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arxiv: 2604.19568 · v2 · submitted 2026-04-21 · 💻 cs.GR

Recognition: no theorem link

SpUDD: Superpower Contouring of Unsigned Distance Data

Ningna Wang , Xiana Carrera , Christopher Batty , Oded Stein , Silvia Sell\'an
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:58 UTC · model grok-4.3

classification 💻 cs.GR
keywords unsigned distance functionssurface reconstructionpower diagramssuperpower contourmesh reconstructionimplicit surfacesdiscrete datageometry processing
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The pith

Discrete unsigned distance samples can reconstruct arbitrary surfaces using superpower contours from power diagrams that converge to the true surface.

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

Unsigned distance functions allow representing surfaces that are open, non-orientable, or non-manifold, but reconstructing them from discrete samples is hard because most methods need signed data or gradients. This paper studies the power diagram formed by these distance samples and defines a new concept, the superpower contour. It proves that the superpower contour converges to the actual surface as the density of samples increases. The contour then serves as a starting point for an algorithm that generates a polygonal mesh close to the unknown surface. This strategy works better than other approaches one might try for this specific reconstruction problem.

Core claim

We study the power diagram generated by the distance samples and propose a novel theoretical concept, the superpower contour, which we prove converges to the true surface in the limit of sampling density. We use this superpower contour as an initial surface proxy and design an algorithm that leverages it to produce a polygonal mesh approximating the unknown true geometry. Our method vastly outperforms other conceivable strategies for the discrete unsigned distance reconstruction task, and sets the stage for future work on this mathematically rich problem.

What carries the argument

The superpower contour, a surface proxy obtained from the power diagram of the unsigned distance samples, proven to converge to the true surface with denser sampling.

If this is right

  • The superpower contour converges to the true surface in the limit of increasing sampling density.
  • The algorithm produces a polygonal mesh that approximates the unknown true geometry.
  • The method handles input that is both discrete and unsigned, succeeding where sign-dependent methods fail.
  • It vastly outperforms other conceivable strategies for discrete unsigned distance reconstruction.

Where Pith is reading between the lines

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

  • This approach could enable surface reconstruction from raw distance measurements in applications like robotics or scanning where orientation is unavailable.
  • The convergence result might be used to develop sampling strategies that adaptively add points to improve the contour accuracy.
  • Similar diagram-based techniques could be investigated for other types of implicit surface data in geometry processing.

Load-bearing premise

The power diagram computed from the discrete unsigned distance samples must produce a superpower contour that converges to the true surface under the paper's sampling conditions.

What would settle it

A counterexample where the superpower contour or the resulting mesh fails to approach the true surface geometry despite arbitrarily high sampling density of the unsigned distance function.

Figures

Figures reproduced from arXiv: 2604.19568 by Christopher Batty, Ningna Wang, Oded Stein, Silvia Sell\'an, Xiana Carrera.

Figure 1
Figure 1. Figure 1: Prior methods focus on reconstructing meshes from either [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Our method (right) manages to reconstruct surfaces from coarse, [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Our method quantitatively (using Chamfer error [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Intuitively, the superpower contour S P is built by filtering out the power faces that intersect any of the distance spheres. This produces a structure that serves as a proxy of the true unknown geometry Ω. treat {(𝑝𝑖 , 𝑑𝑖)} as a set of seeds, and make use of the power distance from any spatial position 𝑥 ∈ R 3 to each of the seeds, traditionally defined as 𝜋𝑖(𝑥) = ∥𝑥 − 𝑝𝑖 ∥ 2 − 𝑑 2 𝑖 . (1) Intuitively, 𝜋𝑖… view at source ↗
Figure 5
Figure 5. Figure 5: Kohlbrenner and Alexa [2025a] introduce the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Like the power contour P, our superpower contour S P converges to the true surface, while maintaining a roughly constant face area ratio. specific case of closed surfaces, SP contains the power contour P defined by Kohlbrenner and Alexa [2025a], as we formalize below. Theorem 2. Given a set of signed distance data (𝑝1, 𝑠1), . . . , (𝑝𝑛, 𝑠𝑛) to a closed surface Ω, their power contour P is a subset of the su… view at source ↗
Figure 8
Figure 8. Figure 8: Our algorithm’s pipeline: from the input unsigned distance data (left), [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Our superpower contour’s function as a proxy for the unknown [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Our optional thinning step removes some of the small, undesirable [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Our method’s computational bottleneck consists of three steps, all [PITH_FULL_IMAGE:figures/full_fig_p007_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Our use of the superpower contour causes our optimization to [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: We avoid the artifacts of prior work at a modest runtime cost. [PITH_FULL_IMAGE:figures/full_fig_p009_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Reach for the Spheres [Sellán et al. 2023] can reconstruct the sim￾plest closed shapes from unsigned distance inputs, but fails for complex geometries and shapes with non-zero genus. 50³ grid 100³ grid PoNQ autoencoding 75³ grid Our reconstruction PoNQ autoencoding PoNQ autoencoding Our reconstructionOur reconstruction OUT OF MEMORY [PITH_FULL_IMAGE:figures/full_fig_p010_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Executing the authors’ provided implementation for open surfaces [PITH_FULL_IMAGE:figures/full_fig_p010_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: By broadening the shapes that can be reconstructed from discrete [PITH_FULL_IMAGE:figures/full_fig_p011_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: , our method can be used to extend discrete CSG trees to open, non-manifold and intersecting surfaces. 2003 SDF grid UDF source SpUDD (ours) Our method can handle closed, open, non-manifold shapes [PITH_FULL_IMAGE:figures/full_fig_p011_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Visual comparison with 5 state-of-the-art methods using [PITH_FULL_IMAGE:figures/full_fig_p012_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Our algorithm is moderately robust to small amounts of noise, but [PITH_FULL_IMAGE:figures/full_fig_p012_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Visual comparison with 5 state-of-the-art methods using [PITH_FULL_IMAGE:figures/full_fig_p013_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Visual comparison with 5 state-of-the-art methods using [PITH_FULL_IMAGE:figures/full_fig_p015_23.png] view at source ↗
read the original abstract

Unsigned distance functions offer a powerful and flexible implicit surface representation that, unlike their signed counterparts, allow for surfaces that are open, non-orientable, or non-manifold. We consider the problem of reconstructing arbitrary surfaces from a finite set of samples of unsigned distance data. Existing methods for mesh reconstruction from distance data rely on sign information, accurate gradients, a corresponding continuous distance function, or extensive data-dependent training. However, they fail when applied to input that is both discrete and unsigned. Inspired by this challenge, we study the power diagram generated by the distance samples and propose a novel theoretical concept, the superpower contour, which we prove converges to the true surface in the limit of sampling density. We use this superpower contour as an initial surface proxy and design an algorithm that leverages it to produce a polygonal mesh approximating the unknown true geometry. Our method vastly outperforms other conceivable strategies for the discrete unsigned distance reconstruction task, and sets the stage for future work on this mathematically rich problem.

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 / 2 minor

Summary. The paper introduces the superpower contour extracted from the power diagram of discrete unsigned distance samples to an arbitrary surface. It claims to prove that this contour converges to the true surface in the limit of increasing sampling density, then leverages the contour as an initial proxy to design an algorithm that outputs a polygonal mesh approximating the unknown geometry. The method is asserted to vastly outperform other conceivable strategies for the discrete unsigned distance reconstruction task.

Significance. If the convergence result holds under the stated sampling conditions, the work fills a notable gap in implicit surface reconstruction by enabling mesh generation from unsigned, discrete distance data without sign information, accurate gradients, or training. The theoretical framing around power diagrams and the empirical outperformance claim could influence future methods for open, non-orientable, or non-manifold surfaces.

major comments (2)
  1. [Abstract (and associated proof section)] The central claim rests on a proof that the superpower contour converges to the true surface. The abstract states this proof exists, but no derivation, error analysis, sampling-density assumptions, or explicit conditions are supplied; without these details the load-bearing theoretical result cannot be verified.
  2. [Abstract (and associated experiments section)] The outperformance claim is presented as empirical, yet the abstract and available description provide no experimental setup, baselines, metrics, or quantitative results; this leaves the practical superiority assertion unsupported.
minor comments (2)
  1. The novel term 'superpower contour' requires an early, self-contained mathematical definition before its use in the convergence argument.
  2. Clarify the precise input model (number of samples, distribution, noise model) under which the power diagram is computed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each major comment below and are prepared to revise the paper accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract (and associated proof section)] The central claim rests on a proof that the superpower contour converges to the true surface. The abstract states this proof exists, but no derivation, error analysis, sampling-density assumptions, or explicit conditions are supplied; without these details the load-bearing theoretical result cannot be verified.

    Authors: The convergence proof, including the full derivation, error bounds, and sampling-density assumptions (requiring samples to be sufficiently dense relative to the local feature size so that the power diagram cells intersect the true surface in a controlled manner), appears in Section 4. We acknowledge that the abstract is overly concise and does not preview these elements, and that the proof section would benefit from an explicit statement of all hypotheses at the outset. We will revise the abstract to include a brief summary of the theorem and conditions, and we will add a dedicated subsection in the proof that enumerates the sampling assumptions and sketches the key steps of the convergence argument. revision: yes

  2. Referee: [Abstract (and associated experiments section)] The outperformance claim is presented as empirical, yet the abstract and available description provide no experimental setup, baselines, metrics, or quantitative results; this leaves the practical superiority assertion unsupported.

    Authors: Section 5 presents the experimental evaluation, including the tested datasets, baselines (Marching Cubes on sign-flipped approximations, Poisson reconstruction after gradient estimation, and learning-based unsigned methods), metrics (Hausdorff distance, mean angular error, and mesh quality measures), and quantitative tables showing consistent outperformance. We agree that the abstract does not convey these details. We will expand the abstract to note the experimental protocol and highlight the key quantitative improvements (e.g., reduction in reconstruction error). revision: yes

Circularity Check

0 steps flagged

No significant circularity in convergence proof or reconstruction

full rationale

The paper defines the superpower contour from the power diagram of discrete unsigned distance samples and states a proof that it converges to the true surface as sampling density increases. This is framed as an independent mathematical result under stated sampling conditions rather than a fit, renaming, or self-referential construction. No equations, fitted parameters called predictions, or load-bearing self-citations appear in the abstract or description that would reduce the central claim to its own inputs by construction. The subsequent mesh algorithm is presented as leveraging this proxy, with outperformance claims being empirical and separate. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The abstract introduces the superpower contour without specifying free parameters or additional axioms beyond standard assumptions in computational geometry.

invented entities (1)
  • superpower contour no independent evidence
    purpose: Surface proxy extracted from the power diagram of unsigned distance samples that converges to the true surface
    New theoretical object defined in the paper; no independent evidence or external validation is mentioned in the abstract.

pith-pipeline@v0.9.0 · 5474 in / 1271 out tokens · 41577 ms · 2026-05-11T01:58:21.521825+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    doi:10.1145/258734.258868 Federico Stella, Nicolas Talabot, Hieu Le, and Pascal Fua. 2024. Neural surface detection for unsigned distance fields. InEuropean Conference on Computer Vision. Springer, 394–409. Ningna Wang, Hui Huang, Shibo Song, Bin Wang, Wenping Wang, and Xiaohu Guo

    work page doi:10.1145/258734.258868 2024

  2. [2]

    doi:10.1145/3687763 Ningna Wang, Bin Wang, Wenping Wang, and Xiaohu Guo

    MATTopo: Topology-preserving Medial Axis Transform with Restricted Power Diagram.ACM Transactions on Graphics (TOG)43, 4 (2024). doi:10.1145/3687763 Ningna Wang, Bin Wang, Wenping Wang, and Xiaohu Guo. 2022. Computing Medial Axis Transform with Feature Preservation via Restricted Power Diagram.ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia 202...

    work page doi:10.1145/3687763 2024