HilbertGen-3D:Hilbert--Multifractal Conditioning for Topology-Aware 3D Generation

Ziyi Zhang; Shixuan Li; Yorie Nakahira; Paul Bogdan; Guannan Qu

cs.CVOct 15, 2026

HilbertGen-3D:Hilbert--Multifractal Conditioning for Topology-Aware 3D Generation

Authors: Ziyi Zhang, Shixuan Li, Yorie Nakahira, Paul Bogdan, Guannan Qu

Organizations: Department of Electrical and Computer Engineering, Carnegie Mellon University · Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California

Abstract

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.

Figures & tables

Figure 1: Comparison of point clouds (top row), their persistence diagrams (middle row), and Hilbert-ordered mass profiles associated with the sequences $c_{s}[t]$ . The examples illustrate changes in spatial mass concentration despite similar coarse topological structure.

Figure 2: Overview of HilbertGen-3D. Multi-scale voxel measures are serialized along 3D Hilbert curves to construct Hilbert-ordered sequences, while generalized fractal dimensions form the global descriptor

D

. Separate encoders produce HMC tokens

C

and a global HMC embedding

g_{\mathrm{HMC}}

, respectively. The noisy point cloud

X_{t}

is embedded into geometry tokens

H_{t}

, which are fused with the projected HMC tokens through bottleneck latents

B

. Successive DiT blocks update

B

, with adaptive layer normalization conditioned on the timestep embedding

E_{\mathrm{time}}(t)

and the projected global embedding

P_{g}(g_{\mathrm{HMC}})

. The processed latents update the geometry tokens, and the output head unpatchifies and devoxelizes the predicted features to obtain the noise estimate

\varepsilon_{\theta}(X_{t},t,\mathcal{C})

for DDPM training and reverse-diffusion sampling.

Figure 3: HilbertGen-3d results compared with DiT-3D and TopoDiT-3D. All experiments are conducted under same scale.

Method	Chair				Airplane				Car
	1-NNA ( $50$ is optimal)		COV ( $\uparrow$ )		1-NNA ( $50$ is optimal)		COV ( $\uparrow$ )		1-NNA ( $50$ is optimal)		COV ( $\uparrow$ )
	CD	EMD	CD	EMD	CD	EMD	CD	EMD	CD	EMD	CD	EMD
r-GAN [ 1 ]	83.69	99.70	24.27	15.13	98.40	96.79	30.12	14.32	94.46	99.01	19.03	6.539
l-GAN (CD)	68.58	83.84	41.99	29.31	87.30	93.95	38.52	21.23	66.49	88.78	38.92	23.58
l-GAN (EMD)	71.90	64.65	38.07	44.86	89.49	76.91	38.27	38.52	71.16	66.19	37.78	45.17
PointFlow [ 26 ]	62.84	60.57	42.90	50.00	75.68	70.74	47.90	46.41	58.10	56.25	46.88	50.00

Table 1: Experimental results on 3D point cloud generation across chair, airplane, and car categories. Metric values are reported as percentages. HilbertGen-3D (VAE) uses VAE-generated HMC conditions, while HilbertGen-3D (bank) uses bank-sampled HMC conditions. For 1-NNA, values closer to 50 are better; for COV, higher values are better. The best entries in each column are shown in bold, including ties. Superscripts for both variants report improvements relative to TopoDiT-3D in percentage points: reductions in absolute distance from 50 for 1-NNA, and increases for COV. Positive improvements are green, negative changes are red, and ties are gray.

Method	1-NNA( $50$ is optimal)		COV( $\uparrow$ )
Method	CD	EMD	CD	EMD
Ours	55.57	53.66	60.98	59.27
w/o HMC tokens	57.33	54.76	58.50	57.67
w/o global HMC embedding	55.84	53.58	60.24	58.25
w/o bottleneck fusion	60.00	54.54	55.92	55.83

Table 2: Ablation study on the Airplane category. For 1-NNA, values closer to 50 is better, and for COV, higher values are better.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Lamp				Vessel
	1-NNA ( $50$ optimal)		COV ( $\uparrow$ )		1-NNA ( $50$ optimal)		COV ( $\uparrow$ )
	CD	EMD	CD	EMD	CD	EMD	CD	EMD
TopoDiT-3D	58.83	57.75	50.43	49.13	72.21	78.96	32.81	28.12
Ours-VAE	49.28 +8.11	48.78 +6.53	55.75 +5.32	56.61 +7.48	56.08 +16.13	51.74 +27.22	49.31 +16.50	53.12 +25.00
Ours-Bank	47.49 +6.32	48.78 +6.53	55.60 +5.17	58.19 +9.06	56.51 +15.70	49.05 +28.01	54.17 +21.36	58.33 +30.21

Appendix

Table 3: Results on the additional Lamp and Vessel categories. All metric values are reported as percentages. Ours-VAE uses VAE-generated HMC conditions, while Ours-Bank uses bank-sampled HMC conditions. For 1-NNA, values closer to 50 are better; for COV, higher values are better. The best entries in each column are shown in bold, including ties. Green superscripts report improvements relative to TopoDiT-3D in percentage points: reductions in absolute distance from 50 for 1-NNA, and increases for COV.

Parameter	Setting	1-NNA ( $50$ optimal)		COV( $\uparrow$ )
		CD	EMD	CD	EMD
Resampler depth	1	63.67	55.75	55.35	56.88
	2	66.03	64.99	48.58	46.10
	3	59.45	53.09	56.77	57.98
	4 †	55.57	53.66	60.98	59.27
MLP ratio	2.5	55.96	53.01	61.02	59.02

Appendix

Table 4: Parameter sensitivity analysis on the Airplane category (mean over four evaluation seeds: 0, 11, 12, 13). All runs use the normalized training recipe with a batch size of 128 for 2000 epochs; conditions are sampled from the train-split HMC bank (Ours-Bank) and scored under the validation protocol with

N=664

samples, exact EMD, and no size correction. The baseline setting of each group is marked with

\dagger

. 1-NNA is better when closer to 50, while higher COV is better. The best entry in each column within a group is shown in bold, including ties. All values are multiplied by 100 for readability.

Model	Train	Test	1-NNA(50 optimal)		COV( $\uparrow$ )
Model	Class	Class	CD	EMD	CD	EMD
DiT-3D-S/4	Chair	Chair	51.99	49.94	54.76	57.37
	Chair,Car	Chair	69.87	65.71	45.28	52.20
	Chair,Car,Airplane	Chair	75.80	74.72	41.46	47.89
TopoDiT-3D-S/4	Chair	Chair	49.84	46.46	58.28	60.69
	Chair,Car	Chair	60.46	60.31	56.17	56.92

Appendix

Table 5: Exploration studies on multi-class training. All models are trained on the listed classes and tested on the chair category. HilbertGen-3D (VAE) samples chair HMC conditions from the chair condition VAE, and HilbertGen-3D (Bank) draws them from the chair training bank. For 1-NNA, values closer to 50 are better; for COV, higher values are better. DiT-3D-S/4 and TopoDiT-3D-S/4 results are quoted from TopoDiT-3D.

Method	Model Size	Sampling Time (s/it) $\downarrow$	Training Memory (GB) $\downarrow$
DiT-3D	S/4	8.50	78.4
TopoDiT-3D	S/4	8.20	51.2
Ours	S/4	2.60	51.0

Appendix

Table 6: Computational cost comparison under the S/4 configuration. Sampling time is reported in seconds per iteration. Training memory denotes the peak GPU memory usage during training.

Method	Model Size	Epoch Time (s/epoch) $\downarrow$	Relative Speedup
Topodit	S/4	11.8	1.00 $\times$
Ours	S/4	2.3	5.13 $\times$

Appendix

Table 7: Training time comparison on the chair category under the S/4 configuration. Epoch time is reported in seconds per epoch.

Figure 4: HilbertGen-3d samples on Chair.

Figure 5: HilbertGen-3d samples on airplane.

Figure 7: HilbertGen-3d samples on vessel.

Figure 8: HilbertGen-3d samples on lamp.

Figure 9: Qualitative visualizations of denoising process for chair generation.

Figure 10: Qualitative visualizations of denoising process for car generation.

Figure 11: Qualitative visualizations of denoising process for airplane generation.

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