Extracting Projective Information from Single Views of 3d Point Sets

A number of recent papers have argued that invariants do not exist for three dimensional point sets in general position 3, 5, 22]. This has often been misinterpreted to mean that invariants cannot be computed for any three dimensional structure. This paper proves by example that although the general statement is true, invariants do exist for structured three dimensional point sets. Projective invariants are derived for two classes of object: the rst is for points that lie on the vertices of polyhedra, and the second for objects that are projectively equivalent to ones possessing a bilateral symmetry. The rst approach to computing invariants is not restricted to measurements on physical poly-hedra, but extends to more general structures whose features can be embedded within conceptual polyhedra. This is a result of using an algebraic framework of constraints between points and planes, and yet not having to observe any actual plane faces. This is called caging. The motivations for computing such invariants are twofold: rstly they can be used for recognition; secondly they can be used to compute projective structure. Examples of invariants computed from real images are given.