# Rethinking the Approximation Error in 3D Surface Fitting for Point Cloud Normal Estimation

arXiv: https://arxiv.org/abs/2303.17167 · HTML: https://mlsanity.com/paper/2303.17167/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 45 references that are in the map.
- First posted on arXiv 30 Mar 2023.

## Builds on (most cited first, 21 of 21)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies](https://mlsanity.com/paper/2206.04670/) (2022)
- [Deep Closest Point: Learning Representations for Point Cloud Registration](https://mlsanity.com/paper/1905.03304/) (2019)
- [PREDATOR: Registration of 3D Point Clouds with Low Overlap](https://mlsanity.com/paper/2011.13005/) (2020)
- [Neural-Pull: Learning Signed Distance Functions from Point Clouds by Learning to Pull Space onto Surfaces](https://mlsanity.com/paper/2011.13495/) (2020)
- [PCPNET: Learning Local Shape Properties from Raw Point Clouds](https://mlsanity.com/paper/1710.04954/) (2017)
- [PUGeo-Net: A Geometry-centric Network for 3D Point Cloud Upsampling](https://mlsanity.com/paper/2002.10277/) (2020)
- [Nesti-Net: Normal Estimation for Unstructured 3D Point Clouds using Convolutional Neural Networks](https://mlsanity.com/paper/1812.00709/) (2018)
- [DeepFit: 3D Surface Fitting via Neural Network Weighted Least Squares](https://mlsanity.com/paper/2003.10826/) (2020)
- [Deep Feature-preserving Normal Estimation for Point Cloud Filtering](https://mlsanity.com/paper/2004.11563/) (2020)
- [AdaFit: Rethinking Learning-based Normal Estimation on Point Clouds](https://mlsanity.com/paper/2108.05836/) (2021)
- [Refine-Net: Normal Refinement Neural Network for Noisy Point Clouds](https://mlsanity.com/paper/2203.12514/) (2022)
- [HSurf-Net: Normal Estimation for 3D Point Clouds by Learning Hyper Surfaces](https://mlsanity.com/paper/2210.07158/) (2022)
- [Normal Estimation for 3D Point Clouds via Local Plane Constraint and Multi-scale Selection](https://mlsanity.com/paper/1910.08537/) (2019)
- [GraphFit: Learning Multi-scale Graph-Convolutional Representation for Point Cloud Normal Estimation](https://mlsanity.com/paper/2207.11484/) (2022)
- [FBNet: Feedback Network for Point Cloud Completion](https://mlsanity.com/paper/2210.03974/) (2022)
- [Point Cloud Upsampling via Cascaded Refinement Network](https://mlsanity.com/paper/2210.03942/) (2022)
- [Contrastive Learning for Joint Normal Estimation and Point Cloud Filtering](https://mlsanity.com/paper/2208.06811/) (2022)
- [Fast and Accurate Normal Estimation for Point Cloud via Patch Stitching](https://mlsanity.com/paper/2103.16066/) (2021)

## Built on by (most cited first, 13 of 13)

- [Learning Signed Hyper Surfaces for Oriented Point Cloud Normal Estimation](https://mlsanity.com/paper/2305.05873/) (2023)
- [NeuralGF: Unsupervised Point Normal Estimation by Learning Neural Gradient Function](https://mlsanity.com/paper/2311.00389/) (2023)
- [CMG-Net: Robust Normal Estimation for Point Clouds via Chamfer Normal Distance and Multi-scale Geometry](https://mlsanity.com/paper/2312.09154/) (2023)
- [Arbitrary-Scale Point Cloud Upsampling by Voxel-Based Network with Latent Geometric-Consistent Learning](https://mlsanity.com/paper/2403.05117/) (2024)
- [PFF-Net: Patch Feature Fitting for Point Cloud Normal Estimation](https://mlsanity.com/paper/2511.21365/) (2025)
- [Learning Normals of Noisy Points by Local Gradient-Aware Surface Filtering](https://mlsanity.com/paper/2507.03394/) (2025)
- [LiSu: A Dataset and Method for LiDAR Surface Normal Estimation](https://mlsanity.com/paper/2503.08601/) (2025)
- [PointNorm-Net: Self-Supervised Normal Prediction of 3D Point Clouds via Multi-Modal Distribution Estimation](https://mlsanity.com/paper/2304.04884/) (2023)
- [E3-Net: Efficient E(3)-Equivariant Normal Estimation Network](https://mlsanity.com/paper/2406.00347/) (2024)
- [Refining 3D Point Cloud Normal Estimation via Sample Selection](https://mlsanity.com/paper/2406.18541/) (2024)
- [OCMG-Net: Neural Oriented Normal Refinement for Unstructured Point Clouds](https://mlsanity.com/paper/2409.01100/) (2024)
- [Camera Movement Estimation and Path Correction using the Combination of Modified A-SIFT and Stereo System for 3D Modelling](https://mlsanity.com/paper/2503.17668/) (2025)
- [MambaH-Fit: Rethinking Hyper-surface Fitting-based Point Cloud Normal Estimation via State Space Modelling](https://mlsanity.com/paper/2510.09088/) (2025)

## Abstract (arXiv)

Most existing approaches for point cloud normal estimation aim to locally fit a geometric surface and calculate the normal from the fitted surface. Recently, learning-based methods have adopted a routine of predicting point-wise weights to solve the weighted least-squares surface fitting problem. Despite achieving remarkable progress, these methods overlook the approximation error of the fitting problem, resulting in a less accurate fitted surface. In this paper, we first carry out in-depth analysis of the approximation error in the surface fitting problem. Then, in order to bridge the gap between estimated and precise surface normals, we present two basic design principles: 1) applies the Z-direction Transform to rotate local patches for a better surface fitting with a lower approximation error; 2) models the error of the normal estimation as a learnable term. We implement these two principles using deep neural networks, and integrate them with the state-of-the-art (SOTA) normal estimation methods in a plug-and-play manner. Extensive experiments verify our approaches bring benefits to point cloud normal estimation and push the frontier of state-of-the-art performance on both synthetic and real-world datasets.