Understanding the Shape Properties of Trihedral Polyhedra
Résumé
This paper presents a framework for the computation of projective invariants of trihedral polyhedra. Invariants for such shapes, which are composed of triples of planes meeting at vertices, were first discussed by Rothwell, {\em et al.} in~\cite{rothwell-forsyth-etal:93}. However, they treated only four degree of freedom objects (dof.), and not polyhedra in general. We extend their results to arbitrary dof. figures by showing that more complex shapes can be broken down into sets of connected four dof. polyhedra. Although the more general shapes do not possess projective properties as a whole (when viewed by a single camera), each subpart does yield a projective description. Furthermore, planar projective invariants can be measured which link together the subparts. Consequently, we are able to provide local-global descriptions for general trihedral polyhedra. The original projective description also involved a difficult mathematical analysis. We demonstrate that a set of {\em butterfly invariants} can be used in equivalence to the original formulation. We also provide a novel algebraic formulation of the butterfly invariant which is simpler both to implement and to understand than previous approaches. We therefore provide a more straightforward route to the computation of polyhedral invariants. Finally, we demonstrate the recovery of polyhedral shape descriptions from images by exploiting the local-global nature of the invariants. Local measures can be extracted far more reliably in real images due to the problems of feature segmentation. Additionally, with the aid of a model base, we can extend the local descriptions to global measures and move towards the recognition of entire polyhedra.