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Neural Marching Cubes

Zhiqin Chen, Hao Zhang

arXiv:2106.11272v3cs.GRcs.CVcs.LG

TL;DR

Classical Marching Cubes uses cube-local templates and heuristic assumptions that limit recovery of features spanning nearby cubes. NMC redesigns tessellations and learns mesh positions and topologies from training data; it reports improved feature preservation and local topology reconstruction compared with other isosurfacing methods.

  • Problem

    Classical MC variants use local tessellations and assumptions such as trilinearity, while sharp features can depend on coherence across neighboring cubes.

    Source: §1 · p. 2

  • Method

    NMC redesigns tessellations and learns vertex positions and local mesh topologies from training meshes using a compact per-cube representation and a 3D network.

    Source: §1 · pp. 2–3

  • Results

    NMC shows superior performance preserving sharp and soft edges, corners, and thin structures compared with other isosurfacing algorithms using uniform-grid signed distances or binary occupancies.

    Source: §5 · p. 13

  • Takeaways & Limitations

    A limited receptive field lets the network learn local features and generalize to new shapes and datasets.

    Source: §1 · p. 3

    NMC is sensitive to rotation, which the authors mainly attribute to training on ABC shapes that are mostly aligned with coordinate axes.

    Source: §5 · p. 13

Abstract

from arXiv · show

We introduce Neural Marching Cubes (NMC), a data-driven approach for extracting a triangle mesh from a discretized implicit field. Classical MC is defined by coarse tessellation templates isolated to individual cubes. While more refined tessellations have been proposed, they all make heuristic assumptions, such as trilinearity, when determining the vertex positions and local mesh topologies in each cube. In principle, none of these approaches can reconstruct geometric features that reveal coherence or dependencies between nearby cubes (e.g., a sharp edge), as such information is unaccounted for, resulting in poor estimates of the true underlying implicit field. To tackle these challenges, we re-cast MC from a deep learning perspective, by designing tessellation templates more apt at preserving geometric features, and learning the vertex positions and mesh topologies from training meshes, to account for contextual information from nearby cubes. We develop a compact per-cube parameterization to represent the output triangle mesh, while being compatible with neural processing, so that a simple 3D convolutional network can be employed for the training. We show that all topological cases in each cube that are applicable to our design can be easily derived using our representation, and the resulting tessellations can also be obtained naturally and efficiently by following a few design guidelines. In addition, our network learns local features with limited receptive fields, hence it generalizes well to new shapes and new datasets. We evaluate our neural MC approach by quantitative and qualitative comparisons to all well-known MC variants. In particular, we demonstrate the ability of our network to recover sharp features such as edges and corners, a long-standing issue of MC and its variants. Our network also reconstructs local mesh topologies more accurately than previous approaches.

1 INTRODUCTION

Classical Marching Cubes relies on cube-local templates and assumptions such as trilinearity, which can yield poor estimates near sharp features that depend on neighboring cubes. Neural Marching Cubes redesigns tessellations and learns vertex positions and local topologies from training meshes, using a compact per-cube representation and a 3D network.

1 INTRODUCTION

  • Classical MC’s trilinearity assumption can yield poor estimates of the implicit field, especially near sharp features.Its tessellations and decisions are confined to individual cubes, while sharp edges reveal coherence across neighboring cubes.Source: §1 · p. 2
  • NMC redesigns tessellation templates and learns vertex positions and local mesh topologies from training meshes to use context from nearby cubes.The approach targets geometric features such as sharp edges and corners.Source: §1 · p. 2
  • NMC’s per-cube representation compactly encodes applicable topological cases and vertex positions for neural processing.Added vertices are associated with corner vertices, enabling tessellations to be generated by design guidelines.Source: §1 · p. 2, Figure 2 · p. 2
  • A 3D ResNet trained on ground-truth meshes predicts topology and geometry, with a limited receptive field intended to support generalization to new shapes and datasets.The network uses separate topological and geometric losses on the representation’s binary and float parts.Source: §1 · p. 3
  • NMC recovers sharp features without additional inputs beyond a uniform grid of implicit-field values and more faithfully reconstructs local topologies near thin structures and closeby surface sheets.The paper also evaluates a simplified NMC-lite variant and training on noisy inputs.Source: §1 · p. 3

2 RELATED WORK

The related work describes limitations of classical MC variants and other isosurfacing methods, then distinguishes NMC from neural geometry and differentiable MC approaches. NMC learns tessellations from training meshes while taking a discrete implicit field as input.

2.1 Marching Cubes (MC) and Variants

  • MC variants address ambiguous or non-manifold tessellations, but trilinear vertex placement can still estimate surface features poorly; earlier sharp-feature methods often required extra intersection or normal information.MC 33 enumerates topological cases under trilinearity, while later work introduced additional face and interior vertices.Source: §2.1 · p. 3

2.2 Other Isosurfacing Algorithms

  • Other isosurfacing algorithms can preserve sharp features, but methods such as dual contouring require gradients or other inputs, while Marching Tetrahedra was not designed for sharp-feature recovery.Dual marching cubes queries the implicit function during reconstruction.Source: §2.2 · p. 3

2.3 Neural Geometry Learning

  • Neural shape representations often need mesh post-processing, while deformable mesh methods depend on fixed templates and cannot alter their topology.NMC instead takes a discretized implicit field and directly outputs a triangle mesh.Source: §2.3 · pp. 3–4
  • Unlike DMC and MeshSDF, NMC learns refined MC tessellations from training meshes and uses a limited receptive field rather than global features.DMC uses only 8 of the original MC’s 15 tessellation templates, whereas NMC targets sharper features and broader topological granularity.Source: §2.3 · p. 4

3 NEURAL MARCHING CUBES

NMC encodes local topology and vertex positions in fixed-format per-cell representations, then designs compatible tessellations and trains networks using meshes prepared in that representation.

3.1 2D NMC: representation in a 2D square

  • The 2D square code separates topology into a boolean part and vertex positions into a float part, adding face vertices to represent geometric features.Its five boolean values encode corner and face signs; the float part stores edge- and face-vertex positions.Source: §3.1 · p. 4

3.2 3D NMC: representation in a 3D cube

  • The 3D cube code represents topology with corner and face signs plus a tunnel flag, and stores added edge, face, and interior vertex positions.For two same-sign connected components, the representation assumes either one tunnel or none; other component counts assume no tunnel.Source: §3.2 · p. 5

3.3 3D NMC: tessellating a 3D cube

  • NMC tessellations follow triangle, sign-separation, and manifold constraints, using 37 unique cases and guidelines for connecting cube interiors.The guidelines require tunnels to contain the small center cube, connect available face vertices to corresponding interior vertices without tunnels, and avoid long triangles.Source: §3.3 · p. 5

3.4 Data preparation

  • Training data preparation assigns each cube a topological case, rejects unrepresentable structures, masks invalid or irrelevant values, and optimizes template vertices against the original mesh.The case test uses a 9×9×9 signed-distance grid; edge vertices are found from mesh intersections, while face and interior vertices are optimized.Source: §3.4 · pp. 5–6

3.5 NMC network and objective functions

  • A 3D ResNet predicts boolean topology and float vertex positions with separate losses; binary voxel inputs use a larger receptive field and an added smoothness term.The smoothness term assumes ground-truth surfaces generally align with coordinate axes and encourages output surfaces to do so.Source: §3.5 · pp. 6, 9

4 RESULTS AND EVALUATION

Across SDF and binary-voxel inputs, NMC generally outperforms established MC variants, with strong feature preservation and more accurate local topologies. Its performance varies with input resolution and training data, while finer tessellations and NMC-lite expose a fidelity–complexity trade-off.

Mesh extraction from SDF grids

  • On ABC SDF inputs, NMC outperforms MC33 and Lopes2003 on all quantitative measures; even 8×-higher-resolution MC33 underperforms except for CD, while NMC is superior at edge preservation.Qualitatively, NMC recovers sharp edges and corners, smooth curves, and thin structures that MC33 and Lopes2003 may reconstruct with incorrect local topologies.Source: §4 · p. 10, Table 1 · p. 10

Organic shapes

  • On 100 FAUST human-body meshes, NMC and its variant outperform MC33 and Lopes2003 while learning to predict smooth and sharp features.The authors suggest that adding more organic shapes to the training set should further improve performance on such inputs.Source: §4 · p. 10, Table 2 · p. 10

Varying input grid resolutions

  • NMC consistently outperforms MC33 across SDF grid resolutions up to 128^3, although quality degrades at lower resolutions and the margin diminishes as resolution increases.The network was trained on meshes obtained at 32^3 and 64^3 resolutions; at lower resolution, a cube may contain geometric varieties beyond NMC’s covered topological cases.Source: §4 · p. 10, Figure 11 · p. 11

Mesh extraction from binary voxels

  • On binary voxels, NMC outperforms other MC variants and the trilinear interpolant on all reported measures except NC, but voxel ambiguities can still prevent faithful reconstruction.The comparison is on the ABC test set; identical voxel grids can correspond to different geometries, such as a rectangular or circular rod.Source: Table 3 · p. 10, §4 · p. 11, Figure 12 · p. 12

Generalizability

  • On Thingi10K test shapes, the ABC-trained network shows a similar performance boost over other methods, but does not significantly outperform them on 32^3 binary-voxel inputs.The authors suggest this exception may reflect overfitting to ABC as the 32^3 input resolution approaches the voxel network’s 15^3 receptive-field size.Source: §4 · p. 11, Table 4 · p. 12, Table 5 · p. 12

Comparison to DMC

  • Even when overfit to one input shape, DMC does not recover sharp features in the comparison, which the authors attribute mainly to its limited set of classical tessellations for simple topologies.The geometry near edges bulges as DMC minimizes point-to-triangle distances without tessellations that support sharp edges.Source: §4 · p. 12, Figure 13 · p. 12

Input and output complexities; NMC vs NMC-lite

  • NMC’s more refined tessellation templates improve reconstruction quality but increase triangle counts; NMC-lite offers a lower-complexity alternative with similar quantitative performance.NMC tends to produce better visual quality and triangle quality, while NMC-lite may miss fine features such as thin structures.Source: §4 · pp. 12–13

Noisy inputs

  • Training NMC on noisy inputs significantly improves results on a noisy test set, whereas a model trained on clean inputs may fail on noisy inputs.The authors report that Chamfer Distance is sensitive to outliers, while F1 is more robust to them.Source: §4 · p. 13, Table 6 · p. 13

5 CONCLUSION, LIMITATION, AND FUTURE WORK

NMC reconstructs geometric features and local mesh topologies more accurately than previous isosurfacing approaches, without requiring normal inputs or mesh postprocessing. Its scope is bounded by rotation sensitivity, unrepresentable cases, and self-intersections, while its representation may extend to other shape tasks.

5 CONCLUSION, LIMITATION, AND FUTURE WORK

  • NMC preserves sharp and soft edges, corners, and thin structures better than other isosurfacing algorithms using uniform-grid signed distances or binary occupancies.The network directly outputs meshes without postprocessing or additional inputs such as normal information.Source: §5 · p. 13
  • NMC is sensitive to rotation because its ABC training shapes are mostly axis-aligned; random rotation augmentation would require longer training and larger networks.The method also cannot represent cases with multiple intersections along a cube edge under its one-intersection assumption.Source: §5 · p. 13, Figure 15 · p. 14, Figure 16 · p. 14
  • On 643-resolution ABC SDF inputs, 32.7% of NMC meshes contain self-intersections, involving 0.0086% of triangles, or about 7.39 triangles per shape.NMC-lite has corresponding values of 29.6%, 0.0078%, and 3.40 triangles per shape.Source: §5 · p. 13
  • The representation may support shape upsampling and generation, while adaptive vertex and triangle allocation could better match feature complexity and control triangle counts.Simple planes need few triangles, whereas curved surfaces need more.Source: §5 · pp. 13–14

A COMPARISON OF DIFFERENT SMOOTHNESS TERMS

The smoothness-term comparison addresses ambiguity in surfaces generated from binary voxels. It compares regularizers with different edge-gradient and angle objectives, and adopts L(5) based on visual and overall quantitative performance.

A COMPARISON OF DIFFERENT SMOOTHNESS TERMS

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