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arxiv: 2606.20856 · v2 · pith:ZETGTPTDnew · submitted 2026-06-18 · 💻 cs.CV · cs.GR· cs.LG

Stochastic Signed Distance Processes

Hiroki Sakuma , Masatoshi Okutomi This is my paper

Pith reviewed 2026-06-26 18:02 UTC · model grok-4.3

classification 💻 cs.CV cs.GRcs.LG
keywords stochastic signed distance processesfirst-passage-time distributionbayesian filteringsigned distance fieldsurface reconstructionvolume renderingmulti-view reconstructionuncertainty quantification
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The pith

Modeling SDF values along rays as a stochastic process induces a first-passage-time distribution that supports probabilistic surface rendering.

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

The paper aims to show that treating the signed distance field along each ray as a stochastic process allows derivation of a first-passage-time distribution for the surface intersection. This leads to a probabilistic formulation of surface rendering where pixel colors arise from a mixture over possible intersection points. Bayesian filtering provides the probability mass for each sampling interval along the ray, with an approximation suitable for efficient computation. NeuS is shown to be a special case within this framework. If valid, the approach would deliver improved surface reconstruction quality along with direct uncertainty estimates from the distribution.

Core claim

Stochastic Signed Distance Processes model the SDF along each ray as a stochastic process that induces a first-passage-time distribution. The first-passage probability for each sampling interval is derived using Bayesian filtering and a practical approximation for parallel rendering, leading to better performance than baselines in surface reconstruction and uncertainty quantification on the DTU and MobileBrick datasets.

What carries the argument

Stochastic Signed Distance Processes (SSDP), which represent the signed distance function along a ray as a stochastic process to generate a first-passage-time distribution for the ray-surface intersection point.

If this is right

  • The first-passage probability per interval follows from Bayesian filtering on the stochastic process.
  • NeuS volume rendering is recovered as a special case of the stochastic formulation.
  • Surface reconstruction accuracy exceeds that of deterministic baselines on DTU and MobileBrick.
  • Uncertainty in the reconstructed surfaces is quantified directly from the first-passage distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stochastic process view could extend to other ray-based rendering tasks beyond surface reconstruction.
  • Approximations in the Bayesian filtering step may trade off exactness for speed in ways that affect uncertainty calibration on unseen data.
  • Integration with learned priors on the stochastic process parameters might further reduce reliance on photometric loss alone.

Load-bearing premise

The signed distance values along a ray can be modeled as samples from a stochastic process whose first-passage distribution allows a tractable Bayesian filtering approximation.

What would settle it

A direct comparison showing that the first-passage probability approximation produces higher reconstruction error or poorer uncertainty calibration than standard deterministic SDF rendering on the DTU dataset would falsify the advantage.

Figures

Figures reproduced from arXiv: 2606.20856 by Hiroki Sakuma, Masatoshi Okutomi.

Figure 1
Figure 1. Figure 1: Overview of our method. We model the SDF along each ray as a stochastic process [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the downward zero-crossing [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualization examples on the DTU dataset. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization examples on the MobileBrick dataset. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Network architecture of our method. The OU network [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trade-off between CD/CRPS and training time on the DTU dataset. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visualization examples on the DTU dataset. [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visualization examples on the DTU dataset. [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visualization examples on the DTU dataset. [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Visualization examples on the MobileBrick dataset. [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Visualization examples on the MobileBrick dataset. [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Visualization examples on the MobileBrick dataset. [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
read the original abstract

Multi-view surface reconstruction is a core problem in computer vision. One prominent line of work represents the surface implicitly as a signed distance field (SDF), optimizing it based on the photometric loss between rendered and observed pixel colors. These approaches typically employ SDF-based volume rendering to obtain a differentiable relaxation of discontinuous visibility along rays, thereby reducing reliance on silhouette supervision. In this paper, we reformulate SDF-based volume rendering as probabilistic surface rendering, where each pixel color is modeled as a mixture distribution induced by the random first ray-surface intersection. To this end, we introduce Stochastic Signed Distance Processes (SSDP), which model the SDF along each ray as a stochastic process, inducing a first-passage-time distribution for each ray. We then derive the first-passage probability for each sampling interval based on Bayesian filtering, together with its practical approximation for parallel rendering. We further show that NeuS, an existing SDF-based volume rendering method, arises as a special case of our formulation. Experiments on the DTU and MobileBrick datasets demonstrate that our method outperforms baselines in both surface reconstruction and uncertainty quantification, supporting the effectiveness of our first-passage formulation. Our code is available at https://github.com/skmhrk1209/SSDP.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper introduces Stochastic Signed Distance Processes (SSDP) to reformulate SDF-based volume rendering in multi-view surface reconstruction as probabilistic surface rendering. SDF values along each ray are modeled as a stochastic process inducing a first-passage-time distribution; first-passage probabilities per sampling interval are derived via Bayesian filtering together with a practical approximation enabling parallel rendering. NeuS is recovered as a special case. Experiments on DTU and MobileBrick report improved surface reconstruction and uncertainty quantification over baselines, with code released.

Significance. If the approximation is shown to be faithful, the work supplies a principled probabilistic generalization of existing deterministic volume rendering methods, with potential benefits for calibrated uncertainty in implicit representations. Open-sourcing the code is a positive contribution to reproducibility.

major comments (2)
  1. [Method (Bayesian filtering and practical approximation)] The practical approximation to the Bayesian-filtering derivation of interval first-passage probabilities (described in the method section following the stochastic process definition) is load-bearing for the central claim. If the truncation or discretization correlates with ray depth or surface curvature, the induced occupancy probabilities can bias the photometric loss and the extracted zero level set; the manuscript should supply either an error bound or an ablation quantifying this effect on the reported DTU/MobileBrick gains.
  2. [Experiments] Table or figure reporting quantitative results on DTU and MobileBrick: the claimed outperformance in surface reconstruction and uncertainty quantification must be accompanied by error bars or multiple-run statistics to establish that the probabilistic formulation, rather than implementation details, drives the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and describe the revisions we will undertake.

read point-by-point responses

  1. Referee: [Method (Bayesian filtering and practical approximation)] The practical approximation to the Bayesian-filtering derivation of interval first-passage probabilities (described in the method section following the stochastic process definition) is load-bearing for the central claim. If the truncation or discretization correlates with ray depth or surface curvature, the induced occupancy probabilities can bias the photometric loss and the extracted zero level set; the manuscript should supply either an error bound or an ablation quantifying this effect on the reported DTU/MobileBrick gains.

    Authors: We agree that the faithfulness of the practical approximation is central to the claims and merits explicit validation. The approximation is derived to preserve the essential properties of the Bayesian filtering result while enabling parallel evaluation. In the revised manuscript we will add an ablation that compares the approximate interval probabilities against a higher-fidelity numerical reference on representative DTU and MobileBrick scenes, reporting the resulting differences in photometric loss and surface metrics. This will directly quantify any systematic effects related to depth or curvature. revision: yes

  2. Referee: [Experiments] Table or figure reporting quantitative results on DTU and MobileBrick: the claimed outperformance in surface reconstruction and uncertainty quantification must be accompanied by error bars or multiple-run statistics to establish that the probabilistic formulation, rather than implementation details, drives the improvement.

    Authors: We concur that statistical characterization of the reported gains is necessary. The current tables reflect single training runs, which were chosen for computational practicality. In the revision we will repeat the full experimental protocol (our method and all baselines) across multiple random seeds on both DTU and MobileBrick, and will augment the tables with mean and standard-deviation values for the key reconstruction and uncertainty metrics. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no load-bearing reduction to self-citation or fitted inputs.

full rationale

The paper introduces SSDP as a new stochastic-process model for SDF along rays, derives first-passage probabilities via standard Bayesian filtering, supplies a practical approximation for rendering, and recovers NeuS as a special case. No equations or claims reduce by construction to author-fitted quantities or prior self-citations; the central probabilistic formulation stands on its own stated assumptions and is externally falsifiable via the reported DTU/MobileBrick experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields limited visibility into free parameters or invented entities; the core modeling choice is treated as a domain assumption.

axioms (1)
  • domain assumption The signed distance function along a ray can be modeled as a stochastic process whose first-passage distribution is derivable by Bayesian filtering.
    This premise is required for the entire probabilistic rendering reformulation.

pith-pipeline@v0.9.1-grok · 5743 in / 1192 out tokens · 24031 ms · 2026-06-26T18:02:09.827280+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

27 extracted references

  1. [1]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Deepsdf: Learning continuous signed distance functions for shape representation , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

  2. [2]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Occupancy networks: Learning 3d reconstruction in function space , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

  3. [3]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Dist: Rendering deep implicit signed distance function with differentiable sphere tracing , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

  4. [4]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Sdfdiff: Differentiable rendering of signed distance fields for 3d shape optimization , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

  5. [5]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Differentiable volumetric rendering: Learning implicit 3d representations without 3d supervision , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

  6. [6]

    Advances in Neural Information Processing Systems , volume=

    Multiview neural surface reconstruction by disentangling geometry and appearance , author=. Advances in Neural Information Processing Systems , volume=

  7. [7]

    European Conference on Computer Vision , pages=

    NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis , author=. European Conference on Computer Vision , pages=. 2020 , organization=

    2020

  8. [8]

    Advances in Neural Information Processing Systems , volume=

    NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction , author=. Advances in Neural Information Processing Systems , volume=

  9. [9]

    Advances in neural information processing systems , volume=

    Volume rendering of neural implicit surfaces , author=. Advances in neural information processing systems , volume=

  10. [10]

    Proceedings of the IEEE/CVF international conference on computer vision , pages=

    Unisurf: Unifying neural implicit surfaces and radiance fields for multi-view reconstruction , author=. Proceedings of the IEEE/CVF international conference on computer vision , pages=

  11. [11]

    Proceedings of the IEEE/CVF International Conference on Computer Vision , pages=

    UNIS: A Unified Framework for Achieving Unbiased Neural Implicit Surfaces in Volume Rendering , author=. Proceedings of the IEEE/CVF International Conference on Computer Vision , pages=

  12. [12]

    Journal of computational physics , volume=

    Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations , author=. Journal of computational physics , volume=. 1988 , publisher=

    1988

  13. [13]

    Transactions of the American mathematical society , volume=

    Viscosity solutions of Hamilton-Jacobi equations , author=. Transactions of the American mathematical society , volume=

  14. [14]

    Proceedings of the 37th International Conference on Machine Learning , pages=

    Implicit geometric regularization for learning shapes , author=. Proceedings of the 37th International Conference on Machine Learning , pages=

  15. [15]

    Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages=

    Objects as volumes: A stochastic geometry view of opaque solids , author=. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages=

  16. [16]

    2013 , publisher=

    Stochastic geometry and its applications , author=. 2013 , publisher=

    2013

  17. [17]

    ACM Transactions on Graphics (TOG) , volume=

    A non-exponential transmittance model for volumetric scene representations , author=. ACM Transactions on Graphics (TOG) , volume=. 2021 , publisher=

    2021

  18. [18]

    Physical review , volume=

    On the theory of the Brownian motion , author=. Physical review , volume=. 1930 , publisher=

    1930

  19. [19]

    Proceedings of the National Academy of Sciences , volume=

    On continuous martingales , author=. Proceedings of the National Academy of Sciences , volume=

  20. [20]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Mip-nerf 360: Unbounded anti-aliased neural radiance fields , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

  21. [21]

    ACM transactions on graphics (TOG) , volume=

    Instant neural graphics primitives with a multiresolution hash encoding , author=. ACM transactions on graphics (TOG) , volume=. 2022 , publisher=

    2022

  22. [22]

    Seminal graphics: pioneering efforts that shaped the field , pages=

    Marching cubes: A high resolution 3D surface construction algorithm , author=. Seminal graphics: pioneering efforts that shaped the field , pages=

  23. [23]

    Proceedings of the fourth Eurographics symposium on Geometry processing , volume=

    Poisson surface reconstruction , author=. Proceedings of the fourth Eurographics symposium on Geometry processing , volume=

  24. [24]

    ACM SIGGRAPH 2023 conference proceedings , pages=

    Nerfstudio: A modular framework for neural radiance field development , author=. ACM SIGGRAPH 2023 conference proceedings , pages=

    2023

  25. [25]

    jit: A just-in-time compiler for differentiable rendering , author=

    Dr. jit: A just-in-time compiler for differentiable rendering , author=. ACM Transactions on Graphics (TOG) , volume=. 2022 , publisher=

    2022

  26. [26]

    2014 IEEE Conference on Computer Vision and Pattern Recognition , pages=

    Large scale multi-view stereopsis evaluation , author=. 2014 IEEE Conference on Computer Vision and Pattern Recognition , pages=. 2014 , organization=

    2014

  27. [27]

    Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages=

    Mobilebrick: Building lego for 3d reconstruction on mobile devices , author=. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages=