From Extrinsic to Intrinsic: Geodesic-Guided Representation Learning for 3D Geometric Data
This work addresses the problem of capturing intrinsic shape identity and manifold topology in 3D geometric data, which is a fundamental challenge for the 3D vision and geometry processing community.
PRISM introduces a new 3D representation learning paradigm that learns isometric embeddings by recovering the intrinsic surface geodesic metric, achieving superior performance in geodesic distance prediction and downstream tasks like shape recognition, surface parameterization, and non-rigid correspondence.
Geometric analysis fundamentally distinguishes between \textit{extrinsic} and \textit{intrinsic} perspectives. The dominant paradigm in current 3D representation learning relies on either extrinsic spatial structures or high-level semantics, struggling to capture the essence of shape identity and underlying manifold topology. To bridge this gap, we introduce a novel 3D representation learning paradigm, namely \textbf{PRISM}, for \textbf{P}re-training, which learns isometric embeddings by \textbf{R}ecovering the \textbf{I}ntrinsic \textbf{S}urface geodesic \textbf{M}etric. PRISM incorporates a topology-enforcing objective that explicitly constrains the structure of latent space, alongside a specialized two-stage training recipe mitigating sample imbalance inherent in the distribution of geodesic distances. Experiments demonstrate that our approach shows satisfactory accuracy, robustness, and high efficiency in geodesic distance prediction and achieves superior performance across diverse downstream tasks, including shape recognition, surface parameterization, and non-rigid correspondence. The code will be publicly available at https://github.com/AidenZhao/PRISM.