Neural Dual Contouring

1 INTRODUCTION

• Earlier data-driven meshing methods mostly learned point-set triangulations, while NMC learned local shape priors from voxel grids and used more complex tessellation templates.The introduction notes that NMC typically outputs 4–8 times as many triangles and incurs 100× or more compute time to reconstruct a mesh.Source: §1 · pp. 1–2
• NDC predicts one vertex per grid cell and a quad for each intersected grid edge, providing a simpler structure for reproducing sharp features than NMC’s 23 vertices per cell.Classical dual contouring uses a handcrafted Quadratic Error Function; NDC replaces it with a learned vertex-location function that uses training-data context and does not require input gradients.Source: §1 · p. 2
• NDC’s learned vertex prediction avoids dual contouring’s gradient-input requirement by learning useful gradients or context from example surfaces.The network can therefore operate on voxel grids without gradients supplied as input.Source: §1 · p. 2
• UNDC predicts edge crossings without relying on corner signs, allowing sign-agnostic reconstruction of open surfaces rather than requiring signed-distance inputs and closed, watertight outputs.The broader input set includes signed and unsigned distance fields, binary voxels, and unoriented point clouds; the stated representational-complexity reduction over NMC is 23:1.Source: §1 · pp. 2–3
• NDC outperforms MC33 and NMC variants on reconstruction quality, feature preservation, triangle quality, and inference time; against NMC it uses 4–8 times fewer mesh elements and 3–20 times less inference time.The evaluation trains on ABC and tests on ABC, Thingi10K, FAUST, MGN, and Matterport3D; point-cloud experiments also report UNDC outperforming classical and learned methods.Source: §1 · p. 2

2 RELATED WORK

• Prior mesh-reconstruction work is extensive; this section focuses on isosurfacing, point-cloud reconstruction, and the dual-contouring formulation used in the paper.The authors refer readers to surveys for broader coverage [Berger et al. 2017; De Araújo et al. 2015].Source: §2 · p. 3

2.1 Isosurfacing and differentiable reconstruction

• NMC [Chen and Zhang 2021] learns feature recovery for marching cubes, while NDC combines learning with dual contouring without requiring gradients and targets multiple input types.Differentiable methods such as DMC, MeshSDF [Remelli et al. 2020], and DefTet [Gao et al. 2020] prioritize end-to-end differentiability; the authors contrast their global-feature predictions with NDC’s local model.Source: §2.1 · p. 3

2.2 Mesh reconstruction from point clouds

• Point-cloud reconstruction methods use varied priors and differ in their support for unoriented inputs and open surfaces; UNDC combines reconstruction steps in one learned process and produces open meshes directly from unoriented point clouds.The authors also describe UNDC as non-interpolatory and, with noise augmentation in training, insensitive to sampling non-uniformity and noise.Source: §2.2 · pp. 3–4

2.3 Dual Contouring (DC)

• Dual contouring samples signed-distance values at lattice vertices, finds edge zero-crossings and their gradients, then creates quad faces dual to those crossings.Unlike marching cubes’ topology enumeration and lookup tables, dual contouring uses simpler Boolean operations and better preserves sharp details, but assumes gradients are available.Source: §2.3 · p. 4

3 METHOD

• NDC predicts grid-cell vertices and corner signs with learned networks, retaining classical dual contouring’s face-generation logic without requiring normal inputs.Source: §3 · p. 4
• UNDC predicts dual faces directly rather than signs, allowing unsigned inputs, thin sub-voxel sheets, and open surfaces for partial data.Source: §3 · pp. 4–5

3.1 Encoders

• Separate encoders process distance fields, voxel occupancy, and point clouds; occupancy uses a wider receptive field to address quantization ambiguity.Source: §3.1 · p. 5

3.2 Training data preparation

• Aliased normals can make classical-DC vertex targets poorly positioned; rotations, mirroring, and sign inversion with L2 training yield zero-mean residuals.With proper data augmentation, NDC can produce meshes with better tessellation quality than the classical-DC training meshes.Source: §3.2 · p. 5, Figure 5 · p. 5

3.3 Training losses

• Networks train separately with narrow-band supervision: L2 fits pseudoground-truth vertices, while BCE trains NDC signs and an analogous loss trains UNDC crossings.Source: §3.3 · p. 5

3.4 Post-processing

• For sparse or noisy point clouds, small isolated UNDC holes are closed by flipping boundary-edge states to add quads, with negligible GPU inference-time impact.Intersection points and normals used to build pseudoground truth are not test-time inputs.Source: §3.4 · pp. 5–6, Figure 7 · p. 6

4.3 Reconstruction from SDF

• On SDF inputs, NDC and UNDC outperform MC33 and DC-est in CD and F1, and lead the compared methods in sharp-feature preservation and surface and triangle quality.The feature-preservation comparison has an exception for EF1 on FAUST; NDC and UNDC also show fewer inaccurate normals at small thresholds on ABC.Source: Table 2 · p. 7, §4.3 · p. 8
• NDC and UNDC have vertex and triangle counts similar to MC33, while NMC variants are larger and slower; simpler tessellation can merge tube walls or miss within-cell structures.At higher SDF resolution, the accuracy gap narrows, but NMC* and NMC-lite* still use more vertices and triangles and take longer to process shapes.Source: §4.3 · pp. 8–9
• SDF tests on Thingi10K and FAUST show generalization from ABC training, although MC33 has better predicted-normal quality on FAUST’s organic shapes.NDC and UNDC produce fewer small-angle triangles in the reported comparisons.Source: Table 3 · p. 7, Table 4 · p. 7, §4.3 · p. 9

4.4 Reconstruction from UDF

• With unsigned-distance inputs, UNDC reconstructs shapes, including open cloth surfaces, with results similar to SDF in the reported datasets but usually worse because signs are absent.The paper reports no comparison to prior UDF methods because it identifies UNDC as the only method known to reconstruct from UDF.Source: Figure 10 · p. 8, §4.4 · p. 9

4.5 Reconstruction from binary voxels

• Binary voxels are harder than SDF grids; NDC and UNDC outperform MC33 across most reported metrics and are faster, smaller, and better at preserving edges and corners than NMC variants.NMC* and NMC-lite* underfit this input, with worse CD and F1 than NDC and UNDC.Source: §4.5 · pp. 9–10, Table 5 · p. 9, Figure 12 · p. 9

4.6 Reconstruction from point clouds

• On point-cloud inputs, UNDC leads the compared methods on reconstruction-quality metrics and inference time, while producing few vertices and triangles and sharp features.UNDC does not use normals, although normals are supplied to methods that require them; its tests use 4,096 input points and a 64^3 output grid.Source: §4.6 · p. 10, Table 6 · p. 10, Figure 13 · p. 10

4.7 Reconstruction from noisy real scans

• On raw Matterport3D scans, UNDC produces less surface noise than Poisson, while Poisson can preserve some details but may create bubbles or trim objects incorrectly.At the shown default setting, UNDC tends to leave more holes; the comparison has no quantitative results because the dataset’s reference meshes were reconstructed by Poisson.Source: §4.7 · p. 11, Figure 14 · p. 11
• On synthetic point clouds with varying density and noise, UNDC outperforms Poisson in the reported quantitative results; its 128^3 outputs tend to be worse than 64^3 outputs.The authors attribute the higher-resolution drop to relatively greater point sparsity and noise at smaller cell sizes.Source: §4.7 · p. 11, Table 7 · p. 11, Figure 15 · p. 12

5 CONCLUSIONS

• Across multiple datasets, NDC and UNDC outperform learned and traditional state-of-the-art methods, while UNDC reconstructs open surfaces and thin structures from unsigned distance fields or unoriented point clouds.NDC uses approximately the same number of vertices and triangles as classical marching cubes, a 3–7× reduction compared with NMC.Source: §5 · p. 12

Limitations

• Dual contouring’s one-vertex-per-cell design can produce non-manifold meshes; some cases are rare and can be fixed by splitting vertices or edges or tunnelling through them.UNDC can also produce open surfaces and non-manifold fins, which may trouble downstream tasks requiring manifoldness.Source: §5 · p. 12
• NDC is not completely orientation-invariant and can generate coherence artifacts on sharp edges as objects rotate, particularly when cube-corner SDF values are near zero.The authors attribute these artifacts to neural-network continuity across transitions whose correct output would change discontinuously.Source: §5 · p. 12

Future works

• The authors identify integration with neural representations inferred from multiple images and differentiable rendering for reconstructing one-sided surfaces as future directions [Mildenhall et al. 2020].The latter possibility concerns sparse images captured from cameras inside a scene.Source: §5 · pp. 12–13

Supplementary Material

• The supplement contains details on network architectures, mask definitions, training protocols, and experimental methods.Source: Supplementary Material · p. 14
• It includes additional qualitative and quantitative results that did not fit in the main paper.Source: Supplementary Material · p. 14
• The material supplements the paper titled “Neural Dual Contouring.”Source: Supplementary Material · p. 14

1 POINT CLOUD NETWORK ARCHITECTURE DETAILS

• For each input point, the network forms a local cluster from its eight nearest neighbors, avoiding radius selection and cluster-size issues.It uses a KD-tree to find the neighbors.Source: §1 · p. 14
• The local PointNet concatenates relative coordinates with point features, applies two fully connected layers with leaky ReLU, and max-pools across each cluster.Hidden layers have 128 channels, and the residual block uses fully connected layers.Source: §1 · p. 14
• Grid-cell centers query nearby point clusters via KNN, but features are computed only for cells within three Manhattan-distance units of the closest input point.The network uses the same loss functions as UNDC.Source: §1 · p. 14

2 MASKS DEFINITIONS

• The mask definitions assume each grid cell has size one unit.Source: §2 · p. 14
• For NDC, MV marks cells whose ground-truth SDF corner vertices have differing signs; for UNDC, MV marks cells containing an edge intersected by the ground-truth shape.Other MV entries are zero, and both definitions apply to all input types.Source: §2 · p. 14
• For SDF-grid input, NDC uses input signs directly and therefore does not need MS; for binary voxels, MS marks vertices of occupied cells adjacent to unoccupied cells in the 3×3×3 neighborhood.Source: §2 · p. 14
• For UNDC, MF marks SDF/UDF-grid edges whose endpoints both have signed distances below one, while binary-voxel edges are marked when all four adjacent cells are occupied.For point-cloud input, MF marks edges in cells within three Manhattan-distance units of the closest point.Source: §2 · p. 14

3 TRAINING DETAILS

• Networks train for 400 epochs with SDF/voxel inputs or 250 epochs with point-cloud inputs, using batch size 1 per shape.Optimization uses Adam at learning rate 0.0001; the rate is halved every 100 epochs.Source: §3 · p. 14
• For noisy real scans, the network replaces three 3×3×3 convolution layers with eight residual blocks and trains on the ABC set.Training augmentation includes random scaling and translation, alongside the paper's other augmentations.Source: §3 · p. 14
• Noisy-scan training adds Gaussian noise with σ=0.5 to input point clouds, assuming each output-grid cell is a unit cube.Source: §3 · p. 14

4 DETAILS OF THE METHODS IN EXPERIMENT “RECONSTRUCTION FROM POINT CLOUDS”

• Ball-pivoting connects existing points into oriented triangles, while Screened Poisson builds an implicit field from points and normals.The experiments use Open3D implementations; Poisson uses octree maximum depth 8.Source: §4 · p. 14
• SIREN fits a neural implicit function to each shape from scratch and requires point normals [Sitzmann et al. 2020].The authors report a 10% chance of a shell covering the output, sometimes requiring retraining; reported inference time assumes first-try success.Source: §4 · p. 15
• Local implicit grid reconstructs overlapping point-cloud blocks separately with optimized neural implicit fields, then combines them [Jiang et al. 2020].It requires point normals; experiments include released ShapeNet-pretrained weights and a version trained on ABC.Source: §4 · p. 15
• ConvONet reconstructs an implicit field without point normals, and the experiments compare its 3-plane and grid configurations plus ShapeNet-pretrained weights [Peng et al. 2020].The three models use 256^3 output resolution; the 3-plane features use 128^2 resolution and the grid uses 64^3 resolution.Source: §4 · p. 15
• ConvONet's local-backbone variant produces artifacts in featureless regions, while pretrained ConvONet tends to turn single-face walls into thin volumetric plates.The authors attribute the local variant's failure to ConvONet needing a non-local receptive field to determine inside versus outside.Source: §4 · p. 15

5 ROBUSTNESS TO TRANSLATION AND ROTATION

• All methods are tested on a shape while it is moved and rotated inside the sampling grid.Source: §5 · p. 15

6 ADDITIONAL QUALITATIVE RESULTS

• Additional qualitative results visualize mesh reconstruction from SDF grids, binary occupancy grids, and point clouds.Source: §6 · p. 15
• The visualizations use 64^3-resolution implicit fields and point clouds containing 4,096 points each.Source: §6 · p. 15
• FAUST results are omitted because the human-body meshes are all very similar to the example shown in the paper.Source: §6 · p. 15

• Figures 18 and 19 show distance-field and binary-occupancy reconstructions, with surface artifacts noted in NMC-lite* and NMC*.Source: Figure 18 · p. 16, Figure 19 · p. 17
• Tables 9–11 report SDF/UDF-input comparisons on ABC, Thingi10K, and FAUST, respectively.Source: §7 · p. 15, Table 9 · p. 19, Table 10 · p. 19
• Tables 12–14 report binary-occupancy-input comparisons on ABC, Thingi10K, and FAUST, respectively.Source: §7 · p. 15, Table 12 · p. 20, Table 13 · p. 20
• Tables 15–17 report point-cloud comparisons on ABC, Thingi10K, and FAUST; Figure 20 shows reconstructions from 4,096-point clouds.For the table comparisons, “+n” marks methods that additionally require point normals.Source: §7 · p. 15, Figure 20 · p. 18, Table 15 · p. 21