Dual Quaternion Rotational and Translational Equivariance in 3D Rigid
  Motion Modelling
Guilherme Vieira Eleonora Grassucci Marcos Eduardo Valle Danilo Comminiello
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.