12 - Real projective spaces and projective shape spaces

Summary

The real projective space ℝP^m is the manifold of all lines through the origin in ℝ^m+1. It may be identified with the sphere S^m modulo the identification of a point p with its antipodal point -p. For m = 1 or 2, this is the axial space. Extrinsic and intrinsic inferences on ℝP^m are developed first in this chapter. This aids in the nonparametric analysis on the space of projective shapes, identified here, subject to a registration, with (ℝP^m)^k-m-2.

Introduction

Consider a k-ad picked on a planar image of an object or a scene in three dimensions. If one thinks of images or photographs obtained through a central projection (a pinhole camera is an example of this), a ray is received as a landmark on the image plane (e.g., the film of the camera). Because axes in three dimensions comprise the projective space ℝP², the k-ad in this view is valued in ℝP². For a k-ad in three dimensions to represent a k-ad in ℝP², the corresponding axes must all be distinct. To have invariance with regard to camera angles, one may first look at the original noncollinear three-dimensional k-ad u and achieve affine invariance by its affine shape (i.e., by the equivalence class Au, A ∈ GL(3, ℝ)), and finally take the corresponding equivalence class of axes in ℝP² to define the projective shape of the k-ad as the equivalence class, or orbit, with respect to projective transformations on ℝP².