# Analytic Marching: An Analytic Meshing Solution from Deep Implicit Surface Networks

arXiv: https://arxiv.org/abs/2002.06597 · HTML: https://mlsanity.com/paper/2002.06597/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Feb 2020.

## Built on by (most cited first, 16 of 16)

- [Deep Marching Tetrahedra: a Hybrid Representation for High-Resolution 3D Shape Synthesis](https://mlsanity.com/paper/2111.04276/) (2021)
- [Neural Marching Cubes](https://mlsanity.com/paper/2106.11272/) (2021)
- [RGBD2: Generative Scene Synthesis via Incremental View Inpainting using RGBD Diffusion Models](https://mlsanity.com/paper/2212.05993/) (2022)
- [Surface Reconstruction from Point Clouds: A Survey and a Benchmark](https://mlsanity.com/paper/2205.02413/) (2022)
- [Learning Parallel Dense Correspondence from Spatio-Temporal Descriptors for Efficient and Robust 4D Reconstruction](https://mlsanity.com/paper/2103.16341/) (2021)
- [HelixSurf: A Robust and Efficient Neural Implicit Surface Learning of Indoor Scenes with Iterative Intertwined Regularization](https://mlsanity.com/paper/2302.14340/) (2023)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [Spelunking the Deep: Guaranteed Queries on General Neural Implicit Surfaces via Range Analysis](https://mlsanity.com/paper/2202.02444/) (2022)
- [ARMesh: Autoregressive Mesh Generation via Next-Level-of-Detail Prediction](https://mlsanity.com/paper/2509.20824/) (2025)
- [Learning and Meshing from Deep Implicit Surface Networks Using an Efficient Implementation of Analytic Marching](https://mlsanity.com/paper/2106.10031/) (2021)
- [Reachable Polyhedral Marching (RPM): An Exact Analysis Tool for Deep-Learned Control Systems](https://mlsanity.com/paper/2210.08339/) (2022)
- [Implicit Conversion of Manifold B-Rep Solids by Neural Halfspace Representation](https://mlsanity.com/paper/2209.10191/) (2022)
- [A Review of Deep Learning-Powered Mesh Reconstruction Methods](https://mlsanity.com/paper/2303.02879/) (2023)
- [Marching Neurons: Accurate Surface Extraction for Neural Implicit Shapes](https://mlsanity.com/paper/2509.21007/) (2025)
- [Polyhedral Complex Derivation from Piecewise Trilinear Networks](https://mlsanity.com/paper/2402.10403/) (2024)
- [TetraSDF: Analytic Isosurface Extraction with Multi-resolution Tetrahedral Grid](https://mlsanity.com/paper/2511.16273/) (2025)

## Abstract (arXiv)

This paper studies a problem of learning surface mesh via implicit functions in an emerging field of deep learning surface reconstruction, where implicit functions are popularly implemented as multi-layer perceptrons (MLPs) with rectified linear units (ReLU). To achieve meshing from learned implicit functions, existing methods adopt the de-facto standard algorithm of marching cubes; while promising, they suffer from loss of precision learned in the MLPs, due to the discretization nature of marching cubes. Motivated by the knowledge that a ReLU based MLP partitions its input space into a number of linear regions, we identify from these regions analytic cells and analytic faces that are associated with zero-level isosurface of the implicit function, and characterize the theoretical conditions under which the identified analytic faces are guaranteed to connect and form a closed, piecewise planar surface. Based on our theorem, we propose a naturally parallelizable algorithm of analytic marching, which marches among analytic cells to exactly recover the mesh captured by a learned MLP. Experiments on deep learning mesh reconstruction verify the advantages of our algorithm over existing ones.