Articulated and Restricted Motion Subspaces and Their Signatures

Articulated objects represent an important class of objects in our everyday environment. Automatic detection of the type of articulated or otherwise restricted motion and extraction of the corresponding motion parameters are therefore of high value, \eg in order to augment an otherwise static 3D reconstruction with dynamic semantics, such as rotation axes and allowable translation directions for certain rigid parts or objects. Hence, in this paper, a novel theory to analyse relative transformations between two motion-restricted parts will be presented. The analysis is based on linear subspaces spanned by relative transformations. Moreover, a signature for relative transformations will be introduced which uniquely specifies the type of restricted motion encoded in these relative transformations. This theoretic framework enables the derivation of novel algebraic constraints, such as low-rank constraints for subsequent rotations around two fixed axes for example. Lastly, given the type of restricted motion as predicted by the signature, the paper shows how to extract all the motion parameters with matrix manipulations from linear algebra. Our theory is verified on several real data sets, such as a rotating blackboard or a wheel rolling on the floor amongst others.