# Explicit Neural Surfaces: Learning Continuous Geometry With Deformation Fields

arXiv: https://arxiv.org/abs/2306.02956 · HTML: https://mlsanity.com/paper/2306.02956/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 43 references that are in the map.
- First posted on arXiv 5 Jun 2023.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction](https://mlsanity.com/paper/2106.10689/) (2021)
- [Volume Rendering of Neural Implicit Surfaces](https://mlsanity.com/paper/2106.12052/) (2021)
- [GRAF: Generative Radiance Fields for 3D-Aware Image Synthesis](https://mlsanity.com/paper/2007.02442/) (2020)
- [UNISURF: Unifying Neural Implicit Surfaces and Radiance Fields for Multi-View Reconstruction](https://mlsanity.com/paper/2104.10078/) (2021)
- [AtlasNet: A Papier-Mach'e Approach to Learning 3D Surface Generation](https://mlsanity.com/paper/1802.05384/) (2018)
- [Deep Marching Tetrahedra: a Hybrid Representation for High-Resolution 3D Shape Synthesis](https://mlsanity.com/paper/2111.04276/) (2021)
- [Modular Primitives for High-Performance Differentiable Rendering](https://mlsanity.com/paper/2011.03277/) (2020)
- [Extracting Triangular 3D Models, Materials, and Lighting From Images](https://mlsanity.com/paper/2111.12503/) (2021)
- [CodeNeRF: Disentangled Neural Radiance Fields for Object Categories](https://mlsanity.com/paper/2109.01750/) (2021)
- [NeuMesh: Learning Disentangled Neural Mesh-based Implicit Field for Geometry and Texture Editing](https://mlsanity.com/paper/2207.11911/) (2022)
- [Deep Geometric Prior for Surface Reconstruction](https://mlsanity.com/paper/1811.10943/) (2018)
- [Voxurf: Voxel-based Efficient and Accurate Neural Surface Reconstruction](https://mlsanity.com/paper/2208.12697/) (2022)
- [Learning Smooth Neural Functions via Lipschitz Regularization](https://mlsanity.com/paper/2202.08345/) (2022)
- [PermutoSDF: Fast Multi-View Reconstruction with Implicit Surfaces using Permutohedral Lattices](https://mlsanity.com/paper/2211.12562/) (2022)
- [FiG-NeRF: Figure-Ground Neural Radiance Fields for 3D Object Category Modelling](https://mlsanity.com/paper/2104.08418/) (2021)
- [Multi-View Mesh Reconstruction with Neural Deferred Shading](https://mlsanity.com/paper/2212.04386/) (2022)
- [Shape Reconstruction by Learning Differentiable Surface Representations](https://mlsanity.com/paper/1911.11227/) (2019)
- [Intrinsic Neural Fields: Learning Functions on Manifolds](https://mlsanity.com/paper/2203.07967/) (2022)

## Built on by (most cited first, 10 of 10)

- [Neural Geometry Processing via Spherical Neural Surfaces](https://mlsanity.com/paper/2407.07755/) (2024)
- [Beyond Pixel Histories: World Models with Persistent 3D State](https://mlsanity.com/paper/2603.03482/) (2026)
- [CanFields: Consolidating Diffeomorphic Flows for Non-Rigid 4D Interpolation from Arbitrary-Length Sequences](https://mlsanity.com/paper/2406.18582/) (2024)
- [Sur2f: A Hybrid Representation for High-Quality and Efficient Surface Reconstruction from Multi-view Images](https://mlsanity.com/paper/2401.03704/) (2024)
- [Fine-Grained Multi-View Hand Reconstruction Using Inverse Rendering](https://mlsanity.com/paper/2407.05680/) (2024)
- [MeshFeat: Multi-Resolution Features for Neural Fields on Meshes](https://mlsanity.com/paper/2407.13592/) (2024)
- [Spatially-Adaptive Hash Encodings For Neural Surface Reconstruction](https://mlsanity.com/paper/2412.05179/) (2024)
- [REFRAME: Reflective Surface Real-Time Rendering for Mobile Devices](https://mlsanity.com/paper/2403.16481/) (2024)
- [HVOFusion: Incremental Mesh Reconstruction Using Hybrid Voxel Octree](https://mlsanity.com/paper/2404.17974/) (2024)
- [Disentangled Geometry and Appearance for Efficient Multi-View Surface Reconstruction and Rendering](https://mlsanity.com/paper/2508.17436/) (2025)

## Abstract (arXiv)

We introduce Explicit Neural Surfaces (ENS), an efficient smooth surface representation that directly encodes topology with a deformation field from a known base domain. We apply this representation to reconstruct explicit surfaces from multiple views, where we use a series of neural deformation fields to progressively transform the base domain into a target shape. By using meshes as discrete surface proxies, we train the deformation fields through efficient differentiable rasterization. Using a fixed base domain allows us to have Laplace-Beltrami eigenfunctions as an intrinsic positional encoding alongside standard extrinsic Fourier features, with which our approach can capture fine surface details. Compared to implicit surfaces, ENS trains faster and has several orders of magnitude faster inference times. The explicit nature of our approach also allows higher-quality mesh extraction whilst maintaining competitive surface reconstruction performance and real-time capabilities.