and Robin-style boundary conditions on particles.
Coupling VWoS with volumetric path tracing in the same medium, carrying memory across both, makes joint light-transport and diffusion simulations such as photochemical cloud effects possible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A symmetric extension of the paper's idea would adapt memory-conditioned sampling to volumetric rendering: VWoS requires full memory while volumetric path tracing is memoryless, so importing the empty-ball/sampled-particle conditioning into free-flight sampling might reduce bias for paths whose segments overlap.
The proof gap in Section 5.2 could be closed by deriving the conditional closest-point density for a conjunction of many overlapping empty-ball events; a testable prediction is that any bias from this gap grows with walk length and overlap, so stress tests in dense media with large particle radii would reveal it.
The performance bottleneck the paper identifies, the thinning of Algorithm 1, is the same structure as delta tracking in rendering, so null-collision, progressive-majorant, and multiple-importance-sampling schemes from that literature should transfer directly to VWoS and could be benchmarked.
For particle models beyond fixed-radius spheres, the memory-as-sufficient-statistic encoding would need re-derivation, since particles not determined by a center and radius would not be summarized by C(M_k) in the same way.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know about arXiv:2506.08237. First, it opens a genuinely new problem setting — linear elliptic PDEs in domains whose geometry is a stochastic Poisson Boolean medium — and delivers two estimators (VWoS, VWoSt) that target the mean solution directly, skipping both ensemble averaging and homogenization. Second, the load-bearing piece, memory-aware conditional closest point sampling, is plausible and likely correct but not fully proven, and there is one provably wrong formula in the appendix. Read it as a strong paper with a real rigour gap, not a broken one.

What is actually new: replacing closest point queries with conditional closest point sampling, where the walk remembers the empty balls it carved and the particles it hit. That memory mechanism is the intellectual core. The authors show that, unlike in volume rendering where a memoryless renewal approximation is standard, dropping memory here produces strongly biased estimates and worse runtime (Section 7.3). That is a genuine, non-obvious finding. The derivation of the boundary integral equation in participating media is clean, the estimator has no fitted constants, the experimental validation against ensemble averaging is honest, and the homogenization comparisons make the bias story concrete. The code is open source. For my money the membrane and cloud demos are suggestive rather than decisive, but that is normal for a methods paper.

Soft spots, in proportion. The proof of the single conditioning event in Appendix A is wrong as printed: Equation 32 subtracts an integral over the full boundary of B(x,r)∩(Q⊕R), but differentiating the void probability gives only the spherical-shell term ∫_{∂B(x,r)∩A} λ. For r beyond the extent of A, the printed term is nonzero while the correct one vanishes. Since Algorithm 1 with the zeroed density implements the correct conditional process, this is a display bug, but it needs fixing. The larger gap: the extension from one conditioning set to the full walk memory — a Palm-type characterization with overlapping empty balls and exact sampled centers — is asserted in Section 5.2, not proved. I am not as worried as the stress-test note about zero-probability conditioning: treating the sampled particles as deterministic atoms is exactly the right Palm interpretation, and the standard independence properties of the Poisson process make the claim believable. But "believable" is not "proved," and the unbiasedness of both estimators rests on it. The ensemble-averaging comparisons are decent evidence but cannot exclude small bias, and both methods share the ε-shell approximation.

Who this is for: Monte Carlo PDE people and volume rendering people, who will recognize the structure. The paper deserves a serious referee, and the right outcome is a major revision: supply the Palm argument (or a counterexample), fix Equation 32, and the central claim will stand or fall cleanly.

Referee Report

3 major / 5 minor

Summary. The paper introduces the problem of solving linear elliptic boundary value problems in participating media, where the domain is a deterministic volume V perforated by random microparticle geometry O following the Poisson Boolean model. The authors define the mean solution as the expectation of the PDE solution over particle configurations and develop two estimators, volumetric walk on spheres (VWoS) and volumetric walk on stars (VWoSt), that recursively sample the closest boundary point using the medium density without constructing explicit particle configurations. The derivation parallels volume rendering: the boundary integral equation is combined with the law of total expectation, conditioning on the walk memory M_k of previously sampled closest points. The memory is summarized by empty balls E(M_k) and sampled particle centers C(M_k), and conditional closest point sampling is implemented by thinning the medium density outside E(M_k)⊕R and treating C(M_k) as deterministic geometry. Experiments compare VWoS and VWoSt with ensemble averaging and homogenization on several scenes, reporting matching mean estimates, speedups over ensemble averaging, and reduced bias relative to homogenization. The paper also demonstrates a coupled VWoS/volumetric path tracing application for photochemistry in clouds and provides an open-source implementation.

Significance. If the correctness of the conditional-sampling construction is established, this is a significant contribution to Monte Carlo PDE simulation and to the graphics community more broadly. The paper introduces a new problem formulation, connects it to the mature volume rendering literature, and provides two algorithms whose estimates have no fitted constants and whose per-walk cost does not require simulating entire particle configurations. The experimental validation against ensemble averaging on scenes with varying particle size and density is a concrete strength, as is the open-source release. The speedups over ensemble averaging and the demonstration of homogenization bias are convincing. The main risk is the rigor of the memory-sufficiency argument: the paper proves conditional sampling only for a single empty-ball event, yet the algorithms rely on a full-memory Palm characterization that is asserted rather than proven, and the proof as printed contains a concrete boundary-integral error. These issues are load-bearing for the unbiasedness claim but appear fixable within the manuscript's scope.

major comments (3)

[Appendix A, Eq. (32)] The displayed conditional closest-center density in Eq. (32) is not the derivative of the exponentiated integral in the preceding line. For A = Q⊕R, the derivative of ∫_{B(x,r)∩A} λ with respect to r is ∫_{∂B(x,r)∩A} λ dA, not ∫_{∂(B(x,r)∩A)} λ dA; the latter incorrectly includes the stationary portion ∂A ∩ B(x,r), which contributes zero to the derivative. In particular, when r is large enough that A ⊂ B(x,r), the printed second boundary integral is nonzero while the correct term vanishes. As printed, Eq. (32) therefore does not equal the unconditional density computed with the thinned density λ(·|Q), so the proof of the conditional-sampling lemma is incorrect. The statement of the lemma is standard and can be repaired, but the corrected proof must appear in the revision.
[Section 5.2] The summary of the walk memory by the empty-ball set E(M_k) and the sampled-particle set C(M_k) is a nontrivial sufficient-statistic claim, and the manuscript only asserts it. A memory event is a conjunction of many empty-ball events together with Palm-type conditions that fix exact centers on the boundaries of dilated empty balls; these are zero-probability events, and a rigorous treatment must show that the conditional point process is a Poisson process on V \ (E(M_k)⊕R) with atoms at C(M_k). Appendix A proves the analogous statement for a single empty-ball event only. Because Algorithm 4 and the unbiasedness of the estimator (19) depend on this characterization, the revision should supply the missing proof or a precise citation to such a Palm characterization.
[Section 5.1, Eq. (18)] The derivation of the volumetric boundary integral equation conditions on the exact closest-point events M_k, which have probability zero under the Poisson Boolean model, and writes pcp_k(y_k|M_k) as a density with respect to Lebesgue measure on V. The manuscript does not specify the regular conditional distribution being used, nor the Jacobian relating the surface-supported closest-point distribution to the volume density in Eq. (18). This is intertwined with the missing memory proof in Section 5.2 and should be clarified; otherwise the chain of equalities leading to the estimator (19) is not fully rigorous.

minor comments (5)

[Section 5.1 heading] The section heading 'Boundary integral eqation in participating media' contains a typo ('eqation' should be 'equation').
[Appendix A, Eq. (29) and Section 5.1, Eq. (15)] The notation pcp_x is used both for a density on the sphere conditional on radius and for a density with respect to Lebesgue measure on V; the Jacobian relating the two is not stated and should be made explicit.
[Section 4, Algorithm 1] The convention that the closest point is set to x when the sampled distance r_{∂O} is negative (i.e., when x lies inside a particle) is used in Algorithm 1 but introduced only implicitly; it should be stated in the main text near Eq. (7).
[Figure 1 caption] The text '1x time 1.15x time' is unclear; the runtime comparison between the deterministic solve and VWoS should be described in words or with a clear annotation.
[Table 2] The entry 'texture(x)' for the membrane experiment is not defined; please specify the texture map or refer to a supplement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the VWoS/VWoSt estimators are derived from the standard boundary integral equation and the definition of the mean solution as an expectation over Poisson Boolean model configurations, with no fitted parameters and no self-citation chain carrying the central claim.

full rationale

The central derivation (Section 5.1) starts from the standard BIE (Eq. 2), applies expectation over the PBM particle configuration to define the mean solution (Eq. 5), and uses the law of total expectation to produce the participating-media BIE (Eq. 18). Eq. 19 is literally the recursive single-sample Monte Carlo estimator of Eq. 18, so the estimator is equivalent to its defining integral equation by construction — this is a standard Monte Carlo derivation, not a hidden circularity. The conditional closest point sampling in Section 5.2 is a substantive mathematical claim about memory sufficiency, which may be right or wrong, but it is not circular: it does not assume the estimator it is used to justify. Appendix A's Eq. 32, even if its printed boundary term were erroneous, would be a correctness bug, not circular reasoning. Comparisons against ensemble averaging and homogenization are external benchmarks, not fitted inputs. Self-citations (Sawhney and Crane 2020; Sawhney et al. 2023; Miller et al. 2024b; Sawhney and Miller 2024; Sawhney 2021) are to prior algorithms and software used as building blocks or implementation tools; none is invoked as a uniqueness theorem or as the sole justification for the target result. The paper's own Section 7.3 experiments show that memory is needed, which is consistent with, rather than circularly assumed by, the derivation. Overall the claimed prediction (unbiased estimation of the mean solution without ensemble averaging) has independent content and is not forced by definition or self-citation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central estimator is derived from the standard BIE and the PBM, with no fitted constants. The only hand-chosen parameters are the termination threshold epsilon and the sampling majorant, both standard numerical parameters. The load-bearing assumptions are the PBM model itself and the correctness of memory-based conditional sampling.

free parameters (2)

epsilon-shell width = 1e-4 to 1e-3 (Table 1)
Walk termination threshold; chosen one order of magnitude smaller than particle radius, standard in WoS, not fitted to match target solutions.
majorant density lambda_bar = not specified, global or per-point majorant
Controls the thinning acceptance rate in Algorithm 1; a sampling efficiency parameter, not a model parameter.

assumptions (4)

standard math Poisson point process void probability and independence (Properties P1 and P2)
Used to derive closest point sampling densities (Equations 8 and 11) and the conditional sampling proof (Appendix A).
standard math Mean value property / Poisson kernel for the Laplace equation on a ball
The boundary integral equation (2) and the WoS recursion rely on this classical result.
domain assumption Existence, uniqueness, and continuity of the Dirichlet BVP solution for each particle configuration
Needed for the mean solution E_O[u] to be well-defined and for the epsilon-shell approximation to be valid.
domain assumption Microparticle geometry follows the Poisson Boolean model with independent spherical particles of fixed radius R
The PBM is the modeling assumption for microparticle geometry; the algorithms' correctness is specific to this model.

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Cite this review

Pith. "Pith review of Solving partial differential equations in participating media." pith.science (2026). https://pith.science/paper/XFHOVGMB

@misc{pith2026250608237,
  author       = {Pith},
  title        = {Pith review of: Solving partial differential equations in participating media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XFHOVGMB}},
  note         = {Machine review of arXiv:2506.08237}
}

read the original abstract

We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling this problem by treating the domain as a participating medium that models microparticle geometry stochastically, through aggregate statistical properties (e.g., particle density). We first introduce the problem setting of PDE simulation in participating media. We then specialize to exponential media and describe the properties that make them an attractive model of microparticle geometry for PDE simulation problems. We use these properties to develop two new algorithms, volumetric walk on spheres and volumetric walk on stars, that generalize previous Monte Carlo algorithms to enable efficient and discretization-free simulation of linear elliptic PDEs (e.g., Laplace) in participating media. We demonstrate experimentally that our algorithms can solve Laplace boundary value problems with complex microparticle geometry more accurately and more efficiently than previous approaches, such as ensemble averaging and homogenization.

Figures

Figures reproduced from arXiv: 2506.08237 by the authors.

**Figure 2.** We consider domains 𝛺 ≔ 𝑉 \𝑂 equal to the difference between a deterministic volume𝑉 and a random configuration of particles𝑂 contained in the volume. The particle configuration follows the Poisson Boolean model (PBM) whose density 𝜆(𝑥 ) is defined over the volume. and density scales, and throughout the entire domain. We revisit homogenization and show experimental comparisons in Section 7.2. Lastly, homogenization … view at source ↗

**Figure 3.** Ensemble averaging is a simple but expensive method to estimate the mean solution of a PDE in a participating medium, by first sampling many random particle configurations (top row), then solving the PDE on each sampled domain (bottom row), and finally averaging the computed solutions. Our volumetric walk on spheres algorithm directly estimates the mean solution without expensive ensemble averaging. As we show in [… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Computing the closest point 𝑦 𝜕𝑂(𝑥 ) from the closest center 𝑐 (𝑥 ). Closest point distribution and sampling. As WoS interacts with the domain through closest point queries, using it in participating media requires reasoning about the random closest point 𝑦 𝜕𝑂 (𝑥) ∈ …

**Figure 5.** Figure 5: The Poisson Boolean model enables efficient closest point [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: (Left) Walk on spheres jumps along spheres whose radius it determines through closest point queries with respect to the deterministic domain boundary. It terminates when it reaches the 𝜀-shell of the boundary. (Right) Volumetric walk on spheres also jumps along sphere…

**Figure 7.** Figure 7: (a) To sample the closest point 𝑦 𝜕𝛺 at 𝑥 conditionally on the memory M accumulated during a walk, we determine two points: First, we sample the random closest point 𝑦 𝜕𝑂 on the stochastic microparticle geometry, but with the PBM density zeroed out inside the spheres f…

**Figure 8.** Figure 8: Sampling to form a star-shaped region. Because 𝑟𝑘 can be greater than the shortest distance 𝑟 𝜕𝑂 𝑘 = ∥𝑥𝑘 −𝑦 𝜕𝑂 𝑘 ∥ to 𝜕𝑂, forming the star-shaped region St(𝑥𝑘 , 𝑟𝑘 ) requires also determining any additional particles that are closer to 𝑥𝑘 than 𝑟𝑘 . We do so by usin…

**Figure 9.** Figure 9: We compare the outputs of VWoS (fourth row) and homogenization (third row) to the reference mean solution (second row) computed with ensemble averaging, in BVPs with Dirichlet-only boundary conditions. Across a range of medium parameters and boundaries, VWoS reliably p…

**Figure 11.** Figure 11: Statistics for walk length (first row) and memory size (second row) for the mushroom ( [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 12.** Figure 12: We compare the bias-performance trade-off of finite memory of [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: We couple VWoS and VPT to model diffusion and light transport (resp.) in a proof-of-concept atmospheric photochemical system: a cloud [Pharr [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

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Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages

[1]

estimates𝐿o(𝑥+𝑡𝜔,−𝜔|𝑡𝜕𝛺 =𝑡) if𝑡𝜕𝛺 <𝑡𝜕𝑉 , effectively applying single-sample Monte Carlo to the integral term

work page
[2]

forgetting

estimates 𝐿o(𝑥+𝑡𝜕𝑉𝜔,−𝜔|𝑡𝜕𝑉 =𝑡) if𝑡𝜕𝛺≥𝑡𝜕𝑉 . Estimation then proceeds recursively, albeit without memory: VPT approximates 𝐿o(𝑥+𝑡𝜔,−𝜔|𝑡𝜕𝛺 = 𝑡) ≈𝐿o(𝑥+𝑡𝜔,−𝜔)—a so- called renewal assumption [Seyb et al. 2024]—and continues to iterate Equation (37) (after first using the in-scattering equation to convert from outgoing to incident radiance). To make free-flight...

work page 2024
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ACM Trans

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

estimates𝐿o(𝑥+𝑡𝜔,−𝜔|𝑡𝜕𝛺 =𝑡) if𝑡𝜕𝛺 <𝑡𝜕𝑉 , effectively applying single-sample Monte Carlo to the integral term

work page

[2] [2]

forgetting

estimates 𝐿o(𝑥+𝑡𝜕𝑉𝜔,−𝜔|𝑡𝜕𝑉 =𝑡) if𝑡𝜕𝛺≥𝑡𝜕𝑉 . Estimation then proceeds recursively, albeit without memory: VPT approximates 𝐿o(𝑥+𝑡𝜔,−𝜔|𝑡𝜕𝛺 = 𝑡) ≈𝐿o(𝑥+𝑡𝜔,−𝜔)—a so- called renewal assumption [Seyb et al. 2024]—and continues to iterate Equation (37) (after first using the in-scattering equation to convert from outgoing to incident radiance). To make free-flight...

work page 2024

[3] [5]

ACM Trans

A Practical Walk-on-Boundary Method for Boundary Value Problems. ACM Trans. Graph. 42, 4, Article 81 (July 2023), 16 pages. https://doi.org/10.1145/3592109 Eric Veach. 1998. Robust Monte Carlo methods for light transport simulation . Stanford University. Delio Vicini, Wenzel Jakob, and Anton Kaplanyan. 2021. A non-exponential transmit- tance model for vol...

arXiv 1993

[4] [2010]

In ACM SIGGRAPH 2010 Papers (Los Angeles, California) (SIGGRAPH ’10)

A radiative transfer framework for rendering materials with anisotropic structure. In ACM SIGGRAPH 2010 Papers (Los Angeles, California) (SIGGRAPH ’10). Association for Computing Machinery, New York, NY, USA, Article 53, 13 pages. https://doi.org/10.1145/1833349.1778790 Doug L James and Dinesh K Pai. 1999. Artdefo: accurate real time deformable objects. I...

work page arXiv 2021

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ACM Trans

Functional optimization of fluidic devices with differentiable stokes flow. ACM Trans. Graph. 39, 6, Article 197 (Nov. 2020), 15 pages. https://doi.org/10.1145/ 3414685.3417795 Bruce C Faust. 1994. Photochemistry of clouds, fogs, and aerosols. Environmental science & technology 28, 5 (1994), 216A–222A. Jeppe Revall Frisvad, Niels Jørgen Christensen, and H...

work page arXiv 1994

[6] [2022]

Atmospheric Measurement Techniques 15, 12 (2022), 3875–3892

The impact of sampling strategy on the cloud droplet number concentration estimated from satellite data. Atmospheric Measurement Techniques 15, 12 (2022), 3875–3892. David W Hahn and M Necati Özisik. 2012. Heat conduction. John Wiley & Sons. Samuel R Hall, Kirk Ullmann, Michael J Prather, Clare M Flynn, Lee T Murray, Arlene M Fiore, Gustavo Correa, Sarah ...

work page 2022

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In ACM SIGGRAPH 2023 Conference Proceedings (Los Angeles, CA, USA) (SIGGRAPH ’23)

Progressive null-tracking for volumetric rendering. In ACM SIGGRAPH 2023 Conference Proceedings (Los Angeles, CA, USA) (SIGGRAPH ’23) . Associa- tion for Computing Machinery, New York, NY, USA, Article 31, 10 pages. https: //doi.org/10.1145/3588432.3591557 Jonathan T Moon, Bruce Walter, and Stephen R Marschner. 2007. Rendering discrete random media using ...

arXiv 1992