Book contents
- Frontmatter
- Contents
- Commonly used notation
- Perface
- 1 Introduction
- 2 Examples
- 3 Location and spread on metric spaces
- 4 Extrinsic analysis on manifolds
- 5 Intrinsic analysis on manifolds
- 6 Landmark-based shape spaces
- 7 Kendall's similarity shape spaces ??km
- 8 The planar shape space ??k2
- 9 Reflection similarity shape spaces R??km
- 10 Stiefel manifolds Vk,m
- 11 Affine shape spaces A??km
- 12 Real projective spaces and projective shape spaces
- 13 Nonparametric Bayes inference on manifolds
- 14 Nonparametric Bayes regression, classification and hypothesis testing on manifolds
- Appendix A Differentiable manifolds
- Appendix B Riemannian manifolds
- Appendix C Dirichlet processes
- Appendix D Parametric models on Sd and ??k2
- References
- Index
12 - Real projective spaces and projective shape spaces
Published online by Cambridge University Press: 05 May 2012
- Frontmatter
- Contents
- Commonly used notation
- Perface
- 1 Introduction
- 2 Examples
- 3 Location and spread on metric spaces
- 4 Extrinsic analysis on manifolds
- 5 Intrinsic analysis on manifolds
- 6 Landmark-based shape spaces
- 7 Kendall's similarity shape spaces ??km
- 8 The planar shape space ??k2
- 9 Reflection similarity shape spaces R??km
- 10 Stiefel manifolds Vk,m
- 11 Affine shape spaces A??km
- 12 Real projective spaces and projective shape spaces
- 13 Nonparametric Bayes inference on manifolds
- 14 Nonparametric Bayes regression, classification and hypothesis testing on manifolds
- Appendix A Differentiable manifolds
- Appendix B Riemannian manifolds
- Appendix C Dirichlet processes
- Appendix D Parametric models on Sd and ??k2
- References
- Index
Summary
The real projective space ℝPm is the manifold of all lines through the origin in ℝm+1. It may be identified with the sphere Sm 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 ℝPm 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 (ℝPm)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.
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- Chapter
- Information
- Nonparametric Inference on ManifoldsWith Applications to Shape Spaces, pp. 147 - 155Publisher: Cambridge University PressPrint publication year: 2012