Neural Geometry Processing via Spherical Neural Surfaces

TL;DR

The paper tackles geometry processing on neural surface representations by introducing Spherical Neural Surfaces (SNS), a differentiable, sphere-based encoding for genus-0 shapes that eliminates the need for meshing. It derives and computes continuous differential-geometry operators on SNS, including normals, the First and Second Fundamental Forms, curvature, and a continuous Laplace-Beltrami operator $Δ_ ext{Σ}$, plus a neural procedure to extract the lowest spectral modes via a Rayleigh-quotient formulation. The approach supports neural spectral analysis, heat flow, and mean-curvature flow on neural surfaces, with robust, discretization-free estimates validated against analytical solutions and mesh baselines. Limitations include the genus-0 restriction and current runtime, with future work aiming at multiple canonical domains, local parametrizations, and dynamic surfaces for end-to-end optimization of geometric energies.

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.

Figures (14)

• Figure 1: We show area-distortion colourmaps for four different spherical embeddings of the frog mesh, and demonstrate the effect of different patterns of distortion on the accuracy of fitting, and normals and curvature estimates.
• Figure :
• Figure :
• Figure :
• Figure :