Nonparametric inference on manifolds : with applications to shape spaces

著者

書誌事項

Nonparametric inference on manifolds : with applications to shape spaces

Abhishek Bhattacharya, Rabi Bhattacharya

（Institute of mathematical statistics monographs, 2）

Cambridge University Press, 2015, c2012

  • : pbk

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注記

Originally published: 2012

Includes bibliographical references (p. 229-234) and index

内容説明・目次

内容説明

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 Frechet 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.

目次

  • 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.

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