Bernstein Bounds for Caustics

Boxuan Li; Chen Wang; Jie Guo; Ling-Qi Yan; Yanwen Guo; Yiming Wang; Yuxuan Guo; Zhimin Fan

REVIEW 3 major objections 5 minor 2 references

Bernstein Bounds for Caustics

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that conservative Bernstein-basis bounds on caustic position and irradiance let a renderer sample specular triangle tuples with bounded variance, rendering sharp caustics up to an order of magnitude faster than existing…

desk verdict Solid and unusually honest caustics-sampling paper whose estimator algebra checks out; the main caveat is that the conservativeness proofs are deferred to a supplemental I haven't seen. read the letter →

arxiv 2504.19163 v1 pith:GNFQ62B4 submitted 2025-04-27 cs.GR

Zhimin Fan , Chen Wang , Yiming Wang , Boxuan Li , Yuxuan Guo , Ling-Qi Yan , Yanwen Guo , Jie Guo This is my paper · ORCID

classification cs.GR

keywords specularcausticsBernsteinpolynomialsrationalfunctionboundsimportancesamplingvariancecontrolunbiasedrenderingtriangletuplechains

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

To render sharp specular caustics without enumerating every candidate reflection/refraction path, the paper proposes stochastically sampling the triangle tuples that can carry light, and it argues this can be done without losing unbiasedness or surrendering variance control. The key move is to bound, for each tuple, where the caustic can land on the receiver and how much irradiance it can deliver; these bounds come from a property of rational functions expressed on the Bernstein basis. From the bounds the paper constructs sampling probabilities and an estimator whose population variance is bounded by a known proxy computed from the same bounds. If the construction is correct, a renderer can safely skip most low-contribution triangles and spend its time solving for paths inside the promising few, yielding up to an order of magnitude faster equal-time rendering of sharp caustics than existing unbiased methods.

What carries the argument

The central object is the Bernstein bound for rational functions: for $f=p/q$ written in Bernstein form over a box, the minimum and maximum of the coefficient ratios $b_{\boldsymbol{i}}(p)/b_{\boldsymbol{i}}(q)$ bracket the true range whenever all denominator coefficients share one sign, and subdivision makes the bracket converge quadratically. Around this, the paper builds three components: a rational formulation of specular vertex positions via rational coordinate mapping, with a remainder variable absorbing the non-rational refraction square root; a rational formulation of irradiance through the generalized geometric term, using implicit differentiation to avoid loose bounds near total internal reflection; and an optimized multi-tuple sampler whose probabilities $P_T = \min(\gamma \widetilde{E}(T), 1)$ arise from minimizing the variance proxy under a budget on expected sample count. The machinery converts light transport over a continuum of paths into bounded boxes that drive stochastic search.

What would settle it

Render the Plane or Slab scene with remainder variables enabled, and for every admissible path compare the irradiance bound $\widetilde{E}$ against the true irradiance computed by explicit path tracing. If any solution has $\widetilde{E} < E$ (the paper's own Fig. 18 shows such red regions only when remainder variables are disabled), the conservative-bound premise fails; a corollary check is to confirm that positions sampled with $P_T = \min(\gamma \widetilde{E}(T), 1)$ reproduce the reference mean at high sample counts.

Watch

Extended reading notes

Core claim

The paper's central claim is that the set of specular triangle tuples contributing to a shading point can be sampled rather than exhaustively searched, with variance controlled by precomputed bounds. It derives conservative position bounds (the range of receiver barycentric coordinates) and irradiance bounds (the range of received irradiance) for each tuple by writing both as rational functions of the first specular vertex's barycentric coordinates and applying the Bernstein coefficient-ratio inequality: the range of $f=p/q$ over the domain is contained between the minimum and maximum of the coefficient ratios $b_{\boldsymbol{i}}(p)/b_{\boldsymbol{i}}(q)$ when the denominator coefficients share a sign. Non-rational pieces, principally the square root in the refraction law, are absorbed into extra remainder variables so the bound stays valid. The estimator sums $E(T)/P_T$ over sampled tuples with $P_T = \min(\gamma \widetilde{E}(T), 1)$, and the second moment is bounded by $\sum_{T \in \mathcal{U}} \widetilde{E}(T)^2 / P_T$, so variance is controllable by construction. The paper reports that in equal-time comparisons this removes an order of magnitude of noise relative to unbiased manifold-sampling and deterministic-search baselines on one- and two-bounce specular chains.

Load-bearing premise

The method's load-bearing premise is that every computed bound really contains the true range: the denominator sign handling, the error term that compensates for the non-rational square root in refraction, the degree-reduction correction, and (as the paper discloses) the assumption of at most one solution per tuple must all hold; if any of them understates the error, a genuinely contributing tuple can get zero sampling probability, breaking both unbiasedness and the variance guarantee.

Editorial extensions

If this is right

Rendering sharp caustics no longer requires enumerating all triangle tuples; the search domain can be sampled with bounded variance.
Existing root-finding routines, whether deterministic or stochastically initialized, can be dropped into the sampled tuples, so the gain is orthogonal to solver choice.
Because low-contribution tuples get small but nonzero probability, the estimator stays unbiased while concentrating work where the precomputed irradiance bound is high.
For one- and two-bounce specular chains, equal-time images approach reference with roughly an order of magnitude lower relative MSE than previous unbiased methods.
The variance proxy gives a principled knob connecting precomputation budget, expected samples per shading point, and rendered noise.

Reading between the lines

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

The Bernstein-bound construction is not tied to triangle tuples; a plausible extension is to replace sample-fitted guiding distributions with conservative functional bounds, removing the initial uniform-distribution burn-in that path guiding needs.
If a proof of bound-tightness convergence under subdivision were supplied, the subdivision depth and threshold parameters could be set automatically; the paper explicitly leaves this open.
The remainder-variable trick absorbs any approximation error expressed as a bounded interval, so it could extend the same pipeline to near-specular BSDFs, non-planar surfaces, or emissive profiles without re-deriving the estimator.
For chains longer than two bounces the bottleneck is tuple growth; a testable variant is to apply bound-driven culling hierarchically during tuple extension to see whether high-contribution triples can be isolated before exhaustive precomputation explodes.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Dear [Colleague],

Quick take: this is a genuine advance in specular caustics sampling, and the central estimator algebra is sound — but the proof of the most load-bearing piece (the error envelopes that keep the bounds conservative) lives in a supplemental I couldn't see, and the m=1 assumption under-bounds some tuples.

What's new: instead of enumerating specular triangle tuples, the authors derive conservative range bounds for each tuple's receiver position and irradiance using Bernstein coefficient ratios (Eq. 3), then use those bounds to drive a multi-tuple sampler with P_T = min(gamma * Ehat(T), 1). The rational formulations come from their prior Specular Polynomials work, and the bounding theorem is Narkawicz et al., but the assembly into a variance-controllable sampler is new. I checked the estimator algebra in Sec. 5: Eq. (24) is unbiased provided Ehat is an upper bound and P_T > 0 for contributing tuples; Eq. (21) correctly multiplies the per-path bound by solution count; the KKT optimization for P_T checks out. The equal-time results in Figs. 13/14 are strong, and the authors are unusually honest about scope: short chains only, no Fresnel/visibility in the bound, and no proof of bound-tightness convergence.

Soft spots, in decreasing order of importance: - The remainder-variable construction (Eq. 12–13) and the SVD degree reduction (Sec. 6.1) are where conservativeness could break, but their error envelopes are deferred to the supplemental. The main text just refers to it. If either envelope understates the error, Ehat can dip below E(T), P_T can vanish, and the estimator is biased. The paper's own Fig. 18 shows this is not a hypothetical issue without remainder variables. - Also, footnote 5 concedes m=1, even though Fig. 8 shows tuples with 2–3 solutions. That doesn't bias the estimator, but it means Ehat under-estimates total tuple irradiance for those tuples, so the variance proxy (Eq. 25) is not a guaranteed upper bound at scale. The authors are upfront about this, but it is a gap in the central claim. - No code or data is released, and the "equal time" comparisons are against some biased photon methods (SPPM, UPSMCMC) that are not apples-to-apples, though the comparisons against deterministic search and manifold sampling look fair.

Bottom line: if the supplemental proofs hold, this is a solid, publishable methods paper with a real contribution. The main text alone is not fully self-contained at the critical point, so I'd want the supplemental checked carefully. I'd send it to peer review, and I'd like to see the m=1 handling tightened or empirically justified beyond the tested scenes.

[Name]

Referee Report

3 major / 5 minor

Summary. This paper proposes a precomputation-driven method for rendering caustics by stochastically sampling specular triangle tuples. The authors derive conservative bounds on the receiver barycentric coordinates and irradiance of each triangle tuple using Bernstein coefficient ratios of rational function formulations (Eq. (3)), introduce remainder variables to handle square roots and degree reduction, and design per-tuple sampling probabilities P_T = min(gamma Ehat(T), 1) that optimize an upper bound on the estimator second moment. The estimator (Eq. (24)) sums selected tuples weighted by 1/P_T; the paper argues unbiasedness and a variance proxy depending only on Ehat and P_T, and reports equal-time comparisons showing an order-of-magnitude speedup over prior unbiased methods on one- and two-bounce caustic scenes. The manuscript is candid about limitations: visibility and Fresnel terms are ignored, bounds require user-controlled subdivisions, and the number of admissible paths per tuple is set to m=1 in experiments.

Significance. If the conservative-bounding claims hold, the paper fills a genuine gap: it provides a self-contained, error-bounded sampling scheme for specular triangle tuples with a theoretically expressible variance proxy, in contrast to point-sampling and online-learning approaches. The main-text algebra is coherent: Eq. (3) is a known Bernstein bound for rational functions, Eq. (21) consistently multiplies per-path irradiance bounds by a solution count (with m=1 used in experiments), and Eqs. (24)-(28) correctly derive unbiasedness and the variance proxy conditional on Ehat being a true upper bound. The paper also deserves credit for explicitly disclosing the m=1 assumption, the absence of a convergence-rate proof, and the dependence on the supplemental document for the envelopes in the remainder-variable and degree-reduction steps. The central risk is that the load-bearing conservativeness of Ehat is deferred rather than proven in the reviewed artifact; if any envelope understates error or m>1 occurs, the variance-control claim is weakened, though unbiasedness can survive.

major comments (3)

[§4.3, Eq. (12)-(13)] The replacement of sqrt(beta_i) by r(beta_i) plus a remainder variable xi_i, with the linear approximation r(beta)=a beta+b, is stated to have error bounds whose proof and closed forms are deferred to the supplemental document. These envelopes are load-bearing: they guarantee that the position bound, and hence the irradiance bound Ehat used in P_T (Eq. (28)) and in the variance proxy (Eq. (25)), is conservative. If the envelope understates the approximation error, Ehat can fall below the true irradiance for a contributing tuple, P_T can become zero, and the estimator in Eq. (24) becomes biased. The main text should at least state the theorem with explicit hypotheses on the range of beta_i and indicate exactly where the proof appears, rather than deferring the entire justification.
[§6.1, degree reduction] The SVD-based degree reduction converts high-degree Bernstein polynomials (above degree about 40) to low-degree approximants and adds a new remainder variable to maintain bounding validity, but no derivation or error-bound statement is given in the main text; the text also states that each reduction eliminates all existing remainder variables. Because degree reduction is applied to the very quantities whose bounds later drive P_T, the conservativeness of Ehat depends on this omitted analysis. Please provide the explicit error-envelope construction and its proof (or a precise pointer to the relevant supplemental section), and justify that eliminating previous remainder variables cannot invalidate the bound.
[§4.5, Eq. (21) and footnote 5] Equation (21) defines Ehat(T) = m max_i E_i^k with m=1 in experiments, while the true total tuple irradiance E(T) is a sum over all admissible paths and Fig. 8 shows tuples with 2 and 3 solutions. With m=1, Ehat is not an upper bound on E(T). As footnote 5 acknowledges, this does not bias Eq. (24) as long as P_T>0, but it breaks the claimed variance bound in Eq. (25): the proxy sum_T Ehat^2/P_T can understate the true second moment. The abstract and Section 5.3 describe the variance as controllable, so this assumption should either be removed by using a provable upper bound on m, or explicitly carried as a condition on the scenes for which the variance-control claim is made.

minor comments (5)

[Figure 17 caption] The caption says that all ratios are greater than zero, but the surrounding text states that a valid bound requires the ratio Ehat/E to be no smaller than 1; please correct the caption to say that all displayed ratios are at least 1.
[Abstract and §6.1] The abstract says the proposed primitive sampling is intrinsically unbiased, while Section 6.1 also combines it with a deterministic, possibly biased root-finding scheme; please clarify that unbiasedness is a property of the tuple-sampling estimator conditional on the root-finding weighting.
[Table 2] The column header 'Normal (K)' is not defined; it appears to refer to normal interpolation, but the caption should spell this out, and the meaning of 'K' should be stated.
[§1, complexity statements] The phrase 'sublinear to the number of triangles' is used for both precomputation growth and tuple counts; please be precise about which quantity grows sublinearly, since Table 3 reports different growth rates for precomputation time, rendering time, and memory.
[Eq. (11)] Equation (11) would benefit from explicit parentheses around dot products to avoid ambiguity between the groupings (n_i·n_i)(d_{i-1}·d_{i-1}) and (d_{i-1}·n_i)^2.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Bernstein bound construction is analytic and anchored in an external range theorem; self-citations and deferred proofs do not make the central claim circular.

full rationale

The paper's central derivation is non-circular. The conservative range bound (Eq. 3) is an external theorem from Narkawicz et al. (2012), applied to rational expressions for vertex positions and irradiance. The rational coordinate mapping (Eqs. 6-9) and multivariate specular polynomial constraints (Eq. 17) are prior published results, not consequences of the target bounds. The sampling probability (Eq. 28) is an explicit function of the precomputed analytic bound Ehat and a user parameter gamma; it is not fitted to reference images or to the estimated quantity. Unbiasedness of Eq. (24) is the standard condition P_T > 0 for all contributing tuples. The m=1 assumption in Eq. (21) is disclosed in footnote 5 as a possible source of extra variance, and Fig. 8 validates it independently of rendering results. The only deferred material (error envelopes for remainder variables and degree reduction) is a proof obligation, not a circular step: those envelopes are auxiliary bounds, not the target irradiance or position bounds. Self-citations to Fan et al. (2024) and Wang et al. (2020) provide formulas and a finiteness assertion, but the load-bearing bounding argument rests on external mathematics and deterministic derivation, so the paper does not reduce to its own inputs.

Assumptions & free parameters 6 free parameters · 8 assumptions · 2 invented entities

The central claim rests on an external range-bounding theorem (Narkawicz et al. 2012), prior-work rational mappings and specular constraints (Fan et al. 2024), paper-specific constructions whose validity is deferred to a supplemental (linear sqrt fit in Eq. 13 and SVD degree reduction), the empirical single-solution assumption m = 1, and the Fresnel and visibility at most 1 simplification. No new physical entities are introduced; the remainder variables are auxiliary coordinates whose sole job is to keep the derived bounds conservative. None of the listed items includes the target result, so nothing forces the conclusion by construction.

free parameters (6)

gamma (sampling parameter) = user-set per scene; 3, 30, 300 in Fig. 11
Sets P_T = min(gamma * Ehat(T), 1) and thereby the expected number of samples W; controls the variance versus render-time tradeoff (Sec. 5.4, Fig. 11).
sigma (position-bound area threshold) = 10^-4
Stops subdivision when the position-bound area is below sigma; Fig. 19 shows results for sigma from 10^-2 to 10^-6 (Sec. 6.1).
alpha (irradiance ratio threshold) = 2 for single scattering, 10 for multiple
Stops subdivision when the irradiance bound ratio Ebar/E is below alpha; scene-dependent by chain type (Sec. 6.1).
maximal subdivision depth = varies per scene (20, 12, 1, 3, 2 in Table 2)
Per-scene depth limit for domain subdivision; balances bound tightness against precomputation cost (Sec. 6.1, Table 2).
grid resolution for bound storage = 512 x 512
Uniform grid chosen for splatting irradiance bounds onto receivers; resolution affects precomputation and query cost (Sec. 6.1).
SVD degree-reduction coefficients = degree threshold near 40
High-degree polynomials are replaced by low-degree approximants fitted by linear regression (SVD); validity restored by added remainder variables (Sec. 6.1).

assumptions (8)

standard math The Bernstein coefficient-ratio bound (Eq. 3) conservatively bounds rational functions when all denominator coefficients share one sign.
External theorem cited to Narkawicz et al. 2012 and Garloff et al. 2012; the entire position and irradiance bounding machinery (Secs. 4.2 to 4.4) rests on it. Not proved inside the paper.
domain assumption The rational coordinate mapping (Eqs. 6 to 9) and the specular constraint polynomials F and G (Eq. 17) correctly encode specular chains.
Inherited from Specular Polynomials (Fan et al. 2024); Sec. 4.3 states 'refer to the original paper for derivation'. Prior-work grounding, not an assumption of the target result.
ad hoc to paper The on-the-fly linear approximation r(beta) = a*beta + b with error envelope (delta, deltabar) encloses sqrt(beta) on the current domain.
Introduced in Sec. 4.3 (Eqs. 12 and 13); the closed form for a, b, delta, deltabar and the proof are deferred to the supplemental document. If the envelope fails, the position bound is not conservative.
ad hoc to paper The SVD degree-reduction approximants (threshold near degree 40) plus new remainder variables preserve bound validity.
Sec. 6.1: high-degree polynomials are replaced by low-degree fits and a new remainder variable is added to maintain bounding validity; details are in the supplemental.
domain assumption At most one admissible specular path exists per triangle tuple for small triangles (m = 1).
Sec. 4.5 and footnote 5: Eq. (21) sets m = 1. Empirically supported in tested scenes (Fig. 8: 99.99 percent of tuples have 0 or 1 solutions), but not proven; the paper admits variance may exceed expected levels if violated.
domain assumption Fresnel terms and visibility are no greater than 1, so dropping them preserves the irradiance upper bound.
Sec. 4.4: 'we ignore F and V in the discussion since they are no greater than 1. Thus, the upper bound of irradiance is still correct.'
standard math Bernstein bounds tighten under subdivision (quadratic convergence for polynomials).
Cited to Garloff 1986; motivates the piecewise-constant subdivision scheme in Secs. 4.1 and 6.1. The convergence rate for the paper's rational bounds is stated as not guaranteed (Sec. 7).
domain assumption The effective u1 domain initialization captures all u1 that map to u_k in the receiver triangle U^2.
Sec. 6.1: implemented by recursive subdivision 'of at most 100 pieces'; tuples outside this computed domain are not bounded, so completeness depends on this step.

invented entities (2)

Remainder variable xi_i per refractive vertex
purpose: Absorbs the difference between sqrt(beta_i) and the rational linear approximation r(beta_i) so Bernstein bounds remain conservative for refraction (Eq. 12).
An auxiliary mathematical coordinate over [0,1], not a physical quantity; its validity rests on the error envelope (delta, deltabar), the proof of which is deferred to the supplemental document.
Remainder variables attached after SVD degree reduction
purpose: Restore bounding validity after replacing high-degree polynomials with low-degree regression fits (Sec. 6.1).
Same device as the vertex remainder variable; details of the error control are deferred to the supplemental document.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Bernstein Bounds for Caustics." pith.science (2026). https://pith.science/paper/GNFQ62B4

@misc{pith2026250419163,
  author       = {Pith},
  title        = {Pith review of: Bernstein Bounds for Caustics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GNFQ62B4}},
  note         = {Machine review of arXiv:2504.19163}
}

read the original abstract

Systematically simulating specular light transport requires an exhaustive search for primitive tuples containing admissible paths. Given the extreme inefficiency of enumerating all combinations, we propose to significantly reduce the search domain by sampling such tuples. The challenge is to design proper sampling probabilities that keep the noise level controllable. Our key insight is that by bounding the range of irradiance contributed by each primitive tuple at a given position, we can sample a subset of primitive tuples with potentially high contributions. Although low-contribution tuples are assigned a negligible probability, the overall variance remains low. Therefore, we derive vertex position and irradiance bounds for each primitive tuple, introducing a bounding property of rational functions on the Bernstein basis. When formulating position and irradiance expressions into rational functions, we handle non-rational components through remainder variables to maintain validity. Finally, we carefully design the sampling probabilities by optimizing the upper bound of the variance, expressed only using the position and irradiance bound. The proposed primitive sampling is intrinsically unbiased. It can be seamlessly combined with various unbiased and biased root-finding techniques within a local primitive domain. Extensive evaluations show that our method enables fast and reliable rendering of complex caustic effects.

Figures

Figures reproduced from arXiv: 2504.19163 by the authors.

**Figure 1.** Rendering sharp caustics reflected by complex geometry (0.35M triangles), where existing methods perform slowly. Consequently, even if deterministically [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** Overview of the precomputation pass. We trace light beams passing through each triangle tuple and leverage their position and irradiance bounds to sample these triangle tuples. Note that our discussion of irradiance is primarily about its upper bound only. high variance. Although it is possible to eventually find all solutions, the variance could be arbitrarily high [Fan et al. 2023]. Instead, we adopt stochastic s… view at source ↗

**Figure 5.** With remainder variables, we correctly bound the range of √︁ 𝛽, though looser than the proper range (left). Note that the actual range of 𝛽 is smaller, so the bound could be tighter (right). [Möller and Trumbore 1997]: 𝒖𝑖+1 = [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Visualizing the bound of caustics cast by a single triangle reflector. We show the bound and its ratio with reference in the base 10 logarithmic space. Note that the axes are barycentric coordinates 𝑢𝑘 and 𝑣𝑘 on the receiver. Thus, the position bounds of different piec…

**Figure 6.** Figure 6: Illustration of the generalized geometric term (GGT). Light emitted in the differential solid angle dΩ0 passing through specular surfaces finally hits the differential area d𝐴𝑘 on the receiver. Note that we use dΩ0 instead of the projected solid angle dΩ ⊥ 0 because we…

**Figure 8.** Figure 8: The number of solutions per tuple. In the Plane scene, nearly all tuples have no more than 1 solutions, so we set 𝑚 ≤ 1 for rendering experiments. Left: the maximal number of solutions per tuple for each shading point. Right: distributions of the number of solutions pe…

**Figure 9.** Figure 9: 2D examples. (a) The incident light (yellow) hits a specular triangle (orange, with interpolated normals). Reflected rays (blue) hit the receiver (black, bottom). (b) We show reference as solid curves and our bound as a shaded region. We perform subdivision on the doma…

**Figure 11.** Figure 11: The impact of the sampling parameter 𝛾. All methods use 4 spp for path tracing. Higher gamma reduces variance by increasing sampling probability. Notably, the probability easily reaches 1 (thus no variance) for high-energy tuples, while noise still persists for low-en…

**Figure 12.** Figure 12: Handling area light sources. We introduce an extra variable 𝜁 to represent the position along the line light source. The rational functions have two variables 𝑢1 and 𝜁 . We succeeded in bounding the irradiance distribution of all points on the area light source. The c…

**Figure 13.** Figure 13: Equal-time comparisons on single scattering. Precomputation takes 2.1 sec and 2.4 sec, respectively. We visualize irradiance bounds (in the base 10 logarithmic space) summed over tuples. We compare with Specular Polynomials (SP) [Fan et al. 2024], Manifold Path Guidin…

**Figure 14.** Figure 14: Equal-time comparisons on double scattering. Precomputation time is included, which takes 23 sec and 25 sec, respectively. We combine our triangle sampling with deterministic (Det, biased) and stochastic (Stoc, unbiased) initialization for Newton’s iteration-based roo…

**Figure 15.** Figure 15: Equal-time (30 sec) comparisons on handling area light sources. We compare our method with path tracing (PT), bidirectional path tracing (BDPT), Practical Path Guiding (PPG), and Manifold Exploration Metropolis light transport (MEMLT) in a Plane scene lit by an area …

**Figure 17.** Figure 17: Visualization of the ratio between the bound and the true irradiance for each solution in the Plane and Slab scenes. The image illustrates the overall situation, where the ratio is averaged per pixel. The accompanying histogram represents the ratios for each solution…

**Figure 18.** Figure 18: Without remainder variables, there exists some solutions whose bound 𝐸˜ is slightly lower than the true irradiance 𝐸 (red). By using remainder variables, all solutions are properly bounded (blue). The ratio 𝐸˜/𝐸 is displayed on a logarithmic scale with base 10. Inter…

**Figure 20.** Figure 20: The advantage of our method is not significant in cases already well [PITH_FULL_IMAGE:figures/full_fig_p014_20.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[2]

Manifold Path Guiding for Importance Sampling Specular Chains.ACM Trans. Graph. 42, 6, Article 257 (Dec 2023), 14 pages. Rida T. Farouki. 2012. The Bernstein polynomial basis: A centennial retrospective. Comput. Aided Geom. Des. 29 (2012), 379–419. J. Garloff. 1986. Convergent bounds for the range of multivariate polynomials. In Interval Mathematics 1985,...

work page 2012
[2023]

In ACM SIGGRAPH 2023 Courses (Los Angeles, California) (SIGGRAPH ’23)

Path tracing in Production: The Path of Water. In ACM SIGGRAPH 2023 Courses (Los Angeles, California) (SIGGRAPH ’23). Association for Computing Machinery, New York, NY, USA, Article 12, 66 pages. Zhimin Fan, Jie Guo, Yiming Wang, Tianyu Xiao, Hao Zhang, Chenxi Zhou, Zhenyu Chen, Pengpei Hong, Yanwen Guo, and Ling-Qi Yan. 2024. Specular Polynomials. ACM Tr...

work page 2023

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [2]

Manifold Path Guiding for Importance Sampling Specular Chains.ACM Trans. Graph. 42, 6, Article 257 (Dec 2023), 14 pages. Rida T. Farouki. 2012. The Bernstein polynomial basis: A centennial retrospective. Comput. Aided Geom. Des. 29 (2012), 379–419. J. Garloff. 1986. Convergent bounds for the range of multivariate polynomials. In Interval Mathematics 1985,...

work page 2012

[2] [2023]

In ACM SIGGRAPH 2023 Courses (Los Angeles, California) (SIGGRAPH ’23)

Path tracing in Production: The Path of Water. In ACM SIGGRAPH 2023 Courses (Los Angeles, California) (SIGGRAPH ’23). Association for Computing Machinery, New York, NY, USA, Article 12, 66 pages. Zhimin Fan, Jie Guo, Yiming Wang, Tianyu Xiao, Hao Zhang, Chenxi Zhou, Zhenyu Chen, Pengpei Hong, Yanwen Guo, and Ling-Qi Yan. 2024. Specular Polynomials. ACM Tr...

work page 2023