Neural Geometry Processing via Spherical Neural Surfaces

TL;DR

This paper introduces a spherical neural surface representation for genus-0 surfaces that allows direct computation of geometric operators, enabling neural geometry processing without meshing.

Contribution

It proposes a novel seamless spherical neural surface model that directly computes geometric operators, bridging neural representations with classical geometry processing.

Findings

Enables direct computation of surface normals and fundamental forms on neural surfaces.

Demonstrates applications in spectral analysis, heat flow, and curvature flow.

Validates robustness to shape variations and compares with traditional methods.

Abstract

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. In the absence of an analogous toolbox, 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 for genus-0 surfaces and demonstrate how to compute core geometric operators directly on this representation. Namely, we estimate surface normals and first and second fundamental forms of the surface, as well as compute surface gradient, surface divergence and Laplace-Beltrami operator on scalar/vector fields defined on the surface. Our representation is fully seamless, overcoming a key limitation of similar explicit representations such as Neural Surface Maps [Morreale et al. 2021]. These operators, in turn, enable geometry processing directly on the neural representations without any unnecessary meshing. We demonstrate illustrative applications in (neural) spectral analysis, heat flow and mean curvature flow, and evaluate robustness to isometric shape variations. We propose theoretical formulations and validate their numerical estimates, against analytical estimates, mesh-based baselines, and neural alternatives, where available. By systematically linking neural surface representations with classical geometry processing algorithms, we believe that this work can become a key ingredient in enabling neural geometry processing. Code is accessible from the project webpage.