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Flatten Anything: Unsupervised Neural Surface Parameterization

Qijian Zhang, Junhui Hou, Wenping Wang, Ying He

arXiv:2405.14633v2cs.CVcs.CG

TL;DR

Traditional surface parameterization relies on high-quality meshes and often requires manual cutting, limiting its fit for ordinary 3D data and complex topologies. FAM learns global free-boundary mappings from surface points to adaptive UV coordinates, and the paper reports robustness to noisy inputs alongside challenges on models with complicated interior structures.

  • Problem

    Traditional parameterization targets well-behaved meshes and often requires manual cutting, while ordinary 3D data and complex topologies make those workflows difficult to apply.

    Source: §1 · p. 2

  • Method

    FAM is an unsupervised neural cycle framework that maps discrete surface points to adaptively deformed UV coordinates while learning cutting, unwrapping, and wrapping.

    Source: §1 · pp. 2–3, §3 · p. 4

  • Results

    FAM shows stable performance with 1%, 2%, and 4% Gaussian positional noise, but has inferior UV quality on highly complicated CAD models with rich interior structures.

    Source: §4 · p. 9

  • Takeaways & Limitations

    FAM's point-based approach is applicable to unstructured point clouds and learns cutting seams and UV boundaries without pre-cutting.

    Source: §1 · p. 3, §5 · p. 10

    FAM has inferior UV unwrapping quality on highly complicated CAD models with rich interior structures and multi-layer issues.

    Source: §4 · p. 9

Abstract

from arXiv · show

Surface parameterization plays an essential role in numerous computer graphics and geometry processing applications. Traditional parameterization approaches are designed for high-quality meshes laboriously created by specialized 3D modelers, thus unable to meet the processing demand for the current explosion of ordinary 3D data. Moreover, their working mechanisms are typically restricted to certain simple topologies, thus relying on cumbersome manual efforts (e.g., surface cutting, part segmentation) for pre-processing. In this paper, we introduce the Flatten Anything Model (FAM), an unsupervised neural architecture to achieve global free-boundary surface parameterization via learning point-wise mappings between 3D points on the target geometric surface and adaptively-deformed UV coordinates within the 2D parameter domain. To mimic the actual physical procedures, we ingeniously construct geometrically-interpretable sub-networks with specific functionalities of surface cutting, UV deforming, unwrapping, and wrapping, which are assembled into a bi-directional cycle mapping framework. Compared with previous methods, our FAM directly operates on discrete surface points without utilizing connectivity information, thus significantly reducing the strict requirements for mesh quality and even applicable to unstructured point cloud data. More importantly, our FAM is fully-automated without the need for pre-cutting and can deal with highly-complex topologies, since its learning process adaptively finds reasonable cutting seams and UV boundaries. Extensive experiments demonstrate the universality, superiority, and inspiring potential of our proposed neural surface parameterization paradigm. Our code is available at https://github.com/keeganhk/FlattenAnything.

1 Introduction

Traditional parameterization depends on well-behaved meshes and manual seam placement, limiting its fit for ordinary 3D data and complex topologies. FAM learns global, free-boundary point mappings through an unsupervised cycle framework that jointly models cutting and UV deformation.

1 Introduction

  • Traditional workflows manually cut complex surfaces into disks before applying disk-oriented parameterization, while ordinary 3D data may have irregular triangulations and non-ideal geometry.The paper notes that seam selection remains subjective and that mesh-edge distributions may need repeated adjustment.Source: §1 · p. 2
  • Cutting and flattening should be jointly optimized because treating them separately can produce sub-optimal parameterizations.Source: §1 · p. 2
  • Prior neural approaches include multi-patch local mappings and fixed-boundary rectangular structuring, whereas FAM targets global mapping with an adaptive UV domain.The paper characterizes some fixed-boundary methods as lacking mapping constraints.Source: §1 · pp. 2–3
  • FAM introduces unsupervised global, free-boundary mappings between surface points and adaptively deformed UV coordinates, with learned cutting seams and UV boundaries.Its cycle framework uses sub-networks for cutting, UV deforming, unwrapping, and wrapping.Source: §1 · pp. 2–3
  • FAM operates on discrete surface points and jointly learns cutting, reducing dependence on mesh triangulation and supporting non-ideal geometries and complex topologies.Its neural mappings are described as smooth for points across the target surface, unlike vertex-only parameterization followed by triangle interpolation.Source: §1 · p. 3

2 Related Works

Traditional parameterization methods often rely on mesh connectivity and regular triangulations, while point-cloud parameterization has fewer prior approaches. Neural geometry-reconstruction methods generally do not focus on surface mappings, and existing neural parameterization methods vary in their constraints and chart structure.

2.1 Traditional Parameterization Approaches

  • Classical mesh methods range from fixed-boundary Laplacian mappings to free-boundary methods [33, 19] and alternating optimization, but rely on mesh connectivity and often assume regular triangulations.OptCuts jointly optimizes cutting and mapping distortion; complex topologies and non-ideal meshes remain challenging.Source: §2.1 · p. 3
  • Point-cloud parameterization has relatively few prior works, covering specialized settings such as disk-type, genus-0, and genus-1 surfaces.Later methods include fixed-boundary mappings [5] and free-boundary conformal parameterization.Source: §2.1 · p. 3

2.2 Neural Parameterization Approaches

  • Recent neural methods learn parameterized 3D geometric representations, including grid-deformation and structure-based approaches, yet mappings between 3D surfaces and 2D parameter domains are barely considered [39] [3] [6] [27].Source: §2.2 · p. 4
  • Neural parameterization methods include multi-patch mappings [2], UV-map encoding [25], segmentation [20], and geometry-image approaches [42], with differing constraints and assembly challenges.Nuvo [36] assigns points to multiple charts and omits chart packing, unlike global parameterization.Source: §2.2 · p. 4

3 Proposed Method

FAM learns point-wise surface-to-UV mappings from discrete points without connectivity, using jointly optimized wrapping, unwrapping, cutting, and UV-deformation networks. Its two cycle-mapping branches and training objectives jointly guide the mappings and learned seams.

3 Proposed Method

  • FAM parameterizes unstructured 3D surface points as UV coordinates without requiring connectivity information or ground-truth UV maps.The framework is trained in an unsupervised, per-model manner.Source: §3 · p. 4

3.1 Bi-directional Cycle Mapping

  • Four parameter-sharing subnetworks handle UV deformation, wrapping, surface cutting, and unwrapping within a jointly optimized bi-directional cycle framework.The framework uses cycle mappings to associate the opposite mapping directions and promote bijectivity.Source: §3.1 · p. 5, §3 · p. 5, Figure 2 · p. 5

3.1.1 2D→3D→2D Cycle Mapping

  • The 2D→3D→2D branch deforms a uniform grid, wraps its UV points onto the surface, cuts the reconstruction, then unwraps it back toward the deformed UVs.The reconstructed point set is expected to approximate the target surface; the resulting cycle UVs are expected to match the deformed coordinates row-wise.Source: §3.1.1 · pp. 5–6

3.1.2 3D→2D→3D Cycle Mapping

  • The 3D→2D→3D branch cuts the input, unwraps it to UV coordinates, and wraps those coordinates back to reconstruct the surface.Seam points are identified by comparing each point's UV coordinate with those of its nearest neighbors; points exceeding a distance threshold are selected.Source: §3.1.2 · p. 6

3.2 Training Objectives

  • Training combines UV-neighbor separation, Chamfer-distance reconstruction, cycle consistency, and differential-geometric distortion regularization.The distortion constraint uses mapping derivatives to promote conformal, angle-preserving parameterizations; ground-truth normal constraints can be omitted when normals are unavailable or difficult to compute.Source: §3.2 · pp. 6–7
  • The reconstruction branch does not use an unwrapping loss, because the authors report that optimizing Chamfer distance empirically suppresses clustered point distributions.They also report degraded performance when optimizing Chamfer distance between the two UV sets, possibly because early, suboptimal coordinates interfere with each other.Source: §3.2 · p. 7

4 Experiments

Experiments compare FAM with SLIM on open surfaces and with FBCP-PC on point clouds, then test ablations, robustness, and complex models. FAM outperforms SLIM on the tested open models, while results vary with the comparison and model complexity.

Comparison with SLIM

  • FAM outperforms SLIM qualitatively and quantitatively on all eight tested open-surface models, using conformality where lower is better.The comparison includes UV unwrapping and texture-mapping results.Source: §4 · p. 7, Table 1 · p. 8, Figure 3 · p. 8

Comparison with FBCP-PC

  • FAM does not outperform FBCP-PC on point clouds, but that comparison is unfair because FBCP-PC requires manually supplied ordered boundary-point indices.FAM operates on discrete points without connectivity information.Source: §4 · p. 8, Table 3 · p. 8

Ablation Studies

  • Removing either cycle-mapping branch degrades performance; removing Cut-Net especially distorts complex models, though simpler face and Möbius-strip models show no obvious degradation.The authors attribute the complex-model effect to neural smoothness impeding the creation of cutting seams without learned offsets.Source: §4 · p. 8, Figure 6 · p. 10, Figure 7 · p. 10

Robustness and self-intersection

  • FAM shows stable performance under 1%, 2%, and 4% Gaussian point-position noise; self-intersections remain small in both FAM and SLIM and can be further relieved by UV refinement.The reported self-intersection proportions are not included in the supplied passage.Source: §4 · p. 9, Figure 8 · p. 10, Table 4 · p. 10

Stress tests

  • FAM obtains a basically optimal solution on a Hilbert-space-filling-shaped cylinder, but produces inferior UV unwrapping for CAD models with rich interior structures and multiple layers.The learned seams are generally reasonable and exterior texture mapping relatively regular, while complex interiors remain difficult.Source: §4 · p. 9, Figure 9 · p. 10

5 Conclusion

The conclusion presents FAM as a global free-boundary neural parameterization approach, applicable across geometric and topological complexities and to unstructured point clouds. It also identifies per-model overfitting and unconsidered parameterization properties as limitations.

Conclusion

  • FAM is presented as the first neural approach targeted at global free-boundary surface parameterization.It automatically learns cutting seams and adaptively deforms the UV domain.Source: §5 · p. 10
  • The approach is described as applicable across geometric and topological complexities regardless of mesh triangulation quality, including unstructured point clouds.Source: §5 · p. 10
  • Per-model overfitting prevents using existing UV-unwrapping results as ground truth for training generalizable neural models; symmetry, seam visibility, and seamless parameterization remain unconsidered.Source: §5 · p. 10

A.1 Implementation Details

FAM uses stacked MLPs without batch normalization, with LeakyReLU activations and specified network widths, thresholds, and loss weights. The appendix also reports a conformality comparison on a non-disk-topology model.

Implementation Details

  • The sub-networks use stacked MLPs without batch normalization and LeakyReLU with negative slope 0.01, with distinct widths for Deform-, Wrap-, Cut-, and Unwrap-Nets.Deform-Net uses [2, 512, 512, 512, 64] and [66, 512, 512, 512, 2]; Wrap-Net uses [2, 512, 512, 512, 64] and [66, 512, 512, 512, 6]; Cut-Net uses [3, 512, 512, 64] and [67, 512, 512, 3]; Unwrap-Net uses [3, 512, 512, 2].Source: §A.1 · p. 13
  • The reported settings include K_cut = 3, a cutting threshold of 2% of the UV bounding-box side length, and loss weights 0.01, 1.0, 0.01, and 0.01 for unwrapping, wrapping, cycle, and conformality losses.The unwrapping threshold is ϵ = 0.2 · L(Q)/N, with K_u = 8.Source: §A.1 · p. 13
  • On the nefertiti model, whose topology is not homeomorphic to a disk, SLIM has conformality 0.089 and FAM 0.117; FAM learns its cutting seam automatically.Source: Figure 10 · p. 13

A.2 Discussions about Global and Multi-Chart Surface Parameterization

The discussion contrasts global parameterization's potential for smooth transitions and distributed distortion with multi-chart methods' boundary discontinuities and packing requirements. It also notes that local parameterization suits some highly complicated surfaces.

Global and Multi-Chart Surface Parameterization

  • Global parameterization can provide smooth transitions and distribute mapping distortion across the surface, whereas multi-chart methods typically introduce patch-boundary discontinuities and visual artifacts.The discussion also connects those discontinuities to difficulties in remeshing, morphing, and compression.Source: §A.2 · p. 13
  • Chart-wise UV maps are not a complete multi-chart workflow: chart packing is needed to compose their domains, but recent neural approaches such as DiffSR and Nuvo largely skip it.The Nuvo results are presented as chart-wise UV maps.Source: §A.2 · p. 13, Figure 11 · p. 14
  • Local parameterization remains suitable for highly complicated surfaces such as ShapeNet-style CAD meshes with rich interiors and severe multi-layer issues, where simpler cropped patches can reduce per-chart distortion.Source: §A.2 · pp. 13–14

A.3 Comparison with SLIM on Non-Disk-Type Surfaces with Manually-Specified Cuts

FAM remains highly competitive with SLIM on the non-disk Nefertiti surface, even though SLIM receives a manually specified optimal cut. The comparison is explicitly described as unfair because finding optimal cuts is difficult.

A.3 Comparison with SLIM on Non-Disk-Type Surfaces with Manually-Specified Cuts

  • FAM achieves highly competitive performance against SLIM on the non-disk Nefertiti surface, even under an unfair comparison setup.SLIM was given a manually specified optimal cutting seam.Source: §A.3 · p. 14
  • The main-manuscript SLIM comparisons focus on models with open boundaries and/or disk topologies, where SLIM can be applied.Source: §A.3 · p. 14
  • Finding optimal cuts is a critical but complicated task, making the manually cut Nefertiti comparison unfair to FAM.Source: §A.3 · p. 14

A.4 Running Efficiency

The paper compares FAM and SLIM training and optimization time, reporting satisfactory learning efficiency for FAM. With a fixed number of sampled input points, FAM's costs remain basically unchanged across test models.

A.4 Running Efficiency

  • FAM's training and optimization time costs are compared with SLIM, and the authors report satisfactory learning efficiency.SLIM's official code runs on an Intel Core i7-9700 CPU.Source: §A.4 · p. 14
  • FAM's time costs basically maintain unchanged across testing models because it uses the same number of uniformly sampled input points, N.Source: §A.4 · p. 14
  • The comparison covers training and optimization time costs for different testing models.Source: §A.4 · p. 14

A.5 Comparison with Nuvo [36]

Against OptCuts, FAM is reported to be generally better while offering flexibility beyond well-behaved meshes, GPU parallelism, and smooth parameterization. The Nuvo results also reveal unstable chart assignments and omitted chart packing.

A.5 Comparison with Nuvo [36]

  • Nuvo's chart assignment is unstable, sometimes assigning spatially disconnected surface patches to the same chart.Source: §A.5 · p. 14
  • Nuvo omits chart packing; valid packing must adapt UV-domain positions, poses, and sizes through translation, rotation, and scaling.Source: §A.5 · p. 14

A.6 Comparison with OptCuts [16]

  • OptCuts jointly obtains reasonable surface cuts and UV unwrapping, but its performance is generally sub-optimal compared with FAM.Source: §A.6 · p. 14
  • FAM is described as more flexible than OptCuts because it is not limited to well-behaved meshes and can apply to point clouds.Source: §A.6 · p. 14
  • FAM also exploits GPU parallelism, is easier to implement and tune, and parameterizes smoothly beyond mesh vertices.Source: §A.6 · p. 14
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