Curvature Enthusiasm: Correspondence-Free Interpolation and Matching of Articulated 3D Shapes using Compressed Normal Cycles

We present an unsupervised framework for physically plausible shape interpolation and dense correspondence estimation between 3D articulated shapes. Our approach intentionally focuses upon pose variation within the same identity, which we believe is a meaningful and challenging problem in its own right. Our method uses Neural Ordinary Differential Equations (NODEs) to generate smooth flow fields that define diffeomorphic transformations, ensuring topological consistency and preventing self-intersections while accommodating hard constraints, such as volume preservation. By incorporating a lightweight skeletal structure, we impose kinematic constraints that resolve symmetries without requiring manual skinning or predefined poses. We enhance physical realism by interpolating skeletal motion with dual quaternions and applying constrained optimisation to align the flow field with the skeleton, preserving local rigidity. Additionally, we employ an efficient formulation of Normal Cycles, a metric from geometric measure theory, to capture higher-order surface details like curvature, enabling precise alignment between complex articulated structures and recovery of accurate dense correspondence mapping. Evaluations on multiple benchmarks show notable improvements over state-of-the-art methods in both interpolation quality and correspondence accuracy, with consistent performance across different skeletal configurations, demonstrating broad utility for shape matching and animation tasks.