🤖 AI Summary
To address the lack of native geometric processing capabilities in neural surface representations, this paper introduces spherical neural surface representation—a framework enabling seamless, mesh-free estimation of normals, first and second fundamental forms, gradients, divergence, and the Laplace–Beltrami operator on genus-0 neural surfaces. Our method leverages spherical parameterization with implicit neural representation, employs automatic differentiation to derive differential geometric operators, and establishes a numerical verification framework alongside neural spectral analysis tools. Key contributions include: (1) breaking the conventional “mesh-then-process” paradigm by establishing a systematic theoretical bridge between neural representations and classical differential geometry; (2) enabling geometric processing tasks—including neural heat flow and mean curvature flow—with robustness under isometric deformations; and (3) achieving numerical accuracy comparable to analytical solutions and mesh-based baselines, significantly outperforming existing neural surrogates.
📝 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.