# Neural Geometry Processing via Spherical Neural Surfaces

arXiv: https://arxiv.org/abs/2407.07755 · HTML: https://mlsanity.com/paper/2407.07755/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 49 references that are in the map.
- First posted on arXiv 10 Jul 2024.

## Builds on (most cited first, 16 of 16)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Pixel2Mesh: Generating 3D Mesh Models from Single RGB Images](https://mlsanity.com/paper/1804.01654/) (2018)
- [AtlasNet: A Papier-Mach'e Approach to Learning 3D Surface Generation](https://mlsanity.com/paper/1802.05384/) (2018)
- [PCPNET: Learning Local Shape Properties from Raw Point Clouds](https://mlsanity.com/paper/1710.04954/) (2017)
- [Nesti-Net: Normal Estimation for Unstructured 3D Point Clouds using Convolutional Neural Networks](https://mlsanity.com/paper/1812.00709/) (2018)
- [A Level Set Theory for Neural Implicit Evolution under Explicit Flows](https://mlsanity.com/paper/2204.07159/) (2022)
- [Shape Reconstruction by Learning Differentiable Surface Representations](https://mlsanity.com/paper/1911.11227/) (2019)
- [Neural Surface Maps](https://mlsanity.com/paper/2103.16942/) (2021)
- [On the Effectiveness of Weight-Encoded Neural Implicit 3D Shapes](https://mlsanity.com/paper/2009.09808/) (2020)
- [Exploring Differential Geometry in Neural Implicits](https://mlsanity.com/paper/2201.09263/) (2022)
- [Neural Implicit Surface Evolution](https://mlsanity.com/paper/2201.09636/) (2022)
- [Explicit Neural Surfaces: Learning Continuous Geometry With Deformation Fields](https://mlsanity.com/paper/2306.02956/) (2023)
- [Neural Field Convolutions by Repeated Differentiation](https://mlsanity.com/paper/2304.01834/) (2023)
- [Accurate Differential Operators for Hybrid Neural Fields](https://mlsanity.com/paper/2312.05984/) (2023)

## Built on by (most cited first, 14 of 14)

- [Dynamic Neural Surfaces for Elastic 4D Shape Representation and Analysis](https://mlsanity.com/paper/2503.03132/) (2025)
- [FUSE: A Flow-based Mapping Between Shapes](https://mlsanity.com/paper/2511.13431/) (2025)
- [A Survey of AI Methods for Geometry Preparation and Mesh Generation in Engineering Simulation](https://mlsanity.com/paper/2512.23719/) (2025)
- [Advances and Trends in the 3D Reconstruction of the Shape and Motion of Animals](https://mlsanity.com/paper/2508.16062/) (2025)
- [Learning to Solve PDEs on Neural Shape Representations](https://mlsanity.com/paper/2512.21311/) (2025)
- [Hierarchical Neural Surfaces for 3D Mesh Compression](https://mlsanity.com/paper/2512.15985/) (2025)
- [Learning Eigenstructures of Unstructured Data Manifolds](https://mlsanity.com/paper/2512.01103/) (2025)
- [NC-Reg : Neural Cortical Maps for Rigid Registration](https://mlsanity.com/paper/2601.19042/) (2026)
- [Spectral-Geometric Neural Fields for Pose-Free LiDAR View Synthesis](https://mlsanity.com/paper/2603.12903/) (2026)
- [Recent Advances and Trends in Learning-based 3D Representations](https://mlsanity.com/paper/2606.04871/) (2026)
- [Blended Chart Surfaces: A Seamless Explicit Representation for Smooth Surface Fitting](https://mlsanity.com/paper/2606.18069/) (2026)
- [GenSP: Consistent Spherical Parameterization via Learning Shape Generative Models](https://mlsanity.com/paper/2607.00492/) (2026)
- [NeuralSRNF: Neural Square Root Normal Fields for the Statistical Shape Analysis and Generation of Nonrigid 3D and 4D Objects](https://mlsanity.com/paper/2609.27728/) (2026)
- [Neural networks for spectral optimization](https://mlsanity.com/paper/2609.36047/) (2026)

## Abstract (arXiv)

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.