Statistical Shape Analysis: With Applications in R

by Ian L. Dryden, Kanti V. Mardia

Overview

A thoroughly revised and updated edition of this introduction to modern statistical methods for shape analysis

Shape analysis is an important tool in the many disciplines where objects are compared using geometrical features.  Examples include comparing brain shape in schizophrenia; investigating protein molecules in bioinformatics; and describing growth of organisms in biology.

This book is a significant update of the highly-regarded Statistical Shape Analysis by the same authors. The new edition lays the foundations of landmark shape analysis, including geometrical concepts and statistical techniques, and extends to include analysis of curves, surfaces, images and other types of object data. Key definitions and concepts are discussed throughout, and the relative merits of different approaches are presented.

The authors have included substantial new material on recent statistical developments and offer numerous examples throughout the text. Concepts are introduced in an accessible manner, while retaining sufficient detail for more specialist statisticians to appreciate the challenges and opportunities of this new field. Computer code has been included for instructional use, along with exercises to enable readers to implement the applications themselves in R and to follow the key ideas by hands-on analysis.

• Offers a detailed yet accessible treatment of statistical methods for shape analysis
• Includes numerous examples and applications from many disciplines
• Provides R code for implementing the examples
• Covers a wide variety of recent developments in shape analysis

Shape Analysis, with Applications in R will offer a valuable introduction to this fast-moving research area for statisticians and other applied scientists working in diverse areas, including archaeology, bioinformatics, biology, chemistry, computer science, medicine, morphometics and image analysis.

Product Details

ISBN-13:	9781119072515
Publisher:	Wiley
Publication date:	07/08/2016
Series:	Wiley Series in Probability and Statistics , #995
Sold by:	JOHN WILEY & SONS
Format:	eBook
Pages:	496
File size:	13 MB
Note:	This product may take a few minutes to download.

About the Author

Ian Dryden, University of Nottingham, UK.

Kanti Mardia, University of Leeds and University of Oxford, UK.

Table of Contents

Preface xix

Preface to first edition xxi

Acknowledgements for the first edition xxv

1 Introduction 1

1.1 Definition and motivation 1

1.2 Landmarks 3

1.3 The shapes package in R 7

1.4 Practical applications 8

2 Size measures and shape coordinates 31

2.1 History 31

2.2 Size 33

2.3 Traditional shape coordinates 39

2.4 Bookstein shape coordinates 41

2.5 Kendall’s shape coordinates 49

2.6 Triangle shape coordinates 52

3 Manifolds, shape and size-and-shape 59

3.1 Riemannian manifolds 59

3.2 Shape 61

3.3 Size-and-shape 66

3.4 Reflection invariance 66

3.5 Discussion 67

4 Shape space 69

4.1 Shape space distances 69

4.2 Comparing shape distances 77

4.3 Planar case 82

4.4 Tangent space coordinates 88

5 Size-and-shape space 99

5.1 Introduction 99

5.2 Root mean square deviation measures 99

5.3 Geometry 101

5.4 Tangent coordinates for size-and-shape space 103

5.5 Geodesics 104

5.6 Size-and-shape coordinates 104

5.7 Allometry 107

6 Manifold means 111

6.1 Intrinsic and extrinsic means 111

6.2 Population mean shapes 112

6.3 Sample mean shape 113

6.4 Comparing mean shapes 115

6.5 Calculation of mean shapes in R 118

6.6 Shape of the means 120

6.7 Means in size-and-shape space 121

6.8 Principal geodesic mean 122

6.9 Riemannian barycentres 122

7 Procrustes analysis 125

7.1 Introduction 125

7.2 Ordinary Procrustes analysis 126

7.3 Generalized Procrustes analysis 134

7.4 Generalized Procrustes algorithms for shape analysis 136

7.5 Generalized Procrustes algorithms for size-and-shape analysis 143

7.6 Variants of generalized Procrustes analysis 145

7.7 Shape variability: principal component analysis 150

7.8 Principal component analysis for size-and-shape 166

7.9 Canonical variate analysis 166

7.10 Discriminant analysis 168

7.11 Independent component analysis 169

7.12 Bilateral symmetry 171

8 2D Procrustes analysis using complex arithmetic 175

8.1 Introduction 175

8.2 Shape distance and Procrustes matching 175

8.3 Estimation of mean shape 178

8.4 Planar shape analysis in R 181

8.5 Shape variability 182

9 Tangent space inference 185

9.1 Tangent space small variability inference for mean shapes 185

9.2 Inference using Procrustes statistics under isotropy 197

9.3 Size-and-shape tests 206

9.4 Edge-based shape coordinates 212

9.5 Investigating allometry 212

10 Shape and size-and-shape distributions 217

10.1 The uniform distribution 217

10.2 Complex Bingham distribution 219

10.3 Complex Watson distribution 227

10.4 Complex angular central Gaussian distribution 231

10.5 Complex Bingham quartic distribution 231

10.6 A rotationally symmetric shape family 232

10.7 Other distributions 233

10.8 Bayesian inference 233

10.9 Size-and-shape distributions 237

10.10 Size-and-shape versus shape 237

11 Offset normal shape distributions 239

11.1 Introduction 239

11.2 Offset normal shape distributions with general covariances 252

11.3 Inference for offset normal distributions 255

11.4 Practical inference 258

11.5 Offset normal size-and-shape distributions 259

11.6 Distributions for higher dimensions 264

12 Deformations for size and shape change 269

12.1 Deformations 269

12.2 Affine transformations 272

12.3 Pairs of thin-plate splines 279

12.4 Alternative approaches and history 303

12.5 Kriging 307

12.6 Diffeomorphic transformations 314

13 Non-parametric inference and regression 317

13.1 Consistency 317

13.2 Uniqueness of intrinsic means 318

13.3 Non-parametric inference 322

13.4 Principal geodesics and shape curves 324

13.5 Statistical shape change 332

13.6 Robustness 336

13.7 Incomplete data 338

14 Unlabelled size-and-shape and shape analysis 341

14.1 The Green–Mardia model 342

14.2 Procrustes model 345

14.3 Related methods 348

14.4 Unlabelled points 349

15 Euclidean methods 353

15.1 Distance-based methods 353

15.2 Multidimensional scaling 353

15.3 Multidimensional scaling shape means 354

15.4 Euclidean distance matrix analysis for size-and-shape analysis 357

15.5 Log-distances and multivariate analysis 360

15.6 Euclidean shape tensor analysis 361

15.7 Distance methods versus geometrical methods 362

16 Curves, surfaces and volumes 363

16.1 Shape factors and random sets 363

16.2 Outline data 364

16.3 Semi-landmarks 368

16.4 Square root velocity function 369

16.5 Curvature and torsion 374

16.6 Surfaces 374

16.7 Curvature, ridges and solid shape 375

17 Shape in images 377

17.1 Introduction 377

17.2 High-level Bayesian image analysis 378

17.3 Prior models for objects 380

17.4 Warping and image averaging 382

18 Object data and manifolds 391

18.1 Object oriented data analysis 391

18.2 Trees 392

18.3 Topological data analysis 393

18.4 General shape spaces and generalized Procrustes methods 393

18.5 Other types of shape 395

18.6 Manifolds 396

18.7 Reviews 396

Exercises 399

Appendix 403

References 407

Index 449

Editorial Reviews

"This is really an excellent, masterly, authoritative book about the statistical shape and size-and-shape analysis of landmark data. It provides the conceptual elements and then specific relationships and equations, working toward the various applications. The main results and equations are given in the text. In addition, there is a lot of information about how to use the tools...The book is well written with a well-integrated system of terms, notations, and derivations. Numerous elements on the historical background are provided. The reviewer highly recommends the reading of this book." (Mathematical Reviews/MathSciNet, July 2017)

"Statistical methods applied to shape analysis. Great for biologists, but strong mathematical treatment and accompanying code expands possible applications." (Raspberry Pi, March 2017)

From the Publisher