Learning Rotations 🌀

Mathematical Methods in the Applied Sciences / AGACSE 2021

Citation:

@article{pepe2024learning,
  title={Learning rotations},
  author={Pepe, Alberto and Lasenby, Joan and Chac{'o}n, Pablo},
  journal={Mathematical Methods in the Applied Sciences},
  volume={47},
  number={3},
  pages={1204--1217},
  year={2024},
  publisher={Wiley Online Library}
}

Introduction 📑

In the field of computer vision 🤖, tasks such as pose estimation from images 📸 or 3D point clouds 🌐 are central to many advancements. These problems often involve learning rotations, which is a fundamental aspect of these tasks. However, there is no universally accepted method for parametrizing rotations mathematically. Different representations such as matrices, quaternions, Euler angles, and axis-angle are commonly used, but each of them comes with its own limitations, including gimbal lock, discontinuities, and antipodal symmetry 🌀. These issues can complicate the training of neural networks, potentially leading to significant errors.

This paper proposes a solution by utilizing a geometric algebra (GA) description of rotations. This approach is compared with a 6D continuous representation, and we show that it not only resolves the limitations of previous methods but also outperforms them in terms of regression accuracy and robustness to noise. In this process, we present three detailed case studies that demonstrate the effectiveness of GA in solving these complex problems.

Case Studies 💡

In line with recent literature, we introduce three case studies to illustrate the practical application of GA in rotation learning:

1. A sanity check ✅ – Testing the performance of the proposed method in controlled scenarios to validate its robustness.
2. Pose estimation from 3D point clouds 🌐 – A real-world scenario where GA is applied to estimate poses from 3D data.
3. An inverse kinematic problem 🦾 – Using GA to solve inverse kinematics for robotic systems, showcasing its utility in practical engineering applications.

For each case study, we employ the GA formulation of rotations and compare it with the 6D continuous representation previously presented in the literature. The findings demonstrate that parametrizing rotations as bivectors significantly improves upon the 6D representation by requiring fewer parameters and offering better generalizability.

Why Geometric Algebra? 🤔

Geometric Algebra (GA) offers several advantages over traditional rotation representations:

• Continuity: GA avoids the continuity issues seen with other representations, such as 6D.
• Efficiency: GA requires fewer parameters, making it more efficient to train neural networks for regression tasks.
• Robustness: GA provides enhanced resilience to noise, ensuring better performance in high-noise scenarios 🎯.

Our empirical results show that GA provides a simple, compact, and effective framework for describing rotations in a manner well-suited to deep learning tasks. This approach not only achieves high regression accuracy but also exhibits strong generalizability across various conditions.

Code & Datasets 📊

The code to generate and process the datasets is available in the notebooks, allowing others to replicate and build upon this work. The ready-to-use datasets can be obtained from the authors upon request 📩.

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This paper provides a comprehensive exploration of how Geometric Algebra can be applied to rotation learning, showcasing its practical advantages in various real-world applications.