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arxiv: 2605.02770 · v1 · submitted 2026-05-04 · 💻 cs.GR · cs.CG

Recognition: unknown

Implicit Minimal Surfaces for Bijective Correspondences

Etienne Corman, Mark Gillespie, Robin Magnet, Yousuf Soliman

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Pith reviewed 2026-05-08 02:07 UTC · model grok-4.3

classification 💻 cs.GR cs.CG
keywords bijective correspondencesimplicit mapsGinzburg-Landau functionalminimal surfacessurface mappinggenus zero surfacestangent vector fields
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The pith

A bijection between surfaces is encoded implicitly as the zero set of a complex section on their product space, whose area is minimized by the Ginzburg-Landau functional.

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

The paper shows that any continuous bijection between two genus-zero surfaces corresponds to a two-dimensional mapping surface embedded inside the four-dimensional product of the two surfaces. This embedded surface is stored implicitly as the zero set of a complex-valued section defined over the product space. Distortion of the map is then reduced simply by minimizing the area of that embedded surface, which is achieved by optimizing the Ginzburg-Landau functional of the section. The resulting procedure uses only standard tangent-vector-field operations, requires no mesh surgery, no barrier penalties, and no bijective initialization, while still guaranteeing orientation-preserving bijective output and accepting both point and curve landmarks.

Core claim

A bijection between surfaces defines a two-dimensional mapping surface sitting inside the four-dimensional product space of the two inputs, and this mapping surface can be stored implicitly as the zero set of a complex section. The distortion of the map can be optimized by minimizing the area of this mapping surface, which amounts to minimizing the Ginzburg-Landau functional of the complex section. This yields a simple algorithm for bijective correspondences that avoids combinatorial mesh modifications and barrier functions.

What carries the argument

The zero set of a complex section on the product space, which implicitly represents the mapping surface whose area is minimized through the Ginzburg-Landau functional.

If this is right

  • Bijective maps can be computed from non-bijective initializations such as functional maps without explicit untangling steps.
  • Landmark points and curves can be incorporated directly as constraints on the complex section.
  • The algorithm relies only on existing tangent vector field solvers and requires no custom combinatorial operations.
  • Robustness to noise increases because no barrier functions are needed to enforce bijectivity.

Where Pith is reading between the lines

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

  • The same product-space construction could be tested on surfaces with boundaries to see whether the implicit zero-set representation continues to enforce bijectivity automatically.
  • Different distortion measures might be obtained by replacing the Ginzburg-Landau functional with other energies while retaining the same implicit zero-set representation.
  • The method's stability on noisy inputs suggests it may serve as a post-processing step for maps produced by learning-based correspondence pipelines.

Load-bearing premise

The zero set of the optimized complex section necessarily produces a continuous, bijective, orientation-preserving map without any additional enforcement mechanisms.

What would settle it

An example in which the zero level set of the converged complex section produces a map that either folds, leaves gaps, or reverses orientation on the target surface.

Figures

Figures reproduced from arXiv: 2605.02770 by Etienne Corman, Mark Gillespie, Robin Magnet, Yousuf Soliman.

Figure 1
Figure 1. Figure 1: We introduce a new implicit representation of maps view at source ↗
Figure 2
Figure 2. Figure 2: To illustrate the idea behind our method, we move down a dimen view at source ↗
Figure 3
Figure 3. Figure 3: As in the case of curves, a map 𝜑 between surfaces 𝐴 and 𝐵 defines a surface Σ𝜑 in the product space whose area encodes the distortion of 𝜑. In particular, maps which flip triangles have larger surface area. express the area of Σ𝜑 using the singular values 𝜎1, 𝜎2 of 𝑑𝜑: Area(Σ𝜑 ) = ∫ 𝐴 √︃ (1 + 𝜎 2 1 ) (1 + 𝜎 2 2 ) vol𝐴 . (3) This area based distortion measure is bounded by the Dirichlet en￾ergy1 and the ar… view at source ↗
Figure 5
Figure 5. Figure 5: Our method preserves mapping orientation by encoding surface view at source ↗
Figure 6
Figure 6. Figure 6: Left: A codimension-1 object (like a 2D surface in 3D space) can be en￾coded as the zero level set of a real-valued function 𝑓 . Right: A codimension-2 object (like a 1D curve in 3D space, or a 2D surface in 4D space) can be encoded as the shared zero level set of a pair of real functions 𝑓1, 𝑓2. Equiv￾alently, a codimension-2 object is the zero level set of a single complex function 𝑓 (𝑥 ) = 𝑓1 (𝑥 ) + 𝚤 𝑓… view at source ↗
Figure 7
Figure 7. Figure 7: Left: Geometrically, the product space 𝐴 ×𝐵 is the result of extruding 𝐴 along 𝐵. Algebraically, points of 𝐴 × 𝐵 are pairs (𝑎, 𝑏) for 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵. Right: The boundary operator obeys a product rule on product cells. subtle [Parise et al. 2024], but in certain situations the zeros of minimizers are known to form minimal surfaces [Lin and Riviere 1999; Canevari et al. 2023]. The intuition to keep in mind … view at source ↗
Figure 8
Figure 8. Figure 8: Our algorithm proceeds in four steps. We begin with a pair of meshes view at source ↗
Figure 9
Figure 9. Figure 9: An experimental search for the optimal choice of Ginzburg-Landau view at source ↗
Figure 10
Figure 10. Figure 10: To evaluate the map at a vertex 𝑢 ∈ 𝑉𝐵, we take the 𝑢’th column of 𝑍 as a section 𝑧 (𝑢) on 𝐴. We can visualize this section as a complex function in a chart that covers all but a single face of 𝐴. Now the image 𝜑 (𝑢) of vertex 𝑢 is given by the location of the zero on 𝐴. i i j k initialization intermediate final result j k view at source ↗
Figure 11
Figure 11. Figure 11: Left: In order to evaluate the correspondence, we have to find a root of the interpolated section 𝑧 within some triangle 𝑖𝑗𝑘. But the interpolant is nonlinear and non-convex, so solving directly with Newton’s method might not find the desired root. Right: Instead, we interpolate the face curvature from flat at 𝑡 = 0—in which case the interpolant becomes linear—to the full curvature Ω𝑖 𝑗𝑘 at 𝑡 = 1, solving… view at source ↗
Figure 14
Figure 14. Figure 14: Our implicit representation of the map can be prolongated to a finer view at source ↗
Figure 13
Figure 13. Figure 13: High quality correspondences can be computed even on very coarse view at source ↗
Figure 15
Figure 15. Figure 15: Landmarks are specified using a pinning potential that has local view at source ↗
Figure 16
Figure 16. Figure 16: Enforcing curve to curve correspondences where points are allowed view at source ↗
Figure 20
Figure 20. Figure 20: Landmark-free correspondences computed on a variety of non view at source ↗
Figure 19
Figure 19. Figure 19: Correspondences obtained from closest-point initialization, visual view at source ↗
Figure 23
Figure 23. Figure 23: We replicate the experiment from Schmidt et al. [2020] to evaluate robustness against poor initializations (top row). The RHM method [Ezuz et al. 2019b] becomes trapped in local minima of distortion for each ini￾tialization. While Schmidt et al. [2020] (ISM) performs better and achieves consistent results for the first three initializations, our method produces nearly identical low-distortion mappings acr… view at source ↗
Figure 22
Figure 22. Figure 22: Enforcing curve-to-curve correspondences can be challenging, and view at source ↗
Figure 24
Figure 24. Figure 24: Both our algorithm and the algorithm of Schmidt et al. [2020] (ISM) represent the common subdivision of the overlay mesh induced by the correspondence. It is needed to evaluate distortion energies of the correspondence intrinsically. While both algorithms find similar correspon￾dence, implicit area minimization produces a less distorted map, even at the finer scales. This is reflected in the smoother dist… view at source ↗
Figure 26
Figure 26. Figure 26: We use our matching algorithm to compare a human femur with view at source ↗
Figure 25
Figure 25. Figure 25: Distortion is inevitable between non-isometric shapes. Compared view at source ↗
Figure 28
Figure 28. Figure 28: Computing implicit minimal surfaces scales roughly linearly in view at source ↗
Figure 27
Figure 27. Figure 27: Our method (left) consistently yields lower distortion than maps produced by the hyperbolic orbifold Tutte embedding of Aigerman and Lipman [2016] (HOTE) across all three pairs. In contrast, HOTE does not minimize global mapping distortion and introduces severe artifacts near landmarks. Distortion is measured between the source mesh and the mesh obtained by mapping its vertices onto the target shape. manu… view at source ↗
Figure 29
Figure 29. Figure 29: Coordinate transfer from the horse 𝐴 into the cow 𝐵, computed for three values of 𝜆 = 𝑡 𝜆0 (right). A large value of 𝜆 produces clean corre￾spondences, but the map degrades near regions of high curvature when 𝜆 decreases. The graph (bottom left) plots the average percentage of vertices whose slices have more than one zero, thus breaking bijectivity. We obtain bijective maps once 𝑡 ≳ 50. Inspecting the fai… view at source ↗
read the original abstract

We introduce an implicit representation of continuous, bijective, orientation-preserving maps between genus zero surfaces with or without boundary. The distortion of these maps can easily be minimized by optimizing the Ginzburg-Landau functional - a ubiquitous model in physics and differential geometry - leading to a simple algorithm for computing bijective correspondences using only standard tools of the tangent vector field toolbox. The method avoids combinatorial mesh modifications and does not require barrier functions to enforce bijectivity making it more robust to noise and simpler to implement. Moreover, the algorithm does not assume a bijective initialization and can untangle non-bijective correspondences generated by computationally cheaper methods such as functional maps. It supports the use of both landmark points and landmark curves to guide the correspondence. The key idea is that a bijection between surfaces defines a two-dimensional mapping surface sitting inside the four-dimensional product space of the two inputs, and this mapping surface can be stored implicitly as the zero set of a complex section - essentially a complex function defined on the product space. Now the distortion of the map can be optimized by minimizing the area of this mapping surface, which amounts to minimizing the Ginzburg-Landau functional of the complex section. We demonstrate the practical benefits of our method by comparing to state-of-the-art correspondence algorithms and show that our implicit representation offers improved stability and naturally supports constraints that are difficult to enforce with explicit map representations.

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 claims that continuous bijective orientation-preserving maps between genus-zero surfaces (with or without boundary) can be represented implicitly as the zero set of a complex section on the 4D product space, with distortion minimized by optimizing the Ginzburg-Landau functional of that section; this yields a simple algorithm using standard tangent vector field tools, supports landmarks, untangles non-bijective initial maps, and requires no barrier functions or combinatorial mesh changes.

Significance. If the bijectivity guarantee holds, the approach would offer a notable simplification over explicit map representations in geometry processing, leveraging a well-studied physical model for area minimization while improving robustness to noise and poor initialization. The implicit encoding and ability to handle curve landmarks are practical strengths.

major comments (2)
  1. [Abstract] Abstract and key idea paragraph: the central claim that minimizing the Ginzburg-Landau functional of the complex section automatically produces a bijective map (i.e., the zero set is a graph over both surfaces with covering degree 1) lacks a supporting argument or theorem establishing transversality to the fibers and orientation preservation; the manuscript does not address why a minimizer must intersect every {x}×S2 fiber exactly once rather than producing folds or multiple sheets.
  2. [Method] Method description (implicit representation section): no explicit conditions or post-processing are stated to enforce that the optimized zero set defines a continuous function rather than a relation; the reduction to GL minimization is presented as sufficient, yet the Γ-limit argument for area minimization does not by itself guarantee the graph property on the product manifold without additional topological controls.
minor comments (2)
  1. [Abstract] Notation for the complex section s on S1 × S2 could be introduced with an explicit local coordinate expression or equation to clarify how the two real equations encode the 2D mapping surface.
  2. [Results] The abstract states comparisons to state-of-the-art methods but does not specify the quantitative metrics (e.g., distortion energy, bijectivity failure rate) used in the evaluation; adding these to the results section would strengthen the claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which help us improve the clarity and rigor of our presentation. Below we address the major comments point by point.

read point-by-point responses

  1. Referee: [Abstract] Abstract and key idea paragraph: the central claim that minimizing the Ginzburg-Landau functional of the complex section automatically produces a bijective map (i.e., the zero set is a graph over both surfaces with covering degree 1) lacks a supporting argument or theorem establishing transversality to the fibers and orientation preservation; the manuscript does not address why a minimizer must intersect every {x}×S2 fiber exactly once rather than producing folds or multiple sheets.

    Authors: We acknowledge that a formal theorem guaranteeing bijectivity for every global minimizer is not provided in the current version. Our argument is based on the fact that the Ginzburg-Landau functional Γ-converges to the area of the zero set, and for genus-zero surfaces the minimal area surface connecting the two boundaries in the product space is expected to be a graph due to the convexity of the domain and the orientation induced by the complex structure. However, to address this concern, we will revise the manuscript to include a dedicated paragraph in the implicit representation section explaining the conditions for the zero set to be a graph (transversality and degree 1), supported by a sketch of the topological argument using the Hopf degree theorem or similar. We also note that in practice, with the proposed initialization and landmark constraints, the optimization consistently produces bijective maps without folds, as verified in all experiments. We will add this discussion and supporting references to related work on minimal graphs in product manifolds. revision: yes

  2. Referee: [Method] Method description (implicit representation section): no explicit conditions or post-processing are stated to enforce that the optimized zero set defines a continuous function rather than a relation; the reduction to GL minimization is presented as sufficient, yet the Γ-limit argument for area minimization does not by itself guarantee the graph property on the product manifold without additional topological controls.

    Authors: We agree that the manuscript would be strengthened by explicitly stating the conditions under which the zero set corresponds to a continuous bijective map. In the revised version, we will add a new paragraph in the implicit representation section detailing that the complex section is constructed to have zeros that are transverse to the coordinate fibers by virtue of the non-vanishing gradient condition enforced during optimization, and that the covering degree is controlled to be 1 through the choice of the initial section and the energy minimization which penalizes multiple sheets. While the Γ-limit alone does not suffice, the combination with the specific topology of genus-zero surfaces and the use of a single complex section (rather than multiple) provides the necessary control. We will also mention that no post-processing is needed because the optimization naturally converges to such graphs, but we can add a simple check for bijectivity in the implementation if desired. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses established Ginzburg-Landau area minimization on an implicit product-space representation

full rationale

The paper constructs an implicit representation of maps as zero sets of complex sections on the product manifold and minimizes distortion by applying the Ginzburg-Landau functional, a standard model from physics and differential geometry whose Γ-limit is known to produce area-minimizing surfaces. No equation or claim reduces the bijectivity property or the optimality result to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain; the central steps rely on external, independently established mathematical facts rather than re-deriving the target properties from the method's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Central claim rests on equivalence of bijective maps to zero sets of complex sections and area minimization equaling GL energy minimization on genus-zero surfaces.

axioms (2)
  • domain assumption Minimizing the Ginzburg-Landau functional of the complex section yields a bijective orientation-preserving map of minimal distortion.
    Core optimization principle invoked without derivation.
  • domain assumption The zero set of the complex section defines a continuous 2D mapping surface corresponding to a bijection between genus-zero surfaces.
    Key representational assumption.
invented entities (1)
  • Complex section whose zero set encodes the mapping surface no independent evidence
    purpose: Implicit representation of bijective correspondences
    New encoding to avoid explicit maps and barriers.

pith-pipeline@v0.9.0 · 9427 in / 1253 out tokens · 84770 ms · 2026-05-08T02:07:14.057144+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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    offset connection

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