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arxiv: 2605.30744 · v1 · pith:UI2JTJ25new · submitted 2026-05-29 · 💻 cs.GR

BijectiveRemesh: Maintaining Bijective Mappings for Data Transfer Across Remeshed Manifolds

Leyi Zhu , Michael Tao , Yixin Hu , Daniele Panozzo , Denis Zorin This is my paper

Pith reviewed 2026-06-28 20:43 UTC · model grok-4.3

classification 💻 cs.GR
keywords bijective mappingremeshingdata transfertriangle meshtetrahedral meshatlas compositionshared scaffold
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The pith

A composite map of local bijective atlases maintains exact mappings across any sequence of remeshing operations.

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

The paper presents BijectiveRemesh, an algorithm that keeps a continuous and bijective mapping between an input mesh and its remeshed version. It does this by building the overall mapping as a chain of local bijective atlases, each corresponding to one primitive remeshing step such as edge collapse or vertex smoothing. For 2D surfaces, a Shared Scaffold structure enforces global bijectivity through local orientation preservation. The approach extends to 3D tetrahedral meshes using Steinitz's Theorem and Maxwell-Cremona lifting. This enables precise data transfer like textures or simulation data without approximation.

Core claim

BijectiveRemesh constructs a mathematically rigorous composite map from the input mesh to the output mesh by chaining local bijective atlases defined for each primitive remeshing operation, representing the overall mapping as a composition that preserves bijectivity on both 2D triangle surfaces and 3D tetrahedral meshes.

What carries the argument

The composition of local bijective atlases, one per remeshing primitive, enforced by the Shared Scaffold in 2D and by Steinitz's Theorem with Maxwell-Cremona lifting in 3D.

Load-bearing premise

Local bijective atlases can be constructed for each remeshing primitive and their composition preserves global bijectivity.

What would settle it

A concrete remeshing sequence on a simple mesh where no local bijective atlas exists for one operation or the composed map overlaps or reverses orientation.

Figures

Figures reproduced from arXiv: 2605.30744 by Daniele Panozzo, Denis Zorin, Leyi Zhu, Michael Tao, Yixin Hu.

Figure 1
Figure 1. Figure 1: Bijective surface tracking through tetrahedral mesh simplification. We track axis-aligned planar surfaces (parallel to the 𝑥 𝑦, 𝑥𝑧, and 𝑦𝑧 planes) through tetrahedral mesh simplification on models from the Thingi10K dataset [Zhou and Jacobson 2016]. For each model, surfaces are sampled on the simplified output mesh Moutput (right) and back-tracked to the original mesh Minput (left). Our bijective framework… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison on Thingi10K model #1706476. We visualize the coarse [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: illustrates this shared scaffold framework. Application to Different Operation Types. The shared scaffold framework applies uniformly across various remeshing operations [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: illustrates the local patches for each operation type. Using the open star St ([Munkres 2018]), the local patches we use are: (1) Edge Collapse (𝑖, 𝑗) → 𝑘: P before = St(𝑖) ∪ St(𝑗) (union of 1-rings of both vertices), P after = St(𝑘) (1-ring of merged vertex). (2) Edge Split on edge (𝑖, 𝑗) creating vertex 𝑘: P before = St(𝑖, 𝑗) (triangles incident to the edge), P after = St(𝑘) (1-ring of new vertex). (3) E… view at source ↗
Figure 5
Figure 5. Figure 5: Constructive algorithm for convex polyhedra embedding. (a) The [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Initial valid embedding for both P before and P after. After constructing the convex polyhedron C via Algorithms 1 and 2, vertex 𝑖 is placed at the barycenter of face 𝑓𝑘 (the one-ring of 𝑖 on the boundary). By Lemma A.1, this configuration is valid (non-overlapping) for both the pre-collapse con￾nectivity (with edge (𝑖, 𝑗)) and the post-collapse connectivity (where 𝑖 has collapsed to 𝑗). This provides a va… view at source ↗
Figure 7
Figure 7. Figure 7: Curve tracking via local atlas. The portion of the curve (red) on the [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Candidate facet selection during curve tracking: Given three query [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Texture transfer on the Ogre model with checkerboard pattern. [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: Texture transfer through mesh decimation and refinement. [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: , the tracked curves faithfully preserve their intersection topology: the combinatorial pattern of curve crossings remains iden￾tical between input and output, with no spurious intersections intro￾duced and no existing intersections lost. This topological guarantee is achieved through our topology-preserving multi-curve tracking algorithm (Section 3.2.3), which maintains ordered sequences of in￾tersection… view at source ↗
Figure 12
Figure 12. Figure 12: Stress test for curve tracking under extreme remeshing. Each pair shows input mesh (left) and aggressively simplified output (right) with tracked [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Surface tracking on a tetrahedral body mesh from CT scan data. [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
read the original abstract

We introduce BijectiveRemesh, a robust algorithm for maintaining a continuous, bijective mapping across complex remeshing sequences on both 2D triangle surfaces and 3D tetrahedral meshes. Unlike traditional data transfer methods that rely on interpolation or projection, our approach constructs a mathematically rigorous composite map from the input mesh to the output mesh by chaining local bijective atlases defined for each primitive remeshing operation. Our framework represents the overall mapping as a composition of local bijective atlases, one per remeshing operation. Building upon successive self-parameterization, we introduce a Shared Scaffold structure for 2D triangle meshes that enforces global bijectivity through local orientation preservation. We extend this approach to handle edge splits, edge swaps, and vertex smoothing beyond the original edge collapses. For 3D tetrahedral meshes, we generalize the local atlas construction using Steinitz's Theorem and Maxwell-Cremona lifting to ensure valid embeddings. This enables exact tracking of geometric entities, including points, curves, and surfaces, across remeshing, with applications from texture transfer to volumetric simulations.

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

1 major / 0 minor

Summary. The paper claims to introduce BijectiveRemesh, an algorithm that maintains continuous bijective mappings across remeshing sequences on 2D triangle surfaces and 3D tetrahedral meshes by constructing a composite map from chaining local bijective atlases for each primitive remeshing operation, using a Shared Scaffold for 2D to enforce global bijectivity through local orientation preservation, and Steinitz's Theorem and Maxwell-Cremona lifting for 3D. This enables exact tracking of points, curves, and surfaces for applications including texture transfer and volumetric simulations.

Significance. If the local atlas constructions and composition steps hold, the work would provide a rigorous alternative to interpolation-based transfer methods, with clear utility in adaptive geometry processing and simulation pipelines. The constructive algorithmic framing and invocation of classical embedding theorems constitute a positive aspect of the approach.

major comments (1)
  1. [Abstract] Abstract: the central claim that the method 'constructs a mathematically rigorous composite map' via chaining of local bijective atlases is load-bearing for the entire contribution, yet the manuscript provides no derivations, proofs, or explicit verification that such atlases exist for every listed primitive (edge collapse, split, swap, smoothing) or that the Shared Scaffold (2D) and Steinitz/Maxwell-Cremona steps (3D) preserve global bijectivity under composition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for stronger substantiation of the central bijectivity claims. We address the comment below and will revise the manuscript to incorporate additional formal derivations and verifications.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the method 'constructs a mathematically rigorous composite map' via chaining of local bijective atlases is load-bearing for the entire contribution, yet the manuscript provides no derivations, proofs, or explicit verification that such atlases exist for every listed primitive (edge collapse, split, swap, smoothing) or that the Shared Scaffold (2D) and Steinitz/Maxwell-Cremona steps (3D) preserve global bijectivity under composition.

    Authors: We agree that the abstract's claim of a mathematically rigorous composite map requires explicit supporting derivations and verifications to be fully substantiated. The manuscript describes the algorithmic construction of local bijective atlases for each primitive operation and invokes the Shared Scaffold for 2D (via orientation preservation) together with Steinitz's Theorem and Maxwell-Cremona lifting for 3D. However, it does not currently contain detailed mathematical derivations or step-by-step verification that each local atlas is bijective and that the composition preserves global bijectivity. In the revised manuscript we will add these elements, including explicit proofs of local bijectivity for edge collapse, split, swap, and smoothing, together with an argument showing that the scaffold and theorem-based embeddings ensure the composite map remains bijective. These additions will appear in an expanded methods section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructive algorithm using external theorems

full rationale

The paper describes an algorithmic construction that chains local bijective atlases for remeshing primitives, relying on the Shared Scaffold (2D) and Steinitz's Theorem plus Maxwell-Cremona lifting (3D) to enforce bijectivity. These are standard external mathematical results, not self-citations or internal fits. No equation or step reduces by construction to its own inputs; the central claim is a procedural composition rather than a derived equality that collapses to a parameter or prior self-result. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence and composability of local bijective atlases plus two geometric theorems for 3D; no free parameters are mentioned.

axioms (1)
  • standard math Steinitz's Theorem and Maxwell-Cremona lifting ensure valid embeddings for tetrahedral meshes
    Invoked to guarantee valid local atlases in 3D case
invented entities (1)
  • Shared Scaffold structure no independent evidence
    purpose: Enforces global bijectivity through local orientation preservation for 2D triangle meshes
    New structure introduced to maintain consistency across operations

pith-pipeline@v0.9.1-grok · 5735 in / 1309 out tokens · 26267 ms · 2026-06-28T20:43:33.062551+00:00 · methodology

discussion (0)

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