# Dual Quaternion Rotational and Translational Equivariance in 3D Rigid Motion Modelling

arXiv: https://arxiv.org/abs/2310.07623 · HTML: https://mlsanity.com/paper/2310.07623/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 27 references that are in the map.
- First posted on arXiv 11 Oct 2023.

## Builds on (most cited first, 12 of 12)

- [On human motion prediction using recurrent neural networks](https://mlsanity.com/paper/1705.02445/) (2017)
- [CLIFF: Carrying Location Information in Full Frames into Human Pose and Shape Estimation](https://mlsanity.com/paper/2208.00571/) (2022)
- [QuaterNet: A Quaternion-based Recurrent Model for Human Motion](https://mlsanity.com/paper/1805.06485/) (2018)
- [Predicting 3D Human Dynamics from Video](https://mlsanity.com/paper/1908.04781/) (2019)
- [Socially and Contextually Aware Human Motion and Pose Forecasting](https://mlsanity.com/paper/2007.06843/) (2020)
- [Modeling Human Motion with Quaternion-based Neural Networks](https://mlsanity.com/paper/1901.07677/) (2019)
- [Disentangling Human Dynamics for Pedestrian Locomotion Forecasting with Noisy Supervision](https://mlsanity.com/paper/1911.01138/) (2019)
- [Quaternion Generative Adversarial Networks](https://mlsanity.com/paper/2104.09630/) (2021)
- [A Quaternion-Valued Variational Autoencoder](https://mlsanity.com/paper/2010.11647/) (2020)
- [Acute Lymphoblastic Leukemia Detection Using Hypercomplex-Valued Convolutional Neural Networks](https://mlsanity.com/paper/2205.13273/) (2022)
- [Quaternion Product Units for Deep Learning on 3D Rotation Groups](https://mlsanity.com/paper/1912.07791/) (2019)
- [Predicting Rigid Body Dynamics using Dual Quaternion Recurrent Neural Networks with Quaternion Attention](https://mlsanity.com/paper/2011.08734/) (2020)

## Built on by (most cited first, 4 of 4)

- [Towards Explaining Hypercomplex Neural Networks](https://mlsanity.com/paper/2403.17929/) (2024)
- [PHemoNet: A Multimodal Network for Physiological Signals](https://mlsanity.com/paper/2410.00010/) (2024)
- [Demystifying the Hypercomplex: Inductive Biases in Hypercomplex Deep Learning](https://mlsanity.com/paper/2405.07024/) (2024)
- [V-EfficientNets: Vector-Valued Efficiently Scaled Convolutional Neural Network Models](https://mlsanity.com/paper/2505.05659/) (2025)

## Abstract (arXiv)

Objects' rigid motions in 3D space are described by rotations and translations of a highly-correlated set of points, each with associated x,y,z coordinates that real-valued networks consider as separate entities, losing information. Previous works exploit quaternion algebra and their ability to model rotations in 3D space. However, these algebras do not properly encode translations, leading to sub-optimal performance in 3D learning tasks. To overcome these limitations, we employ a dual quaternion representation of rigid motions in the 3D space that jointly describes rotations and translations of point sets, processing each of the points as a single entity. Our approach is translation and rotation equivariant, so it does not suffer from shifts in the data and better learns object trajectories, as we validate in the experimental evaluations. Models endowed with this formulation outperform previous approaches in a human pose forecasting application, attesting to the effectiveness of the proposed dual quaternion formulation for rigid motions in 3D space.