From Extrinsic to Intrinsic: Geodesic-Guided Representation Learning for 3D Geometric Data

Geometric analysis fundamentally distinguishes between extrinsic and 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 PRISM, for Pre-training, which learns isometric embeddings by Recovering the Intrinsic Surface geodesic Metric. 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. Our code will be made publicly available.

Computers are increasingly asked to understand 3D shapes, but many current methods describe a shape mainly by how it sits in ordinary 3D space or by broad labels such as “chair” or “person.” This can miss something more fundamental: a shape may bend, rotate, or deform while keeping the same underlying surface structure. We introduce PRISM, a training method that helps AI models understand shapes from the surface itself. Instead of only measuring straight-line distances through space, PRISM learns distances along the surface, like the shortest path one could walk over the object. We also guide the model so that its internal representation preserves the surface’s basic structure, and we use a two-stage training strategy to handle the fact that some surface distances appear much more often than others. This gives the model a more reliable sense of shape identity. In experiments, PRISM predicts surface distances accurately and efficiently, and it improves tasks such as recognizing shapes, flattening surfaces into 2D maps, and matching corresponding points across deformable 3D objects.