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
1 - Introduction
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
Digital images today play a vital role in science and technology, and also in many aspects of our daily life. This book seeks to advance the analysis of images, especially digitized ones, through the statistical analysis of shapes. Its focus is on the analysis of landmark-based shapes in which a k-ad, that is, a set of k labeled points or landmarks on an object or a scene, is observed in two or three dimensions, usually with expert help, for purposes of identification, discrimination, and diagnostics.
In general, consider the k-ad to lie in ℝm (usually, m = 2 or 3) and assume that not all the k points are the same. Then the appropriate shape of the object is taken to be the k-ad modulo a group of transformations.
For example, one may first center the k-ad, by subtracting the mean of the k-ad from each of the k landmarks, to remove the effect of location. The centered k-ad then lies in a hyperplane of dimension mk - m, because the sum of each of the m coordinates of the centered k points is zero. Next one may scale the centered k-ad to unit size to remove the effect of scale or size. The scaled, centered k-ad now lies on the unit sphere Sm(k-1)-1 in a Euclidean space (the hyperplane) of dimension m(k - 1) and is now called the preshape of the k-ad.
- Type
- Chapter
- Information
- Nonparametric Inference on ManifoldsWith Applications to Shape Spaces, pp. 1 - 7Publisher: Cambridge University PressPrint publication year: 2012