Manifolds of Projective Shapes
share
The projective shape of a configuration of k points or "landmarks" in RP(d) consists of the information that is invariant under projective transformations. Mathematically, the space of projective shapes for these k landmarks can be described as the quotient space of k copies of RP(d) modulo the action of the projective linear group PGL(d). The main purpose of this paper is to give a detailed examination of the topology of projective shape space, and it is shown how to derive subsets that are in a certain sense maximal, differentiable Hausdorff manifolds which can be provided with a Riemannian metric. A special subclass of the projective shapes consists of the Tyler regular shapes, for which geometrically motivated pre-shapes can be defined, thus allowing for the construction of a natural Riemannian metric.
READ FULL TEXT