On the Topology of Projective Shape Spaces

The projective shape of a configuration consists of the information that is invariant under projective transformations. It encodes the information about an object reconstructable from uncalibrated camera views. The space of projective shapes of k points in d-dimensional real projective space is by definition the quotient space of k copies of that projective space modulo the action of the projective linear group. A detailed examination of the topology of projective shape space is given, and it is shown how to derive subsets that are maximal Hausdorff manifolds. A special case are Tyler regular shapes for which one can construct a Riemannian metric.