# PoNQ: a Neural QEM-based Mesh Representation

arXiv: https://arxiv.org/abs/2403.12870 · HTML: https://mlsanity.com/paper/2403.12870/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 67 references that are in the map.
- First posted on arXiv 19 Mar 2024.

## Builds on (most cited first, 30 of 37)

- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [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)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Deep Learning for 3D Point Clouds: A Survey](https://mlsanity.com/paper/1912.12033/) (2019)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (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)
- [ABC: A Big CAD Model Dataset For Geometric Deep Learning](https://mlsanity.com/paper/1812.06216/) (2018)
- [PolyGen: An Autoregressive Generative Model of 3D Meshes](https://mlsanity.com/paper/2002.10880/) (2020)
- [BSP-Net: Generating Compact Meshes via Binary Space Partitioning](https://mlsanity.com/paper/1911.06971/) (2019)
- [Shape As Points: A Differentiable Poisson Solver](https://mlsanity.com/paper/2106.03452/) (2021)
- [Thingi10K: A Dataset of 10,000 3D-Printing Models](https://mlsanity.com/paper/1605.04797/) (2016)
- [CvxNet: Learnable Convex Decomposition](https://mlsanity.com/paper/1909.05736/) (2019)
- [Flexible Isosurface Extraction for Gradient-Based Mesh Optimization](https://mlsanity.com/paper/2308.05371/) (2023)
- [MeshSDF: Differentiable Iso-Surface Extraction](https://mlsanity.com/paper/2006.03997/) (2020)
- [Neural Kernel Surface Reconstruction](https://mlsanity.com/paper/2305.19590/) (2023)
- [Learning Deformable Tetrahedral Meshes for 3D Reconstruction](https://mlsanity.com/paper/2011.01437/) (2020)
- [MeshUDF: Fast and Differentiable Meshing of Unsigned Distance Field Networks](https://mlsanity.com/paper/2111.14549/) (2021)
- [Neural Dual Contouring](https://mlsanity.com/paper/2202.01999/) (2022)
- [Dual Octree Graph Networks for Learning Adaptive Volumetric Shape Representations](https://mlsanity.com/paper/2205.02825/) (2022)
- [Variational Autoencoders for Deforming 3D Mesh Models](https://mlsanity.com/paper/1709.04307/) (2017)
- [Deep Implicit Moving Least-Squares Functions for 3D Reconstruction](https://mlsanity.com/paper/2103.12266/) (2021)
- [Neural Marching Cubes](https://mlsanity.com/paper/2106.11272/) (2021)
- [Surface Reconstruction from Point Clouds: A Survey and a Benchmark](https://mlsanity.com/paper/2205.02413/) (2022)
- [Dynamic Point Fields](https://mlsanity.com/paper/2304.02626/) (2023)
- [OctField: Hierarchical Implicit Functions for 3D Modeling](https://mlsanity.com/paper/2111.01067/) (2021)

## Built on by (most cited first, 12 of 12)

- [MILo: Mesh-In-the-Loop Gaussian Splatting for Detailed and Efficient Surface Reconstruction](https://mlsanity.com/paper/2506.24096/) (2025)
- [SpaceMesh: A Continuous Representation for Learning Manifold Surface Meshes](https://mlsanity.com/paper/2409.20562/) (2024)
- [DMesh++: An Efficient Differentiable Mesh for Complex Shapes](https://mlsanity.com/paper/2412.16776/) (2024)
- [From Blobs to Spokes: High-Fidelity Surface Reconstruction via Oriented Gaussians](https://mlsanity.com/paper/2604.07337/) (2026)
- [Occupancy-Based Dual Contouring](https://mlsanity.com/paper/2409.13418/) (2024)
- [ShapeShifter: 3D Variations Using Multiscale and Sparse Point-Voxel Diffusion](https://mlsanity.com/paper/2502.02187/) (2025)
- [A Survey of AI Methods for Geometry Preparation and Mesh Generation in Engineering Simulation](https://mlsanity.com/paper/2512.23719/) (2025)
- [NASM: Neural Anisotropic Surface Meshing](https://mlsanity.com/paper/2410.23109/) (2024)
- [MorphoSkel3D: Morphological Skeletonization of 3D Point Clouds for Informed Sampling in Object Classification and Retrieval](https://mlsanity.com/paper/2501.12974/) (2025)
- [QEMesh: Employing A Quadric Error Metrics-Based Representation for Mesh Generation](https://mlsanity.com/paper/2504.05720/) (2025)
- [Patchwork: A compact representation for 3D polygonal shapes](https://mlsanity.com/paper/2605.16266/) (2026)
- [Learning Manifolds in High-D Point Embedding for Anisotropic Surface Approximation from Unstructured Point Clouds](https://mlsanity.com/paper/2607.28855/) (2026)

## Abstract (arXiv)

Although polygon meshes have been a standard representation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning-based applications. In this work, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neural shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the boundary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target surface alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.