# Learning and Meshing from Deep Implicit Surface Networks Using an Efficient Implementation of Analytic Marching

arXiv: https://arxiv.org/abs/2106.10031 · HTML: https://mlsanity.com/paper/2106.10031/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 63 references that are in the map.
- First posted on arXiv 18 Jun 2021.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [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)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [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)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [DISN: Deep Implicit Surface Network for High-quality Single-view 3D Reconstruction](https://mlsanity.com/paper/1905.10711/) (2019)
- [Deep Mesh Reconstruction from Single RGB Images via Topology Modification Networks](https://mlsanity.com/paper/1909.00321/) (2019)
- [On the number of response regions of deep feed forward networks with piece-wise linear activations](https://mlsanity.com/paper/1312.6098/) (2013)
- [BAE-NET: Branched Autoencoder for Shape Co-Segmentation](https://mlsanity.com/paper/1903.11228/) (2019)
- [PatchNets: Patch-Based Generalizable Deep Implicit 3D Shape Representations](https://mlsanity.com/paper/2008.01639/) (2020)
- [Controlling Neural Level Sets](https://mlsanity.com/paper/1905.11911/) (2019)
- [Orthogonal Deep Neural Networks](https://mlsanity.com/paper/1905.05929/) (2019)
- [Blessing of dimensionality: mathematical foundations of the statistical physics of data](https://mlsanity.com/paper/1801.03421/) (2018)
- [Deep Meta Functionals for Shape Representation](https://mlsanity.com/paper/1908.06277/) (2019)
- [A Skeleton-bridged Deep Learning Approach for Generating Meshes of Complex Topologies from Single RGB Images](https://mlsanity.com/paper/1903.04704/) (2019)
- [Exact and Consistent Interpretation for Piecewise Linear Neural Networks: A Closed Form Solution](https://mlsanity.com/paper/1802.06259/) (2018)
- [Analytic Marching: An Analytic Meshing Solution from Deep Implicit Surface Networks](https://mlsanity.com/paper/2002.06597/) (2020)
- [QuickCSG: Fast Arbitrary Boolean Combinations of N Solids](https://mlsanity.com/paper/1706.01558/) (2017)

## Built on by (most cited first, 8 of 8)

- [RGBD2: Generative Scene Synthesis via Incremental View Inpainting using RGBD Diffusion Models](https://mlsanity.com/paper/2212.05993/) (2022)
- [Fast Neural Representations for Direct Volume Rendering](https://mlsanity.com/paper/2112.01579/) (2021)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [ARMesh: Autoregressive Mesh Generation via Next-Level-of-Detail Prediction](https://mlsanity.com/paper/2509.20824/) (2025)
- [What's the Situation with Intelligent Mesh Generation: A Survey and Perspectives](https://mlsanity.com/paper/2211.06009/) (2022)
- [Marching Neurons: Accurate Surface Extraction for Neural Implicit Shapes](https://mlsanity.com/paper/2509.21007/) (2025)
- [Neural Vector Fields: Generalizing Distance Vector Fields by Codebooks and Zero-Curl Regularization](https://mlsanity.com/paper/2309.01512/) (2023)
- [Polyhedral Complex Derivation from Piecewise Trilinear Networks](https://mlsanity.com/paper/2402.10403/) (2024)

## Abstract (arXiv)

Reconstruction of object or scene surfaces has tremendous applications in computer vision, computer graphics, and robotics. In this paper, we study a fundamental problem in this context about recovering a surface mesh from an implicit field function whose zero-level set captures the underlying surface. To achieve the goal, existing methods rely on traditional meshing algorithms; while promising, they suffer from loss of precision learned in the implicit surface networks, due to the use of discrete space sampling in marching cubes. Given that an MLP with activations of Rectified Linear Unit (ReLU) partitions its input space into a number of linear regions, we are motivated to connect this local linearity with a same property owned by the desired result of polygon mesh. More specifically, we identify from the linear regions, partitioned by an MLP based implicit function, the analytic cells and analytic faces that are associated with the function's zero-level isosurface. We prove that under mild conditions, the identified analytic faces are guaranteed to connect and form a closed, piecewise planar surface. Based on the theorem, we propose an algorithm of analytic marching, which marches among analytic cells to exactly recover the mesh captured by an implicit surface network. We also show that our theory and algorithm are equally applicable to advanced MLPs with shortcut connections and max pooling. Given the parallel nature of analytic marching, we contribute AnalyticMesh, a software package that supports efficient meshing of implicit surface networks via CUDA parallel computing, and mesh simplification for efficient downstream processing. We apply our method to different settings of generative shape modeling using implicit surface networks. Extensive experiments demonstrate our advantages over existing methods in terms of both meshing accuracy and efficiency.