Nonparametric Inference on Manifolds
With Applications to Shape Spaces

A systematic introduction to a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes.

Abhishek Bhattacharya (Author), Rabi Bhattacharya (Author)

9781107019584, Cambridge University Press

Hardback, published 5 April 2012

252 pages, 20 b/w illus.
23.4 x 15.6 x 2 cm, 0.52 kg

'… this is an excellent text that will benefit many students in computer science, mathematics, and physics … A significant plus of the book is the library of MATLAB codes and datasets available for download from the authors' site.' Alexander Tzanov, Computing Reviews

This book introduces in a systematic manner a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes. The theory has important and varied applications in medical diagnostics, image analysis, and machine vision. An early chapter of examples establishes the effectiveness of the new methods and demonstrates how they outperform their parametric counterparts. Inference is developed for both intrinsic and extrinsic Fréchet means of probability distributions on manifolds, then applied to shape spaces defined as orbits of landmarks under a Lie group of transformations - in particular, similarity, reflection similarity, affine and projective transformations. In addition, nonparametric Bayesian theory is adapted and extended to manifolds for the purposes of density estimation, regression and classification. Ideal for statisticians who analyze manifold data and wish to develop their own methodology, this book is also of interest to probabilists, mathematicians, computer scientists, and morphometricians with mathematical training.

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
11. Affine shape spaces A?km
12. Real projective spaces and projective shape spaces
13. Nonparametric Bayes inference
14. Regression, classification and testing
i. Differentiable manifolds
ii. Riemannian manifolds
iii. Dirichlet processes
iv. Parametric models on Sd and ?k2
References
Subject index.

Subject Areas: Probability & statistics [PBT]