Dual Quaternion Discrete Linear Canonical Transforms: FFT-Based Fast Algorithms and Joint Rotation--Translation Pose Denoising
Abstract
Rigid-body pose signals couple rotation and translation, yet most existing denoising pipelines treat them separately and analyse them only in the Fourier domain. We show that this separation is unnecessary and often suboptimal. We fuse dual quaternions, which encode rotation and translation in a single singularity-free algebra, with linear canonical transforms, which supply flexible quadratic-phase analysis, to develop one- and two-dimensional dual quaternion discrete linear canonical transforms (DQ-DLCTs) with an explicit kernel ordering. We establish inverse and energy-preservation properties, structured spatial and frequency convolution relations, and, crucially, fast algorithms built on complex FFTs. These algorithms make the DQ-DLCT practical: a two-stage chirp-adaptive scheme then performs joint rotation--translation denoising in a single transform domain. Numerical experiments confirm the accuracy and efficiency of the fast transform and demonstrate strong denoising performance on measured pose sequences and constructed two-dimensional pose fields. The DQ-DLCT thus turns a heavy algebraic construction into a deployable tool for pose signal processing.
Related
- Order-Specified Quaternion CP Decomposition: Models, Algorithms, and Applications
- Diagonalization of skew-Hermitian dual quaternion matrices and ϕ-Hermitian dual quaternion matrices
- An elementary proof of the Artin--Springer theorem for generalized quadratic forms over quaternion algebras
- Counting quaternion extensions by multi-heights
- Genus of quaternion algebras over formally real fields
BibTeX
@misc{wu2026dual,
title = {{Dual Quaternion Discrete Linear Canonical Transforms: FFT-Based Fast Algorithms and Joint Rotation--Translation Pose Denoising}},
author = {Yi Wu and Xuan Chen and Kit Ian Kou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26140},
eprint = {2610.26140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26140},
}
RIS
TY - UNPB
TI - Dual Quaternion Discrete Linear Canonical Transforms: FFT-Based Fast Algorithms and Joint Rotation--Translation Pose Denoising
AU - Yi Wu
AU - Xuan Chen
AU - Kit Ian Kou
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26140
UR - https://arxiv.org/abs/2610.26140
AB - Rigid-body pose signals couple rotation and translation, yet most existing denoising pipelines treat them separately and analyse them only in the Fourier domain. We show that this separation is unnecessary and often suboptimal. We fuse dual quaternions, which encode rotation and translation in a single singularity-free algebra, with linear canonical transforms, which supply flexible quadratic-phase analysis, to develop one- and two-dimensional dual quaternion discrete linear canonical transforms (DQ-DLCTs) with an explicit kernel ordering. We establish inverse and energy-preservation properties, structured spatial and frequency convolution relations, and, crucially, fast algorithms built on complex FFTs. These algorithms make the DQ-DLCT practical: a two-stage chirp-adaptive scheme then performs joint rotation--translation denoising in a single transform domain. Numerical experiments confirm the accuracy and efficiency of the fast transform and demonstrate strong denoising performance on measured pose sequences and constructed two-dimensional pose fields. The DQ-DLCT thus turns a heavy algebraic construction into a deployable tool for pose signal processing.
ER -