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.
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
About the Author :
Ian Dryden, University of Nottingham, UK.
Kanti Mardia, University of Leeds and University of Oxford, UK.
Review :
"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)
