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HilbertGen-3D:Hilbert--Multifractal Conditioning for Topology-Aware 3D Generation
Ziyi Zhang, Shixuan Li, Yorie Nakahira, Paul Bogdan, Guannan Qu
TL;DR
Local voxel- or patch-based 3D generation may miss global structure, motivating conditioning that guides denoising with global information. HilbertGen-3D encodes multi-scale voxel mass with Hilbert and multifractal signals and injects them into a diffusion denoiser. The paper reports HMC as a promising strategy for global coherence and structural fidelity, with a Car-category comparison showing improvements in 1-NNA deviation and coverage over TopoDiT-3D.
Problem
Many 3D point-cloud generators rely on local geometric operators and may produce locally plausible shapes with weaker global coherence, motivating explicit global structural guidance.
Source: §1 · p. 1Method
HilbertGen-3D conditions a diffusion denoiser on Hilbert-serialized voxel mass and multifractal descriptors through global embeddings, multi-scale tokens, and bottleneck fusion.
Source: §3.2 · p. 3, §4.1 · p. 6, §4.2 · p. 7Results
Experiments suggest HMC is promising for global coherence and structural fidelity; for Car, Ours-VAE improves 1-NNA-CD and 1-NNA-EMD over TopoDiT-3D by 6.81 and 8.90 percentage points.
Source: §5.2 · p. 9, §6 · p. 10Takeaways & Limitations
Locality-preserving spatial serialization combined with multi-resolution scaling statistics is presented as an effective alternative to purely local conditioning schemes.
Source: §6 · p. 10
Abstract
from arXiv · showhide
High-fidelity 3D generative modeling underpins game asset creation, medical imaging, and VR/AR environmental design, where geometric structure directly influences perception and decision-making. Diffusion models represent a powerful paradigm for point cloud generation. However, existing approaches rely heavily on local voxel or patch features and often miss global structural organization, and much less work has studied how global conditioning can be used to improve generation quality. In this paper, we introduce HilbertGen-3D, a novel topology-aware conditional diffusion framework based on Hilbert-Multifractal Conditioning (HMC). HMC computes multi-scale voxel measures, serializes them via a 3D Hilbert curve to expose structured spatial dependencies, and estimates the generalized fractal dimensions from scale-dependent partition functions. We encode these signals as global embeddings and multi-scale tokens, and inject them into a diffusion denoiser through an efficient bottleneck fusion module. Across the Chair, Airplane, and Car categories of ShapeNet, our two conditioning variants reduce the absolute deviation of 1-Nearest Neighbor Accuracy (1-NNA) from 50% by up to 8.90 percentage points and increase coverage by up to 5.88 percentage points relative to TopoDiT-3D. This shows that combining local density cues with global Hilbert-multifractal structure provides a simple and effective recipe for high-fidelity and diverse 3D generation.
1 Introduction
HilbertGen-3D proposes HMC to make global structural information explicit in 3D point-cloud denoising. It combines Hilbert-serialized voxel mass with multifractal summaries and injects the resulting signals into a diffusion model.
1 Introduction
- The paper identifies locally driven architectures as a limitation and proposes explicitly guiding denoising with global structural information.The motivation notes potential coherence challenges for shapes with thin parts, holes, branches, or strongly nonuniform mass distributions across scales.Source: §1 · p. 1
- HMC serializes multi-scale voxel mass with a 3D Hilbert mapping and estimates generalized fractal dimensions to summarize how mass varies across scales.Together, the ordered sequences and scale-dependent statistics form the shape representation.Source: §1 · p. 2
- A global multifractal descriptor and windowed Hilbert embeddings become an embedding and conditioning tokens, injected into the denoiser through a memory-efficient bottleneck fusion module.The paper reports improved structural fidelity, sample diversity, and global coherence relative to purely local baselines.Source: §1 · p. 2
2 Related Works
The related-work section positions HMC against both topology-aware conditioning and Hilbert-based serialization. Its distinction is to use Hilbert traversal and multifractal mass descriptors as explicit structural priors for generation.
3D generative models
- Persistent homology encodes components, loops, and voids, whereas HMC represents voxel mass distribution, density heterogeneity, and local spatial neighborhoods.The passage states that HMC can exploit prior information about reference density structure to improve generation quality and diversity.Source: §2 · p. 2
Hilbert curves and structure-preserving serialization for 3D data
- Unlike prior Hilbert-order methods used for tokenization, grouping, or sequence modeling, this method uses serialized multi-scale voxel measures and Hilbert-window embeddings as denoiser conditioning.It combines global multifractal summaries with local Hilbert-window embeddings.Source: §2 · p. 2
3 Preliminaries
The preliminaries introduce conditional DDPMs for point-cloud generation and Hilbert curves as the spatial ordering used by HMC. The sections explain the reverse denoising process and how Hilbert serialization preserves multi-scale spatial organization.
3 Preliminaries
- The preliminaries frame the method around 3D point-cloud DDPMs and introduce the Hilbert curve as the locality-preserving space-filling curve used for conditioning.The conditional input C includes the HMC representation.Source: §3 · p. 2, §3.1 · p. 3
3.1 Denoising Diffusion Probabilistic Model (DDPM) for 3D Point Clouds
- Conditional DDPM generation reverses scheduled Gaussian corruption through learned transitions, with a denoiser predicting injected noise under conditioning C.Training uses uniform timestep weighting as a reweighted noise-prediction surrogate; sampling starts from Gaussian noise and holds C fixed through reverse transitions.Source: §3.1 · p. 3
3.2 Hilbert Curves
- Hilbert traversal orders voxel cells into a sequence, with recursive ordering tending to keep spatially nearby voxels close in that sequence across resolutions.The method uses a discrete 3D traversal; it permutes voxel values and adds no information loss beyond voxelization.Source: §3.2 · pp. 3–4
4 Methodology
HilbertGen-3D conditions diffusion denoising on multi-scale spatial structure encoded by Hilbert-ordered voxel measures and generalized fractal dimensions. Separate token and global-descriptor pathways are fused with geometry features through a bottleneck.
4 Methodology
- The denoiser voxelizes point clouds into patch tokens and predicts noise before unpatchifying and devoxelizing the output.The overview places the HMC conditions alongside the noisy geometry tokens in the diffusion pipeline.Source: §4 · p. 5, Figure 2 · p. 5
4.1 Hilbert serialization and generalized dimensions
- Multi-scale voxel measures are Hilbert-serialized, while generalized dimensions summarize density scaling; separate encoders produce HMC tokens and a global embedding.The token pathway encodes windowed sequences, and the global pathway embeds the generalized-dimension descriptor.Source: §4.1 · pp. 5–6
4.2 Integration into Diffusion Architectures
- A fixed-size Perceiver-style bottleneck fuses geometry tokens with HMC context, keeping DiT computation independent of the full HMC token length.DiT blocks update the bottleneck latents, which then update geometry tokens; HMC tokens provide cross-attention context and the global embedding modulates adaptive layer normalization.Source: §4.2 · pp. 6–7
5 Experiments
Across Chair, Airplane, and Car, both HilbertGen-3D conditioning variants improve 1-NNA relative to TopoDiT-3D, while coverage advantages depend on category and conditioning source. Airplane ablations support the combined design, with metric-specific differences among components.
5.1 Experiment Setup
- Experiments train HilbertGen-3D on ShapeNet Chair, Airplane, and Car categories and evaluate 1-NNA and Coverage (COV); lower 1-NNA scores indicate superior generative performance in both quality and diversity.COV quantifies the proportion of ground-truth reference point clouds that correspond to at least one generated shape.Source: §5.1 · pp. 7–8
5.2 Comparison to Prior Works
- Across Chair, Airplane, and Car, both variants improve 1-NNA relative to TopoDiT-3D, while coverage gains depend on category and conditioning source.VAE conditioning yields stronger 1-NNA on Chair and Car, whereas bank conditioning gives higher coverage there; the pattern reverses on Airplane.Source: §5.2 · pp. 8–9, Table 1 · p. 9
- On Car, Ours-VAE improves 1-NNA-EMD by 8.90 percentage points and Ours-bank improves COV-EMD by 5.88 percentage points over TopoDiT-3D.These are improvements in absolute distance from 50 for 1-NNA and increases for COV.Source: §5.2 · p. 9, Table 1 · p. 9
5.3 Ablation analysis.
- On Airplane, removing Hilbert tokens worsens all four metrics, bottleneck removal most harms CD-based 1-NNA and coverage, and the global embedding does not improve every metric.The full model has the best 1-NNA-CD and both coverage metrics, while the variant without generalized dimensions has the closest 1-NNA-EMD to 50.Source: §5.3 · p. 10
6 Conclusion
HilbertGen-3D uses Hilbert–Multifractal Conditioning (HMC) for point-cloud diffusion, and experiments suggest it can improve global coherence and structural fidelity. The conclusion notes that the approach exploits prior information about overall density distributions, while persistent-homology approaches may perform well otherwise.
6 Conclusion
- Experiments suggest HMC is a promising conditioning strategy for improving global coherence and structural fidelity in generated 3D shapes.The framework combines locality-preserving spatial serialization with multi-resolution scaling statistics as an alternative to purely local conditioning schemes.Source: §6 · p. 10
- HilbertGen-3D is intended for settings where prior information about the overall density distribution can be exploited.The conclusion notes that alternative approaches such as persistent homology may perform well otherwise.Source: §6 · p. 10
- Future directions include distilling the diffusion model to reduce computational cost and using Bayesian techniques to further improve generation accuracy.Source: §6 · p. 10
A Additional results
In the additional Lamp and Vessel experiments, both HMC conditioning variants outperform TopoDiT-3D across all four metrics. Their relative strengths vary by category and metric: neither conditioning source is uniformly superior.
Additional results
- Both HMC variants outperform TopoDiT-3D on all four metrics for Lamp and Vessel.Table 3 reports percentages; values nearer 50 are better for 1-NNA, while higher values are better for COV.Source: Table 3 · p. 14, A Additional results · p. 14
- For Lamp, Ours-VAE reaches 1-NNA-CD 49.28, an 8.11-point reduction in distance from 50 versus TopoDiT-3D, while Ours-Bank leads COV-EMD at 58.19.Ours-VAE also leads COV-CD at 55.75 versus 55.60 for Ours-Bank and 50.43 for TopoDiT-3D; both variants attain 1-NNA-EMD 48.78, a 6.53-point improvement.Source: A Additional results · p. 14
- For Vessel, Ours-Bank leads COV-CD and COV-EMD at 54.17 and 58.33, improving on TopoDiT-3D by 21.36 and 30.21 points.Ours-VAE achieves the best 1-NNA-CD at 56.08, reducing distance from 50 by 16.13 points; Ours-Bank achieves the best 1-NNA-EMD at 49.05, a 27.22-point improvement over TopoDiT-3D.Source: A Additional results · p. 14
- Across Lamp and Vessel, Ours-VAE has better CD-based 1-NNA, while Ours-Bank has higher EMD-based coverage on both and higher CD-based coverage on Vessel.Neither conditioning source is uniformly superior across the reported metrics.Source: A Additional results · p. 14
B Full Parameter Sensitivity Results
Parameter sensitivity results show trade-offs across evaluation criteria rather than one uniformly best setting. The comparisons are limited to Ours-Bank on Airplane under a 2000-epoch recipe.
Resampler depth
- Depth 4 reduces 1-NNA-CD’s distance from 50 by 3.88 percentage points versus depth 3 and raises COV-CD and COV-EMD by 4.21 and 1.29 points, while depth 3 has the better 1-NNA-EMD.Depth 2 performs worse, so the observed depth trend is not monotonic.Source: B Full Parameter Sensitivity Results · p. 14
MLP ratio
- MLP ratio 3 improves all four reported means over ratio 4, while ratio 3.5 has the highest COV-CD, just 0.02 points above ratio 3.The authors retained ratio 4 in the hope that a smaller step size, larger batch, and more epochs would benefit the more complicated model.Source: B Full Parameter Sensitivity Results · pp. 14–15
Number of bottleneck latents
- With 96 bottleneck latents, 1-NNA-EMD and COV-EMD are best, while 128 latents has the best 1-NNA-CD and 96 latents has slightly lower COV-CD.The results show no consistent advantage for the largest bottleneck.Source: B Full Parameter Sensitivity Results · p. 15
Scale selection
- The full scale set {3, 4, 5} favors 1-NNA-CD and both coverage metrics, whereas {3, 5} yields the closest 1-NNA-EMD to 50.Because all tested configurations use multiple scales, these results do not establish an advantage over single-scale conditioning.Source: B Full Parameter Sensitivity Results · pp. 15–16
Moment-order selection
- Moment set {1, 2, 3} gives the best 1-NNA-CD and COV-EMD, while {1} gives the best COV-CD and {1, 2, 3, 4} the best 1-NNA-EMD.Adding moment orders does not improve metrics monotonically; {1} retains the information-dimension descriptor and does not remove the global HMC descriptor.Source: B Full Parameter Sensitivity Results · p. 16
Interpretation and scope
- Groupwise tests do not establish the performance of combined favorable settings, and these results concern Ours-Bank on Airplane under the 2000-epoch recipe.Evaluation seeds do not quantify variation across independently trained models, so small differences should be read as numerical rankings, not evidence of statistical significance.Source: B Full Parameter Sensitivity Results · pp. 14, 16
C HMC-DiT3D for multi-category generation.
In chair generation, HilbertGen-3D is comparatively stable as training expands to additional categories, particularly with bank-sampled conditions. The baselines deteriorate more as categories are added, while VAE-sampled conditions show a slight downward trend.
Multi-category generation
- Adding training categories reduces baseline chair COV-CD and raises 1-NNA-CD: DiT-3D-S/4 changes from 54.76 to 41.46 and 51.99 to 75.80, while TopoDiT-3D-S/4 changes from 58.28 to 50.90 and 49.84 to 68.75.Models were trained on {Chair}, {Chair, Car}, or {Chair, Car, Airplane} and tested on Chair; the class label is the only category signal specified for the baselines.Source: C HMC-DiT3D for multi-category… · p. 16
- For HMC-DiT3D, under bank-sampled conditions, chair 1-NNA-CD stays within 48.81–49.36 and COV-CD within 57.62–57.93 across three training sets, with no clear upward or downward trend.Source: C HMC-DiT3D for multi-category… · p. 17
- For VAE-sampled conditions, chair 1-NNA-CD rises from 50.25 to 51.93 and COV-EMD falls from 54.83 to 53.96 as categories are added.The authors attribute HilbertGen-3D’s relative robustness to chair-specific multi-scale Hilbert measure sequences that describe geometry explicitly.Source: C HMC-DiT3D for multi-category… · p. 17
D Computational Cost Analysis
Under the S/4 configuration, HilbertGen-3D uses less training memory and samples faster than the reported DiT-3D and TopoDiT-3D baselines. On Chair, its per-epoch training time is also substantially shorter than the baseline setting.
Computational cost
- HilbertGen-3D samples in 2.60 s/it, faster than both reported S/4 baselines, and uses less training memory than both.The comparison focuses on S/4 rather than sweeping model scales.Source: D Computational Cost Analysis · p. 17
- On Chair, HilbertGen-3D trains in 2.3 seconds per epoch versus 11.8 seconds for the baseline, a 5.13× speedup and 80.5% less per-epoch time.Together with lower peak memory, the paper reports efficiency gains during training as well as sampling.Source: D Computational Cost Analysis · p. 17
E Result Visualization
The figures provide generated-shape samples across five categories and qualitative denoising-process visualizations for three categories.
E Result Visualization
- Figures 4–8 show HilbertGen-3D samples for Chair, airplane, car, vessel, and lamp.Source: Figure 4 · p. 18, Figure 5 · p. 18, Figure 6 · p. 19
F Qualitative visualizations of denoising process
- Figures 9–11 show qualitative denoising-process visualizations for chair, car, and airplane generation.Source: Figure 9 · p. 21, Figure 10 · p. 21, Figure 11 · p. 22