# Differentiable Surface Triangulation

arXiv: https://arxiv.org/abs/2109.10695 · HTML: https://mlsanity.com/paper/2109.10695/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 72 references that are in the map.
- First posted on arXiv 22 Sep 2021.

## Builds on (most cited first, 14 of 14)

- [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)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [AtlasNet: A Papier-Mach'e Approach to Learning 3D Surface Generation](https://mlsanity.com/paper/1802.05384/) (2018)
- [Soft Rasterizer: A Differentiable Renderer for Image-based 3D Reasoning](https://mlsanity.com/paper/1904.01786/) (2019)
- [DSAC - Differentiable RANSAC for Camera Localization](https://mlsanity.com/paper/1611.05705/) (2016)
- [Thingi10K: A Dataset of 10,000 3D-Printing Models](https://mlsanity.com/paper/1605.04797/) (2016)
- [MeshCNN: A Network with an Edge](https://mlsanity.com/paper/1809.05910/) (2018)
- [Correspondence Learning via Linearly-invariant Embedding](https://mlsanity.com/paper/2010.13136/) (2020)
- [Meshing Point Clouds with Predicted Intrinsic-Extrinsic Ratio Guidance](https://mlsanity.com/paper/2007.09267/) (2020)
- [Neural Surface Maps](https://mlsanity.com/paper/2103.16942/) (2021)
- [PointTriNet: Learned Triangulation of 3D Point Sets](https://mlsanity.com/paper/2005.02138/) (2020)
- [Learning Delaunay Surface Elements for Mesh Reconstruction](https://mlsanity.com/paper/2012.01203/) (2020)

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

- [Few 'Zero Level Set'-Shot Learning of Shape Signed Distance Functions in Feature Space](https://mlsanity.com/paper/2207.04161/) (2022)
- [Unsupervised Occupancy Learning from Sparse Point Cloud](https://mlsanity.com/paper/2404.02759/) (2024)
- [What's the Situation with Intelligent Mesh Generation: A Survey and Perspectives](https://mlsanity.com/paper/2211.06009/) (2022)
- [DMesh: A Differentiable Mesh Representation](https://mlsanity.com/paper/2404.13445/) (2024)
- [TetGAN: A Convolutional Neural Network for Tetrahedral Mesh Generation](https://mlsanity.com/paper/2210.05735/) (2022)
- [Few-Shot Unsupervised Implicit Neural Shape Representation Learning with Spatial Adversaries](https://mlsanity.com/paper/2408.15114/) (2024)
- [Toward Robust Neural Reconstruction from Sparse Point Sets](https://mlsanity.com/paper/2412.16361/) (2024)
- [Point2Quad: Generating Quad Meshes from Point Clouds via Face Prediction](https://mlsanity.com/paper/2504.19545/) (2025)
- [CircNet: Meshing 3D Point Clouds with Circumcenter Detection](https://mlsanity.com/paper/2301.09253/) (2023)
- [Learning Self-Prior for Mesh Inpainting Using Self-Supervised Graph Convolutional Networks](https://mlsanity.com/paper/2305.00635/) (2023)
- [OffsetOPT: Explicit Surface Reconstruction without Normals](https://mlsanity.com/paper/2503.15763/) (2025)
- [Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator](https://mlsanity.com/paper/2511.04804/) (2025)
- [Iskra: A System for Inverse Geometry Processing](https://mlsanity.com/paper/2602.12105/) (2026)
- [Soft Anisotropic Diagrams for Differentiable Image Representation](https://mlsanity.com/paper/2604.21984/) (2026)

## Abstract (arXiv)

Triangle meshes remain the most popular data representation for surface geometry. This ubiquitous representation is essentially a hybrid one that decouples continuous vertex locations from the discrete topological triangulation. Unfortunately, the combinatorial nature of the triangulation prevents taking derivatives over the space of possible meshings of any given surface. As a result, to date, mesh processing and optimization techniques have been unable to truly take advantage of modular gradient descent components of modern optimization frameworks. In this work, we present a differentiable surface triangulation that enables optimization for any per-vertex or per-face differentiable objective function over the space of underlying surface triangulations. Our method builds on the result that any 2D triangulation can be achieved by a suitably perturbed weighted Delaunay triangulation. We translate this result into a computational algorithm by proposing a soft relaxation of the classical weighted Delaunay triangulation and optimizing over vertex weights and vertex locations. We extend the algorithm to 3D by decomposing shapes into developable sets and differentiably meshing each set with suitable boundary constraints. We demonstrate the efficacy of our method on various planar and surface meshes on a range of difficult-to-optimize objective functions. Our code can be found online: https://github.com/mrakotosaon/diff-surface-triangulation.