三维刚体运动建模中的双四元数旋转与平移等变性
Dual Quaternion Rotational and Translational Equivariance in 3D Rigid Motion Modelling
AI总结：
针对实值网络丢失三维刚体运动信息的问题，提出双四元数表示法联合建模旋转与平移，实现等变性，提升人体姿态预测性能。
摘要：
三维空间中物体的刚体运动由一组高度相关的点的旋转和平移来描述，每个点都关联有$x,y,z$坐标，而实值网络将这些坐标视为独立实体，从而丢失了信息。以往的工作利用了四元数代数及其在三维空间中建模旋转的能力。然而，这些代数不能正确编码平移，导致在三维学习任务中性能次优。为克服这些限制，我们采用三维空间中刚体运动的双四元数表示，该表示联合描述点集的旋转和平移，将每个点作为单一实体处理。我们的方法具有平移和旋转等变性，因此不受数据偏移的影响，并能更好地学习物体轨迹，这一点在实验评估中得到了验证。采用该公式的模型在人体姿态预测应用中优于以往方法，证明了所提出的双四元数公式在三维空间刚体运动中的有效性。
英文摘要：
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.
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