pith. sign in

arxiv: 2605.25503 · v1 · pith:KYI5V5RWnew · submitted 2026-05-25 · 💻 cs.CV

Metric--Phase Fields: Decoupling Distance and Sign for Thin-Structure Reconstruction from Unoriented Point Clouds

Jiayi Kong , Xuhui Chen , Chen Zong , Fei Hou , Junhui Hou , Wenping Wang , Ying He This is my paper

Pith reviewed 2026-06-29 22:27 UTC · model grok-4.3

classification 💻 cs.CV
keywords metric phase fieldsimplicit surface reconstructionthin structure reconstructionunoriented point cloudsunsigned distance fieldsneural implicit representationsthin-shell geometry
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The pith

Metric-phase fields decouple unsigned distance from a learnable phase to reconstruct thin structures from unoriented point clouds.

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

The paper introduces Metric-Phase Fields as a way to represent surfaces implicitly by learning two separate components: an unsigned metric field that measures proximity and a smooth phase field that supplies soft inside-outside information only where it is meaningful. This separation avoids the strict topological constraints of signed distance functions, which break on open or thin geometry, and the gradient singularities of unsigned distance functions, which destabilize optimization near the surface. The two fields are joined through a gated-metric formulation plus residual phase injection, producing a signed implicit function whose near-surface gradients remain stable. A learnable sharpness parameter further lets the model adapt the transition width automatically. Experiments on synthetic and scanned thin-shell and thin-plate data show that the resulting surfaces preserve fine layered structures more accurately than either prior approach while supporting reliable extraction.

Core claim

Given an unoriented point cloud, Metric-Phase Fields learn an unsigned metric field r together with a smooth phase field θ; a bounded indicator P = tanh(βθ) with learnable β supplies soft sign cues, and the fields are coupled by a gated-metric formulation with residual phase injection to yield a signed implicit function whose gradients remain well-behaved near the surface, allowing faithful reconstruction of thin and open geometry.

What carries the argument

The metric-phase field pair (unsigned metric r, phase θ) coupled by gated-metric formulation and residual phase injection, with soft indicator P = tanh(βθ).

If this is right

  • Thin-shell and thin-plate geometries can be reconstructed without requiring watertight topology or suffering zero-level gradient collapse.
  • Surface extraction becomes more reliable because near-surface gradients stay non-singular.
  • Training converges more stably than with pure unsigned distance fields because the phase term supplies usable sign information.
  • The learnable β automatically adjusts the sharpness of the phase transition to the local geometry.

Where Pith is reading between the lines

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

  • The same separation may allow hybrid representations that switch between metric and phase cues depending on local topology.
  • Because the phase field is smooth and independent, it could be transferred across different point-cloud densities without retraining the metric component.
  • The formulation suggests a route to layered reconstruction where multiple phase transitions are stacked inside a single metric field.

Load-bearing premise

A smooth phase field can be learned from unoriented points and stably coupled to the metric field via gated-metric formulation and residual injection without introducing optimization instabilities or extraction artifacts.

What would settle it

Train the model on a dataset of closely spaced parallel thin plates; if the extracted surfaces merge across the gap or if gradients vanish at the zero level set, the decoupling claim does not hold.

Figures

Figures reproduced from arXiv: 2605.25503 by Chen Zong, Fei Hou, Jiayi Kong, Junhui Hou, Wenping Wang, Xuhui Chen, Ying He.

Figure 1
Figure 1. Figure 1: Challenges of SDFs and UDFs in surface reconstruction. When the inside–outside sign is ambiguous (e.g., in thin plates, boundary regions, or non-manifold configurations), learning a single SDF becomes ill-conditioned and may introduce artifacts such as inflated thickness. In contrast, UDFs often suffer from optimization difficulties and unstable zero level set extraction due to non-differentiability at u =… view at source ↗
Figure 2
Figure 2. Figure 2: Pipeline. Query points sampled from an unoriented point cloud are processed by a SIREN-based network to predict r(x) and θ(x), which are composed into the MPF ϕ(x). The reconstructed mesh is directly extracted using Marching Cubes. (Optional) Normal alignment. For dense and low-noise point clouds, one can estimate reliable local (unoriented) normals via PCA, although their global orientation may be inconsi… view at source ↗
Figure 3
Figure 3. Figure 3: Toy example: cube + attached thin sheet (non-manifold + boundary). The shape is watertight on the cube, but includes a thin sheet attached to the top face, forming a non-manifold junction at the attachment and a boundary edge at the free end of the sheet. From left to right we show a UDF learned by CAP-UDF (Zhou et al., 2024), an SDF learned by NSH (Wang et al., 2023a), and our MPF composite field ϕ. Each … view at source ↗
Figure 4
Figure 4. Figure 4: Ablation study of the phase scaling parameter β on thin-walled structures. Results are shown for β ∈ {1, 50, 100, 500} at 1k, 4k, and 10k training iterations. Effect of directly parameterizing ϕ(x) = P(x). We investigate a simplified formulation that directly learns the implicit field as ϕ = tanh(β θ(x)), without explicitly modeling a distance component. Although this formulation can partially recover thin… view at source ↗
Figure 5
Figure 5. Figure 5: Result obtained when the shape is represented solely by the phase term, i.e., ϕ(x) = P(x). implicit surfaces. Since ϕ = tanh(βθ) enforces continuous sign transitions everywhere, it cannot correctly terminate a single surface at boundary regions, inevitably producing spurious thin layers near the boundaries. These observations highlight the necessity of explicitly decoupling distance and sign modeling, whic… view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of the components of our MPF representation. Separately showing these fields highlights their complementary contributions to the final reconstruction. not constrained by unit-gradient conditions, it can better adapt to rapid sign changes without interfering with distance learning. This results in a better-conditioned optimization process and improved reconstruction quality for thin and comple… view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative comparison on watertight models (Spikeball, Artichoke, Ship) with complex thin and multi-layered structures under varying sampling levels. Compared to DiGS (Ben-Shabat et al., 2022), StEik (Yang et al., 2023), NSH (Wang et al., 2023a), PG-SDF (Koneputugodage et al., 2024), and I-filtering (Li et al., 2025), our approach produces significantly more robust reconstructions, especially in the prese… view at source ↗
Figure 8
Figure 8. Figure 8: Qualitative comparison on inputs that consist of single-layer point clouds. Compared to DiGS (Ben-Shabat et al., 2022), StEik (Yang et al., 2023), NSH (Wang et al., 2023a), and I-filtering (Li et al., 2025), our method produces thinner reconstructions that more closely adhere to the original input point clouds. CAP-UDF GeoUDF S2DF Ours [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Qualitative comparison on a toy model containing boundary and non-manifold structures. We use this toy model with simple geometry to evaluate iso-surface extraction behavior. All UDF-based baselines are able to recover the overall shape reasonably well. However, they either rely on gradient information or optimization-based procedures to extract the zero level set. Small inaccuracies in UDF values can lead… view at source ↗
Figure 10
Figure 10. Figure 10: Comparisons on a real-scan thin-structure model. The real-scan model contains approximately 3 million input points with thin structures, which are challenging to reconstruction from real scans. Results are extracted at a resolution of 5123 . 500k 150k 150k Input CAP-UDF GeoUDF DEUDF S2DF Ours [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison with UDF-based baselines on graphics benchmarks featuring watertight geometries with fine details. While all methods produce visually plausible reconstructions, our method demonstrates clear advantages in zero level set extraction. For example, on the Dragon model at a resolution of 5123 , extracting the zero level set takes 10.5, 3.6, 5.8, and 15.2 minutes for CAP-UDF, GeoUDF, DEUDF, and S2DF,… view at source ↗
Figure 12
Figure 12. Figure 12: MPF gallery: reconstructions across diverse geometries and topologies, including thin structures and open boundaries that challenge SDF-based methods. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Visual comparison for the ablation study in [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Qualitative comparison of thin-structure reconstruction at varying thickness levels T. SDF-based methods fail to preserve extremely thin geometry, while UDF-based methods suffer from topological artifacts such as holes and splitting. In contrast, MPF consistently reconstructs accurate and complete thin structures across all scales. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of different methods on the scene datasets. SDF-based methods include DiGS, StEik, NSH, I-filtering and PG-SDF; UDF-based methods include CAP-UDF, DUDF, and GeoUDF. Our method demonstrates superior performance in terms of reconstruction quality. Input DiGS StEik NSH PG-SDF I-filtering CAP-UDF DUDF GeoUDF Ours [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Comparison of indoor scene reconstruction results. The first three rows show point maps extracted from RGB-D videos in ScanNet, which contain significant noise, while the subsequent rows use points sampled from the official meshes as input. SDF-based methods tend to produce bulging artifacts, I-filtering achieves good results, and UDF-based methods are prone to fragmented surfaces. Our method produces hig… view at source ↗
Figure 17
Figure 17. Figure 17: The top four rows compare different methods on the SRB dataset, which contains partial missing regions and noise/outlier perturbations due to its real-scanned nature. Our method achieves accurate reconstructions despite these challenges. The bottom three rows show reconstructions with added Gaussian noise (0.5%), highlighting the robustness of our method under noisy conditions. All methods are evaluated u… view at source ↗
read the original abstract

Neural Signed Distance Functions (SDFs) excel at reconstructing watertight manifolds but fail on thin structures and open boundaries due to strict inside--outside constraints. Conversely, Unsigned Distance Fields (UDFs) accommodate general geometries but suffer from gradient singularities at the zero-level set, hindering optimization and extraction. We introduce Metric--Phase Fields (MPFs), a decoupled implicit representation that separates metric proximity from topological phase. Given an unoriented point cloud, MPFs learn (i) an unsigned metric field $r$ and (ii) a smooth phase field $\theta$, for which we derive a bounded phase indicator $P=\tanh(\beta\theta)$ that provides soft inside--outside cues where they are meaningful. We couple the two fields via a gated-metric formulation with a residual phase injection to obtain a signed implicit function with stable near-surface gradients. The phase coefficient $\beta$ is learnable, allowing MPFs to adaptively control the sharpness of the phase transition and the degree of saturation of the soft sign indicator. Experiments on both synthetic and scanned thin-shell and thin-plate shapes demonstrate that MPFs preserve thin and layered structures more faithfully than recent SDF-based methods, while also enabling more robust training and more reliable surface extraction than UDF-based approaches. Check out \href{https://github.com/JIAYI-Scarlett/ICML2026-MPF}{MPFs-GitHub} for source code and test models.

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

3 major / 1 minor

Summary. The paper introduces Metric-Phase Fields (MPFs) as a decoupled implicit representation for thin-structure reconstruction from unoriented point clouds. It learns an unsigned metric field r together with a smooth phase field θ, defines a bounded phase indicator P = tanh(βθ) with learnable β, and couples the fields through a gated-metric formulation plus residual phase injection to produce a signed implicit whose near-surface gradients remain stable. Experiments on synthetic and scanned thin-shell and thin-plate shapes are claimed to show superior preservation of thin and layered structures relative to SDF methods and more robust training/extraction relative to UDF methods.

Significance. If the gated-metric coupling and residual injection can be shown to produce stable gradients without forcing β to extremes or introducing extraction artifacts, the approach would address a recognized limitation of both SDFs (inability to handle open boundaries and thin structures) and UDFs (gradient singularities at the zero level set). A parameter-light, learnable phase that supplies soft sign cues only where meaningful could become a useful primitive for general-geometry implicit reconstruction.

major comments (3)
  1. [Abstract] Abstract: the central experimental claim—that MPFs “preserve thin and layered structures more faithfully than recent SDF-based methods” and enable “more robust training and more reliable surface extraction than UDF-based approaches”—is stated without any quantitative metrics, baselines, error tables, or training-protocol details. This absence makes the headline superiority assertion unverifiable from the manuscript as presented.
  2. [Abstract] Abstract (and the description of the phase indicator): P is defined directly as tanh(βθ) with β learnable. Without an explicit loss term or derivation showing that smoothness of θ is enforced away from the surface and that the gate does not collapse when local sign information is absent (thin shells, open boundaries, layered plates), it remains unclear whether reported advantages arise from the architectural decoupling or simply from adaptive fitting of β.
  3. [Abstract] The gated-metric formulation with residual phase injection is presented as the mechanism that yields stable near-surface gradients. No equation or ablation is supplied that quantifies gradient behavior (e.g., norm histograms or condition numbers) under the proposed coupling, leaving the “stable gradients” and “more reliable extraction” claims without direct supporting evidence.
minor comments (1)
  1. [Abstract] The GitHub link is given but no statement is made about code release, reproducibility package, or whether the reported experiments can be rerun from the supplied models.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive critique. We address each major comment below. Revisions will be made to strengthen the abstract and clarify the supporting derivations and evidence in the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central experimental claim—that MPFs “preserve thin and layered structures more faithfully than recent SDF-based methods” and enable “more robust training and more reliable surface extraction than UDF-based approaches”—is stated without any quantitative metrics, baselines, error tables, or training-protocol details. This absence makes the headline superiority assertion unverifiable from the manuscript as presented.

    Authors: The abstract is intentionally concise and summarizes results whose quantitative details (error tables, baselines, Chamfer distances, training protocols) appear in Section 5 and the supplementary material. We agree the abstract would benefit from one or two key metrics; we will revise it to include representative quantitative figures while remaining within length limits. revision: yes

  2. Referee: [Abstract] Abstract (and the description of the phase indicator): P is defined directly as tanh(βθ) with β learnable. Without an explicit loss term or derivation showing that smoothness of θ is enforced away from the surface and that the gate does not collapse when local sign information is absent (thin shells, open boundaries, layered plates), it remains unclear whether reported advantages arise from the architectural decoupling or simply from adaptive fitting of β.

    Authors: Section 3.2 derives the phase indicator and Section 4.1 specifies the loss terms (including the smoothness regularizer on θ away from the surface and the gated coupling that prevents collapse on open or thin regions). We will move the derivation of the loss and the non-collapse argument into the main text (currently in the supplement) and add a short paragraph clarifying why adaptive β alone cannot explain the observed behavior. revision: yes

  3. Referee: [Abstract] The gated-metric formulation with residual phase injection is presented as the mechanism that yields stable near-surface gradients. No equation or ablation is supplied that quantifies gradient behavior (e.g., norm histograms or condition numbers) under the proposed coupling, leaving the “stable gradients” and “more reliable extraction” claims without direct supporting evidence.

    Authors: The gated-metric equation and residual injection appear in Eq. (7)–(9); an ablation on gradient norms is reported in the supplement (Figure S3). We acknowledge the main text would be stronger with a brief main-paper quantification; we will add a short paragraph and a compact gradient-norm plot to Section 4.3. revision: partial

Circularity Check

0 steps flagged

No significant circularity; proposal is self-contained with empirical validation

full rationale

The paper introduces MPFs as a new decoupled representation (unsigned metric r plus phase field θ, with P = tanh(βθ) and gated coupling), where β is explicitly learnable. No derivation chain reduces a claimed result to its own inputs by construction, no self-citations are load-bearing, and no fitted parameter is relabeled as an independent prediction. Central claims rest on experiments comparing reconstruction fidelity, training robustness, and extraction reliability against SDF/UDF baselines on synthetic and scanned data. The formulation is presented as a modeling choice with stated advantages, not as a theorem forced by prior self-work or definitional equivalence. This is the normal case of a non-circular technical proposal.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The method rests on a learnable sharpness parameter and standard activation functions; full training assumptions and loss terms are not visible in the abstract.

free parameters (1)
  • β
    Learnable coefficient that controls phase transition sharpness and saturation of the soft sign indicator P=tanh(βθ).
axioms (1)
  • standard math tanh provides a bounded soft inside-outside indicator
    Invoked to define the phase indicator P from the phase field θ.
invented entities (1)
  • Metric-Phase Field no independent evidence
    purpose: Decoupled implicit representation separating metric and phase for thin structures
    New representation introduced by the paper; no independent external evidence cited in abstract.

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Reference graph

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