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REVIEW 3 major objections 5 minor 45 references

Gaussian Splatting with Discretized SDF for Relightable Assets

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that encoding the signed distance field as a per-Gaussian scalar, linked to opacity through an SDF-to-opacity transformation, improves relighting and decomposition quality over Gaussian-based inverse rendering baselines…

desk verdict The per-Gaussian discretized SDF is a genuinely simpler and memory-efficient design with good empirical results, but the claimed proof that the projection loss approximates the Eikonal equation does not hold up. read the letter →

arxiv 2507.15629 v1 pith:C7HIFHDY submitted 2025-07-21 cs.GR cs.CV

Zuo-Liang Zhu , Jian Yang , Beibei Wang This is my paper · ORCID
classification cs.GRcs.CV
keywords relighting3DGaussiansplattingdiscretizedSDFinverserenderingBRDFdecompositionprojection-basedconsistencylosssurfacereconstructionsigneddistancefield
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

This paper tries to show that a signed distance field (SDF) can be encoded as a single scalar stored on each Gaussian splat, instead of as an extra neural network, and that this discretized SDF is enough to regularize geometry for inverse rendering. The method links each Gaussian's SDF value $s_i$ to its opacity through the SDF-to-opacity transformation $T_\gamma(s_i)=4 e^{-\gamma s_i}/(1+e^{-\gamma s_i})^2$, so the SDF is rendered by ordinary splatting and requires no ray marching. A projection-based consistency loss, which the paper proves approximates the Eikonal loss, pushes each Gaussian's projected point onto the $\alpha$-blended surface. If the paper's claims hold, relightable assets are reconstructed with better geometry and material decomposition than existing Gaussian-based inverse rendering methods, at about one-fifth of the training memory of the best SDF-network baseline, with simpler optimization and competitive training time.

What carries the argument

The load-bearing object is the discretized SDF: a per-Gaussian scalar SDF value that replaces the continuous SDF network used by prior work. A bell-shaped SDF-to-opacity transformation $T_\gamma$ turns that scalar into the Gaussian's opacity, so SDF constraints act directly on the splatting weights. A median loss ties the transformation width $\gamma$ to the median unsigned distance, forcing Gaussians to converge toward a sharp surface. A projection-based consistency loss, proven as an approximation of the Eikonal loss, projects each Gaussian onto the zero-level set along its normal and compares the projected depth with the $\alpha$-blended depth, with an $\varepsilon$-threshold discarding outliers. A spherical initialization of foreground Gaussians avoids early local minima that otherwise trap geometry.

What would settle it

Train the method on a scene with a known ground-truth mesh, record each Gaussian's stored SDF value $s_i$ together with its position and normal, and check whether the projection consistency loss actually drives the Eikonal residual $|\nabla f|-1$ toward zero inside the thin shell; alternatively, ablate the $\varepsilon$-threshold and observe whether relighting quality degrades sharply, which would show the loss is discarding rather than correcting the outliers it depends on.

Watch

Extended reading notes

Core claim

The central claim is that the discrete nature of 3D Gaussian primitives is not an obstacle to SDF regularization if the SDF itself is discretized onto the Gaussians. Each Gaussian carries an SDF value $s_i$, and its opacity is derived as $o_i = T_\gamma(s_i) = 4 e^{-\gamma s_i}/(1+e^{-\gamma s_i})^2$, making the SDF renderable through standard $\alpha$ blending. Because gradient-based Eikonal losses are unavailable for discrete samples, the paper projects each Gaussian onto the zero-level set via $\mu_{\mathrm{proj}} = \mu_i - s_i \nabla f_i / |\nabla f_i|$ and aligns the projected depth with the $\alpha$-blended aggregated depth, discarding outliers whose difference exceeds a threshold $\varepsilon$. The paper argues this projection loss approximates the Eikonal condition $|\nabla f|=1$ in a thin shell around the surface, and reports that the resulting method outperforms existing Gaussian-based inverse rendering methods on relighting and decomposition metrics across Glossy Blender, TensoIR, Shiny Blender, and real NeILF++ scenes, while using 4G memory versus 22G for the SDF-network baseline.

Load-bearing premise

Each Gaussian's surface normal points along the true gradient of the underlying signed distance field, so the projection $\mu_i - s_i \nabla f_i / |\nabla f_i|$ lands on the real surface and its depth can be trusted as supervision.

Editorial extensions

If this is right

  • Relightable Gaussian assets can be trained without an auxiliary SDF network, removing warm-up or multi-stage schedules that earlier SDF-based Gaussian methods required.
  • Training memory drops by roughly a factor of five compared with the SDF-network baseline, enabling higher-resolution scenes or more primitive counts on the same GPU.
  • The projection-based depth alignment provides a substitute for Eikonal regularization that could apply to other discrete surface representations beyond Gaussian splatting.
  • Improved geometry decomposition transfers directly to relighting quality, including smoother normals on glossy objects and preserved details such as a horse's tail.
  • Real-time rendering is retained, with the reported pipeline running at 143 FPS while using less than 4G of GPU memory during training.

Reading between the lines

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

  • Editorial inference: the proof that depth alignment approximates the Eikonal condition holds only where the Taylor remainder is negligible, so the loss should in theory weaken for Gaussians that drift far from the surface; a testable extension is to anneal the threshold $\varepsilon$ with training progress.
  • Editorial inference: because the projection loss relies on the Gaussian normal as an estimate of the SDF gradient, scenes with strong specularities or heavy occlusion could initially reinforce incorrect normals; an adaptive weighting based on surface visibility might mitigate the reported failures on inter-reflective scenes such as 'Toaster'.
  • Editorial inference: a discretized SDF value per Gaussian could also serve downstream tasks beyond relighting, such as collision detection or mesh extraction, since the SDF no longer lives in a separate network that must be queried independently.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a relightable 3D Gaussian splatting framework based on a 'discretized SDF': each Gaussian stores a scalar signed-distance value s_i, the Gaussian opacity is derived from s_i through an SDF-to-opacity transformation, and geometry is regularized by a projection-based consistency loss that aligns a projected point with the alpha-blended depth. A median loss and a spherical initialization are added to improve convergence. Experiments on Glossy Blender, TensoIR, Shiny Blender, and NeILF++ report higher relighting and geometry quality than GShader, GS-IR, R3DG, and GS-ROR, with lower memory and comparable training time.

Significance. The empirical contribution is practically significant: the method reports improved relighting metrics on standard benchmarks, while using only 4G GPU memory versus 22G for GS-ROR and avoiding an extra SDF network. The availability of code and comparisons against public baselines strengthen the empirical claim. However, the paper's central theoretical assertion, that the projection-based loss is 'proven as an approximation of the Eikonal loss,' is not established by the presented proof, and this gap affects the interpretation of the method as an SDF regularizer rather than a depth-normal consistency heuristic.

major comments (3)
  1. [Supp. Prop. 1 and Sec. 3.3, Eqs. (5)-(6)] The proof of Proposition 1 assumes that the projected point x0 lies on the zero-level set, f(x0)=0, and that the Gaussian normal t_u x t_v equals the normalized SDF gradient. In the actual loss, Eq. (6) minimizes |D_agg - D_proj|, a scalar depth difference, not f(mu_proj)=0; two points on the same ray can have the same depth without the projected point being on the surface. The paper itself states in Fig. 4 that the discretized SDF cannot query f at the projected point and therefore uses depth 'as a bridge.' Thus the key premise of the proof is precisely the condition that the loss does not enforce. Furthermore, the Taylor expansion drops O(s_i^2) and concludes |grad f| ~ 1 only as s_i -> 0, whereas training operates at finite values (median |s|_m < 0.2). This does not justify the claim that L_p approximates the Eikonal condition.
  2. [Sec. 3.3, Eq. (5)] The loss relies on the learned normal n_i = t_u x t_v being the gradient direction of an underlying SDF. This normal is obtained from appearance-driven Gaussian attributes and is not constrained to be a gradient of any SDF. Early in training, or in specular and occluded regions, an incorrect normal makes mu_proj = mu_i - s_i n_i point away from the true surface; minimizing the depth difference then reinforces that incorrect geometry instead of correcting it. The epsilon-threshold discards large residuals rather than using them to correct systematic normal error, and applying the loss only after 1K iterations means that an initially wrong normal can persist. The authors should either add an explicit alignment term between Gaussian normals and SDF gradients or temper the claim that this loss regularizes an SDF.
  3. [Supp. Sec. 1.2 and Supp. Table 5] The threshold epsilon, which is central to the projection-based loss, is reported inconsistently: Sec. 1.2 states epsilon = 0.05, while Table 5 uses epsilon = 0.1 as the chosen value and shows large sensitivity (Horse PSNR ranges from 20.12 to 24.01 across epsilon values). The paper should state the exact configuration used for all reported results and discuss how epsilon was selected, since the current description is ambiguous and the loss is highly sensitive to this hyperparameter.
minor comments (5)
  1. [Abstract and Sec. 1] There are minor language issues, including 'an discretized SDF' in the abstract and 'Chamber distance' for 'Chamfer distance' in Sec. 4.2 and Table 4.
  2. [Sec. 3.3 and Supp. Sec. 1.2] The text refers to the 'median of unsigned distance |s|_m' in Sec. 3.2 but later uses 'SDF median s_m < 0.2' in the supplementary; please use consistent notation for the median SDF magnitude.
  3. [Fig. 4 caption and footnote] The statement 'only the gradient direction on Gaussian is available' is unclear: if |grad f| is not one during optimization, one needs a precise definition of how the 2D Gaussian normal is identified with the normalized SDF gradient. Please clarify this relationship.
  4. [Table 1] The entries in Table 1 are run together and difficult to read; please format the PSNR/SSIM/LPIPS values as separate columns or with clear spacing.
  5. [Sec. 4.1 and Supp. Sec. 1.2] The loss weights in Table 1 of the supplementary are written as '0 .05' and '0 .2'; please fix the spacing for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the projection-based consistency loss is a conditional approximation of the Eikonal condition, and the paper's claims are validated against external benchmarks and independent baselines.

full rationale

The paper's central claim is an empirical performance comparison against external Gaussian and NeRF baselines on standard datasets, so the outcome is not defined by the inputs. The discretized SDF is a parameterization choice, not a result derived from its own conclusion. The projection-based consistency loss in Eq. (6) penalizes depth disagreement between the alpha-blended depth and the depth of a projected Gaussian point, while the supplementary Proposition 1 derives an approximate Eikonal condition from the Taylor expansion under the explicit hypothesis that the projected point lies on the zero-level set. This is a conditional mathematical argument, not a definitional reduction: the loss is not defined in terms of the Eikonal conclusion, and the proof does not assume the result it claims. The practical limitation that depth alignment is only a proxy for enforcing f(mu_proj)=0, and that the Gaussian normal is learned rather than guaranteed to be the SDF gradient, is an approximation or correctness gap, not circularity. Self-citations such as GS-ROR [44] and TensoSDF [14] are used as baselines and related work, not as load-bearing justification of the method's core premise, and no uniqueness theorem is imported from prior work. No fitted parameter is renamed as a prediction, and no known result is merely relabeled. Accordingly, no circular step meeting the required evidentiary standard was found.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The central claim relies on the discretized SDF being a valid proxy for a continuous SDF, which is not proven; several hyperparameters are hand-set; the projection loss's theoretical justification is conditional.

free parameters (4)
  • Projection loss threshold ε = 0.1
    Ablated on Glossy Blender scenes (Supp. Tab. 5) and set to 0.1 for all scenes; performance on the evaluation set is used to choose it.
  • Median opacity target o_m for γ_m = 0.5
    Chosen empirically (Supp. Sec. 3.1) to define γ_m from |s|_m in Eq. 3.
  • Loss weights λ_n, λ_d, λ_γ, λ_p, λ_sm, λ_m = 0.2, 2000, 1, 10, 0.05, 0.2
    Hand-tuned in Supp. Tab. 1; no sensitivity analysis reported.
  • Number of initial Gaussians N for spherical initialization = 10^6
    Chosen for spherical initialization; no ablation reported.
assumptions (4)
  • domain assumption The Gaussian normal n = t_u × t_v is the direction of the SDF gradient ∇f/|∇f|.
    Invoked in Sec. 3.3 and Fig. 4 to justify using the normal for projection; if false, the projection-based loss is misdirected.
  • domain assumption The alpha-blended depth D_agg obtained by splatting defines a surface that should coincide with the zero-level set of the SDF.
    Used in Sec. 3.3 to define the consistency target for the projected point.
  • domain assumption The signed distance encoded per Gaussian behaves locally like a smooth signed distance function, so that Taylor expansion and the Eikonal condition apply.
    The proof of Prop. 1 in the supplementary assumes a differentiable f and uses a Taylor expansion; the learned per-Gaussian scalars are not constrained to be differentiable in space.
  • standard math Standard calculus facts, such as Taylor expansion and vanishing higher-order terms near the surface, are valid.
    Used in the supplementary proof of the projection loss.
invented entities (1)
  • Discretized SDF: a per-Gaussian scalar s_i representing signed distance. independent evidence
    purpose: Regularize Gaussian geometry for inverse rendering without an extra continuous SDF representation.
    The representation's benefit is evaluated on public benchmarks (relighting PSNR, normal MAE, Chamfer distance), providing an external falsifiable handle; however, it is a data structure rather than a physical entity.

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Cite this review

Pith. "Pith review of Gaussian Splatting with Discretized SDF for Relightable Assets." pith.science (2026). https://pith.science/paper/C7HIFHDY

@misc{pith2026250715629,
  author       = {Pith},
  title        = {Pith review of: Gaussian Splatting with Discretized SDF for Relightable Assets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C7HIFHDY}},
  note         = {Machine review of arXiv:2507.15629}
}
read the original abstract

3D Gaussian splatting (3DGS) has shown its detailed expressive ability and highly efficient rendering speed in the novel view synthesis (NVS) task. The application to inverse rendering still faces several challenges, as the discrete nature of Gaussian primitives makes it difficult to apply geometry constraints. Recent works introduce the signed distance field (SDF) as an extra continuous representation to regularize the geometry defined by Gaussian primitives. It improves the decomposition quality, at the cost of increasing memory usage and complicating training. Unlike these works, we introduce a discretized SDF to represent the continuous SDF in a discrete manner by encoding it within each Gaussian using a sampled value. This approach allows us to link the SDF with the Gaussian opacity through an SDF-to-opacity transformation, enabling rendering the SDF via splatting and avoiding the computational cost of ray marching.The key challenge is to regularize the discrete samples to be consistent with the underlying SDF, as the discrete representation can hardly apply the gradient-based constraints (\eg Eikonal loss). For this, we project Gaussians onto the zero-level set of SDF and enforce alignment with the surface from splatting, namely a projection-based consistency loss. Thanks to the discretized SDF, our method achieves higher relighting quality, while requiring no extra memory beyond GS and avoiding complex manually designed optimization. The experiments reveal that our method outperforms existing Gaussian-based inverse rendering methods. Our code is available at https://github.com/NK-CS-ZZL/DiscretizedSDF.

Figures

Figures reproduced from arXiv: 2507.15629 by the authors.

Figure 1
Figure 1. We present a relightable Gaussian splatting framework that introduces a discretized SDF to promote decomposition quality. We [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Overview of our framework. Our model discretizes the underlying continuous SDF into discrete samples and encodes the SDF values of samples on Gaussian primitives. We propose the median loss to facilitate convergence and projection-based consistency loss to regularize the discrete values of SDF samples. Besides, we design a spherical initialization for foreground objects to avoid local minima. and the Gaussian positi… view at source ↗
Figure 5
Figure 5. The projection-based consistency loss minimizes the dif [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: The difference between continuous and discretized SDF [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 6
Figure 6. Figure 6: The comparison of relighting results and normal on the Glossy Blender (the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: The qualitative comparison of relighting results and normal on the NeILF++ dataset. Our method can provide realistic relighting [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: The qualitative comparison of normal on the Shiny Blender dataset. Our method provides globally smooth normal while [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: Ablation of the key components in our method on the Glossy Blender dataset, including the discretized SDF framework with the median loss, the projection-based consistency loss (Proj.), and spherical initialization (Sph.). (The red arrow points out the artifacts.) Compo…
Figure 1
Figure 1. Figure 1: Without the projection-based consistency loss, we ob [PITH_FULL_IMAGE:figures/full_fig_p013_1.png]
Figure 2
Figure 2. Figure 2: Decomposed maps of our method on the Glossy Blender dataset. Our method can provide a reasonable decomposition for [PITH_FULL_IMAGE:figures/full_fig_p015_2.png]
Figure 3
Figure 3. Figure 3: The qualitative comparison with Gaussian-based methods in terms of relighting results and normal on the Glossy Blender dataset. [PITH_FULL_IMAGE:figures/full_fig_p016_3.png]
Figure 4
Figure 4. Figure 4: The qualitative comparison with NeRF-based methods in terms of relighting results and normal on the Glossy Blender dataset. [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: Decomposed maps of our method on the Shiny Blender dataset. The relighting ground truth is unavailable on the dataset, so we [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: The qualitative comparison with Gaussian-based methods in terms of normal on the Shiny Blender dataset. The MAE under the [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Decomposed maps of our method on the TensoIR synthetic dataset. Our method can provide a reasonable decomposition for [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: The qualitative comparison with Gaussian-based methods in terms of relighting results and normal on the TensoIR synthetic [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: The qualitative comparison with NeRF-based methods in terms of relighting results and normal on the TensoIR synthetic dataset. [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: The qualitative comparison in terms of relighting results and normal on the NeILF++ dataset. Our method ensures the smooth [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.