Nonparametric Inference on Manifolds

With Applications to Shape Spaces

Abhishek Bhattacharya, Rabi Bhattacharya

• Publisher: Cambridge University Press
• Year: 2012
• Print ISBN:
  9781107019584
• eBook ISBN:
  9781139335287
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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.

Table of contents

• Cover
• Nonparametric Inference on Manifolds
• Title
• Copyright
• Contents
• Commonly used notation
• Preface
• 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 Σkm
• 8: The planar shape space Σk2
• 9: Reflection similarity shape spaces RΣkm
• 10: Stiefel manifolds Vk,m
• 11: Affine shape spaces AΣkm
• 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
• References
• Index