VNC: A Scale-Space Foundation for Learnable 3D Surface Evolution

Controlling continuous, physically valid 3D surface evolutionis a fundamental challenge in computer vision and graphics. Current neu-ral representations typically rely on unstructured latent spaces or discreteiterations, lacking explicit geometric interpretability and struggling toproduce smooth deformations. While classical scale-space theory offersclear physical interpretability, its numerical irregularity isolates it frommodern deep learning. In this paper, we bridge this gap by establishing alearnable scale-space foundation. Our framework comprises three syner-gistic components: (1) the Variational Neighborhood Curvature (VNC)operator, an efficient, parallelizable, and scale-stable geometric metric,(2) the spacetime-balanced VNC-Flow algorithm, which translates ir-regular physical smoothing into structured deformation trajectories, and(3) a Controllable Variational Autoencoder (C-VAE) that learns thesetrajectories conditioned on normalized evolution time. Departing fromtraditional discrete iterations or unstructured latent interpolations, ourcontinuous neural surrogate induces a structured radial latent organiza-tion. This enables precise control over continuous shape abstraction andestablishes a structural prior for downstream applications.