Geometry Distributions

TL;DR

This work tackles the limitations of traditional 3D representations in handling non-watertight geometries and complex topologies by introducing Geometry Distributions (GeomDist), a diffusion-model-based framework that represents geometry as a distribution of surface points. By solving a forward ODE, it maps Gaussian noise to surface samples on the target geometry, and a backward ODE enables inverse mapping back to noise space, allowing effectively infinite surface points to describe a shape. The approach is trained with a denoising objective on noisy surface samples, uses a specialized architecture with magnitude-preserving layers, and employs continuous resampling to simulate infinite data across epochs. Experiments across remeshing, textured geometry, and dynamic geometry demonstrate high geometric fidelity, with ablations highlighting the importance of network design, noise distributions, and diffusion steps. The method promises practical impact for textured mesh generation, neural surface compression, and realistic rendering, and opens avenues for joint sampling and meshing in neural geometry pipelines.

Abstract

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

Figures (16)

• Figure 1: Our representation can handle 3D geometry with complex details, high genus, sharp features, and non-watertight surfaces: our trained diffusion networks $\mathcal{E}_i$ can transform the samples $\mathbf{X}$ from a Gaussian distribution $\mathcal{N}$ to the geometry ${\mathcal{M}}\subset{\mathbb{R}}^3$. The colors indicate the correspondence between the Gaussian noise and the surface points.
• Figure 2: Compared to signed distance functions (SDFs), our GeomDist can model open and non-watertight objects using significantly fewer network parameters. See the Appendix for the meshing algorithm used in this figure. The SDF is fit using Instant-NGP muller2022instant, with isosurface extraction evaluated on a 512-resolution grid. We observe that SDFs struggle to represent thin structures or non-watertight geometry.
• Figure 3: Compared to vector fields-based method, our GeomDist produces more uniformly distributed samples with higher fidelity. The chamfer distance $(\times 10^3)$ between the samples and target surface is reported below.
• Figure 4: Heatmap of the $L_2$ distance from sampled points to the target surface using different network architectures.
• Figure 5: Inference process for generating 1M points on a lamp mesh (top) and a jellyfish mesh (bottom) from uniform and Gaussian distributions, respectively. Results are shown at timesteps $t = 0, 40, 48, 56, 60, 64$, with a close-up of the generated samples at $t=64$ overlaid on the ground-truth mesh. A complete illustration is available in the accompanying video demo. Both meshes are taken from zhou2016thingi10k.