quaternion algebra

This work addresses redundancy and poor numerical stability in conventional modeling of pose, spatial velocity (twist), and generalized forces (wrench) in rigid-body kinematics. We propose a unified geometric representation framework based on dual quaternions. Methodologically, we systematically establish complete algebraic correspondences among pose transformations, differential twist motions, and wrench mappings, integrating dual-number algebra, quaternion theory, and the se(3) Lie algebra structure. Our key contribution is the first pedagogically oriented, coherent mapping framework unifying these three fundamental kinematic elements—revealing intrinsic advantages over homogeneous matrices in algebraic conciseness, differential consistency, and computational compactness. The approach significantly simplifies formula derivation in dynamics modeling and real-time control, reduces computational redundancy, and enables direct integration into robotic motion planning and control algorithms.

This work addresses the lack of systematic statistical and algorithmic support for hypercomplex algebras—particularly quaternions—in current machine learning methods, which hinders effective modeling of high-dimensional data such as 3D rotations. The study presents the first comprehensive statistical framework for quaternion-valued stochastic processes and introduces generalized linear models grounded in quaternion calculus and mean-square estimation theory, thereby establishing a unified foundation for quaternion-based machine learning. By bridging hypercomplex algebra with modern machine learning, this research provides directly applicable theoretical tools and algorithmic paradigms for modeling 3D rotations and processing high-dimensional signals, significantly advancing the practical adoption of quaternions in machine learning applications.

Traditional quaternion neural networks suffer from complex-domain arithmetic operations, training instability, and low parameter efficiency. To address these issues, this paper proposes Quaternion Approximate Networks (QUAN): it approximates quaternion convolution via real-valued decomposition of the Hamilton product, enabling efficient rotation-equivariant learning entirely within the real domain for the first time; introduces Independent Quaternion Batch Normalization (IQBN) to enhance training stability; and designs a quaternion spatial attention mechanism to improve geometric awareness. QUAN consistently outperforms standard CNNs and existing quaternion CNNs on image classification (CIFAR-10/100, ImageNet) and oriented object detection (COCO, DOTA), achieving higher accuracy while significantly reducing both parameter count and computational cost—making it particularly suitable for resource-constrained robotic vision systems.

This work proposes a dual quaternion-based 6-degree-of-freedom visual object tracking framework to address the sensitivity of conventional P$n$P methods to noise and outliers, as well as their difficulty in handling missing observations. By analyzing system observability through Lie algebra, the approach introduces a measurement model based on unit vectors and relative positions, and uniquely integrates Lie group unscented Kalman filtering with dual quaternions to achieve robust state estimation. The method provides a control-theoretic interpretation of the collinearity-induced degeneracy in P3P and is applicable to non-cooperative, non-smooth motion scenarios. Simulations demonstrate significant improvements over existing P$n$P solvers in both pose accuracy and robustness under occlusion, highlighting its suitability for applications such as visual-inertial navigation and SLAM.

This work addresses the fundamental challenge of recovering absolute poses from noisy pairwise relative transformations in robotics and 3D vision. We propose a novel SE(3) synchronization method based on dual quaternions, which directly models the problem on the unit dual quaternion manifold. Our approach features a two-stage algorithm comprising spectral initialization followed by a Dual Quaternion Generalized Power Method (DQGPM), with iterative projection steps to enforce feasibility. Notably, this is the first method to provide finite-step error bounds and linear error contraction guarantees for SE(3) synchronization, circumventing the need for conventional multi-stage heuristic pipelines. Experiments on both synthetic data and real-world multi-frame point cloud registration demonstrate that our method significantly outperforms prevailing matrix-based approaches, achieving a unified improvement in accuracy, computational efficiency, and theoretical tractability.

This work addresses the issue of temporal inconsistency and jitter in existing video-based 3D human motion capture methods, which often stems from neglecting inter-frame coherence—particularly in online settings where Euler angle representations lead to discontinuous poses. To this end, we propose QuaMo, the first approach to integrate quaternion differential equations (QDEs) into visual motion capture, modeling pose evolution over time under unit-sphere constraints to guarantee continuity. QuaMo further incorporates a state-space model for real-time kinematics and introduces an acceleration-augmented meta-PD controller that adaptively modulates control signals to enhance dynamic response accuracy. Extensive experiments demonstrate that QuaMo outperforms current methods across multiple benchmarks—including Human3.6M, Fit3D, SportsPose, and AIST—achieving high-fidelity, low-distortion, and temporally seamless 3D human motion reconstruction.

To address the challenge of modeling intrinsic coupling among RGB channels in color image classification, this paper proposes the Low-Rank Supported Quaternion Matrix Machine (LSQMM). LSQMM represents RGB images as pure quaternion matrices, leveraging quaternion algebra to inherently preserve inter-channel phase relationships and structural correlations. It introduces, for the first time, a quaternion nuclear norm regularizer integrated with the hinge loss to formulate a low-rank optimization framework. An efficient ADMM-based algorithm is designed for solving the resulting optimization problem. Extensive experiments on multiple benchmark datasets demonstrate that LSQMM significantly outperforms conventional SVMs, Support Matrix Machines, and Tensor Machines. It achieves consistent improvements in classification accuracy, robustness to noise, and computational efficiency—overcoming fundamental limitations of real-valued vector- or tensor-based representations in capturing the intrinsic chromatic coupling structure.

This work addresses the challenge of deploying deep neural networks on edge devices, where dense linear operators incur prohibitive memory bandwidth and computational costs. The authors propose a novel approach that integrates quaternion channel coupling with a block-circulant structure to construct linear and convolutional layers of low shift rank. For the first time, they introduce a complex adjoint representation to enable efficient FFT-based inference. This design achieves substantial improvements in parameter compression and computational efficiency while preserving model accuracy. Experimental results demonstrate that the proposed method attains high compression ratios and competitive accuracy across multiple benchmarks—including CIFAR-10/100, SVHN, and hyperspectral image datasets—with consistently low and stable inference latency on both CPU and GPU platforms.

Existing RoPE methods struggle to preserve translational equivariance under high-dimensional scaling; notably, Spherical RoPE suffers from the non-commutativity of spherical rotations, leading to ambiguous definitions of rotated sequences. To address this, we propose QuatRo—the first quaternion-based rotational positional embedding framework—and generalize it to Clifford Algebra Rotational Embedding (CARE). CARE leverages Clifford spinors to uniformly model rotations in arbitrary dimensions and employs a multivector structure for hierarchical positional encoding. Within the geometric algebra framework, CARE rigorously guarantees translational equivariance while subsuming and extending both Mixed RoPE and Spherical RoPE. Experiments demonstrate that CARE achieves superior theoretical soundness, expressive flexibility, and empirical effectiveness in high-dimensional positional modeling.