Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

E$^3$-Net: Efficient E(3)-Equivariant Normal Estimation Network

Hanxiao Wang, Mingyang Zhao, Weize Quan, Zhen Chen, Dong-ming Yan, Peter Wonka

TL;DR

The paper tackles robust point cloud normal estimation by enforcing $E(3)$-equivariance while preserving computational efficiency. It introduces $E(3)$-equivariant learning through a random-frame training scheme and a frame-averaged predictor, paired with a Gaussian-weighted loss and receptive-aware inference over geodesic patches. Key contributions include the efficient random-frame approach (8x efficiency), a loss that blends regression and angular terms with distance-based weighting, and a geodesic-patch inference strategy with weighted aggregation. Empirically, the method achieves state-of-the-art RMSE across PCPNet, SceneNN, and FamousShape, and demonstrates practical utility for Poisson reconstruction and denoising, signaling strong impact for downstream 3D geometry processing tasks.

Abstract

Point cloud normal estimation is a fundamental task in 3D geometry processing. While recent learning-based methods achieve notable advancements in normal prediction, they often overlook the critical aspect of equivariance. This results in inefficient learning of symmetric patterns. To address this issue, we propose E3-Net to achieve equivariance for normal estimation. We introduce an efficient random frame method, which significantly reduces the training resources required for this task to just 1/8 of previous work and improves the accuracy. Further, we design a Gaussian-weighted loss function and a receptive-aware inference strategy that effectively utilizes the local properties of point clouds. Our method achieves superior results on both synthetic and real-world datasets, and outperforms current state-of-the-art techniques by a substantial margin. We improve RMSE by 4% on the PCPNet dataset, 2.67% on the SceneNN dataset, and 2.44% on the FamousShape dataset.

E$^3$-Net: Efficient E(3)-Equivariant Normal Estimation Network

TL;DR

The paper tackles robust point cloud normal estimation by enforcing -equivariance while preserving computational efficiency. It introduces -equivariant learning through a random-frame training scheme and a frame-averaged predictor, paired with a Gaussian-weighted loss and receptive-aware inference over geodesic patches. Key contributions include the efficient random-frame approach (8x efficiency), a loss that blends regression and angular terms with distance-based weighting, and a geodesic-patch inference strategy with weighted aggregation. Empirically, the method achieves state-of-the-art RMSE across PCPNet, SceneNN, and FamousShape, and demonstrates practical utility for Poisson reconstruction and denoising, signaling strong impact for downstream 3D geometry processing tasks.

Abstract

Point cloud normal estimation is a fundamental task in 3D geometry processing. While recent learning-based methods achieve notable advancements in normal prediction, they often overlook the critical aspect of equivariance. This results in inefficient learning of symmetric patterns. To address this issue, we propose E3-Net to achieve equivariance for normal estimation. We introduce an efficient random frame method, which significantly reduces the training resources required for this task to just 1/8 of previous work and improves the accuracy. Further, we design a Gaussian-weighted loss function and a receptive-aware inference strategy that effectively utilizes the local properties of point clouds. Our method achieves superior results on both synthetic and real-world datasets, and outperforms current state-of-the-art techniques by a substantial margin. We improve RMSE by 4% on the PCPNet dataset, 2.67% on the SceneNN dataset, and 2.44% on the FamousShape dataset.
Paper Structure (17 sections, 1 theorem, 16 equations, 14 figures, 4 tables, 1 algorithm)

This paper contains 17 sections, 1 theorem, 16 equations, 14 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Methodology
  5. Efficient E(3)-Equivariance
  6. Loss Function
  7. Inference
  8. Experimental Results
  9. Comparison to Prior Work
  10. Ablation Studies
  11. Applications
  12. Conclusions, Limitations, and Future Work
  13. Theoretical Proof
  14. Details of the Used Datasets
  15. Assessments on the FamousShape Dataset
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathbf{x}\in\mathbb{R}^{m\times 3}$ be an input point patch and $f\in \mathrm{E}(3)$ be an operation. Then the function $\Phi$ defined in Eq. eq:frame_ave satisfies

Figures (14)

  • Figure 1: An illustration of the E(3)-equivariance, where $\Phi$ is an E(3)-equivariant function predicting a point's normal, and $g$ is an element of the E(3) group. These two operators are commutative, meaning that the order in which they are applied does not affect the final outcome.
  • Figure 2: Illustration of the proposed framework. The red arrows depict the training process, while the black arrows represent the inference phase. The diagram only shows 4 elements in the frame set.
  • Figure 3: The proposed distance weighting scheme for normal estimation. The length of the red arrows indicates the loss weight for the point. From the left to right are the point-based method, $\mathcal{L}^{\text{Val}}$, and $\mathcal{L}^{\text{Gau}}$.
  • Figure 4: Difference between Euclidean (left) and geodesic patches (right). Euclidean patches contain points from both sides, whereas geodesic patches only have points from one side.
  • Figure 5: Qualitative comparisons of normal estimation on the PCPNet and SceneNN dataset. The values below each model indicate the RMSE deviation.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof