🤖 AI Summary
Existing neural surface representations struggle to simultaneously achieve compactness, explicit formulation, global smoothness, topological generality, and reliable differential quantity computation. This work proposes a novel explicit surface representation that dispenses with neural networks entirely: guided by a user-provided coarse proxy mesh, it optimizes local polynomial mappings at each vertex and seamlessly blends neighboring mappings via one-ring coordinate smoothing to produce a globally smooth and differentiable surface. The method innovatively decouples topology from geometric detail, yielding an intrinsically smooth, parameterization-free surface that is equivariant under rigid transformations and uniform scaling—thereby eliminating seam artifacts and parametrization dependencies common in conventional approaches. Experiments demonstrate its effectiveness across diverse topologies and geometric complexities, achieving a superior balance among compactness, simplicity, accessibility of differential quantities, and representational power.
📝 Abstract
A surface representation suitable for geometry processing should be compact and explicit, provide global smoothness guarantees, support a wide range of surface topologies, and offer reliable access to differential quantities such as normals and surface energies, while remaining compatible with modern differentiable optimization. Existing neural representations typically sacrifice one or more of these properties: implicit fields typically require iso-surfacing for downstream use, while explicit neural maps are constrained by canonical-domain parametrizations or exhibit seam artifacts between local charts. We introduce Blended Chart Surfaces, a compact, network-free, explicit representation that is smooth by construction and anchored to user-provided topology. Given a coarse proxy mesh encoding the intended surface topology and approximate geometry, Blended Chart Surfaces jointly optimize for a polynomial map at each proxy vertex using an off-the-shelf optimizer to fit to an implicit target shape, avoiding the need for an input parametrization. Neighboring maps are fused using a smooth 'one-ring coordinate' blending scheme, decoupling topology and coarse geometry (carried by the proxy) from geometric details (carried by the local patches). The surface is globally smooth, fully differentiable, and enables stable evaluation of derivatives, making differential quantities and surface energies directly accessible. Additionally, our construction is equivariant to rigid motions and scaling of the proxy mesh. We evaluate Blended Chart Surfaces on various topologies and geometric complexity, and compare against explicit alternatives including interpolating-function baselines and mesh-displacement MLPs. Across these, Blended Chart Surfaces achieve a favorable trade-off among compactness, simplicity, access to differential quantities, and expressivity while remaining smooth across patch boundaries.