Structured 2D Atlases via Geodesic Optimal Transport for 3D Representation Learning

Recent 3D generation pipelines commonly employ Variational Autoencoders (VAEs) to learn compact latent representations of 3D shapes. However, learning high-resolution 3D representations at scale is highly GPU-memory intensive, and existing Shape VAEs often rely on substantial surface subsampling, coarse 3D discretization, or large model capacity to make training tractable. We propose Structured 2D Atlases, which reorganize dense oriented 3D surface samples into a regular image-space representation while retaining their original 3D positions and surface normals. Our method establishes a bijection between surface samples and fixed Fibonacci-sphere targets using hierarchical optimal transport guided by intrinsic geodesic distances, followed by deterministic equirectangular indexing into square position and normal maps. An $H\times W$ atlas stores $N=HW$ oriented surface samples, enabling an atlas-based transformer Shape VAE to encode and decode 3D surface geometry through a regular 2D domain rather than a 3D-native representation. Experiments show that our atlas-based Shape VAE uses $5.3\times$ to $25.0\times$ fewer parameters than the evaluated 3D-native Shape VAEs, providing a favorable trade-off between model size and fine-scale geometric detail. These results demonstrate that Structured 2D Atlases provide a parameter-efficient alternative for dense 3D surface representation learning.