Blended Chart Surfaces: A Seamless Explicit Representation for Smooth Surface Fitting

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.

Neural Geometry Processing via Spherical Neural Surfaces

Neural surfaces (e.g., neural map encoding, deep implicits and neural radiance fields) have recently gained popularity because of their generic structure (e.g., multi-layer perceptron) and easy integration with modern learning-based setups. Traditionally, we have a rich toolbox of geometry processing algorithms designed for polygonal meshes to analyze and operate on surface geometry. However, neural representations are typically discretized and converted into a mesh, before applying any geometry processing algorithm. This is unsatisfactory and, as we demonstrate, unnecessary. In this work, we propose a spherical neural surface representation (a spherical parametrization) for genus-0 surfaces and demonstrate how to compute core geometric operators directly on this representation. Namely, we show how to construct the normals and the first and second fundamental forms of the surface, and how to compute the surface gradient, surface divergence and Laplace Beltrami operator on scalar/vector fields defined on the surface. These operators, in turn, enable us to create geometry processing tools that act directly on the neural representations without any unnecessary meshing. We demonstrate illustrative applications in (neural) spectral analysis, heat flow and mean curvature flow, and our method shows robustness to isometric shape variations. We both propose theoretical formulations and validate their numerical estimates. By systematically linking neural surface representations with classical geometry processing algorithms, we believe this work can become a key ingredient in enabling neural geometry processing.