# Statistical Shape Analysis: With Applications in R, 2nd Edition

## Books

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.

Statistical 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

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1 Introduction 1

1.1 Definition and Motivation 1

1.2 Landmarks 3

1.3 The shapes package in R 6

1.4 Practical Applications 8

1.4.1 Biology: Mouse vertebrae 8

1.4.2 Image analysis: Postcode recognition 11

1.4.3 Biology: Macaque skulls 12

1.4.4 Chemistry: Steroid molecules 15

1.4.5 Medicine: SchizophreniaMR images 16

1.4.6 Medicine and law: Fetal Alcohol Spectrum Disorder 16

1.4.7 Pharmacy: DNA molecules 18

1.4.8 Biology: Great ape skulls 19

1.4.9 Bioinformatics: Protein matching 22

1.4.10 Particle science: Sand grains 22

1.4.11 Biology: Rat skull growth 24

1.4.12 Biology: Sooty mangabeys 25

1.4.13 Physiotherapy: Human movement data 25

1.4.14 Genetics: Electrophoretic gels 26

1.4.15 Medicine: Cortical surface shape 26

1.4.16 Geology:Microfossils 28

1.4.17 Geography: Central Place Theory 29

1.4.18 Archaeology: Alignments of standing stones 32

2 Size measures and shape coordinates 33

2.1 History 33

2.2 Size 35

2.2.1 Configuration space 35

2.2.2 Centroid size 35

2.2.3 Other size measures 38

2.3 Traditional shape coordinates 41

2.3.1 Angles 41

2.3.2 Ratios of lengths 42

2.3.3 Penrose coefficent 43

2.4 Bookstein shape coordinates 44

2.4.1 Planar landmarks 44

2.4.2 Bookstein-type coordinates for three dimensional data 49

2.5 Kendall’s shape coordinates 51

2.6 Triangle shape co-ordinates 53

2.6.1 Bookstein co-ordinates for triangles 53

2.6.2 Kendall’s spherical coordinates for triangles 56

2.6.3 Spherical projections 58

2.6.4 Watson’s triangle coordinates 58

3 Manifolds, shape and size-and-shape 61

3.1 Riemannian Manifolds 61

3.2 Shape 63

3.2.1 Ambient and quotient space 63

3.2.2 Rotation 63

3.2.3 Coincident and collinear points 65

3.2.4 Filtering translation 65

3.2.5 Pre-shape 65

3.2.6 Shape 66

3.3 Size-and-shape 67

3.4 Reflection invariance 68

3.5 Discussion 69

3.5.1 Standardizations 69

3.5.2 Over-dimensioned case 69

3.5.3 Hierarchies 70

4 Shape space 71

4.1 Shape space distances 71

4.1.1 Procrustes distances 71

4.1.2 Procrustes 74

4.1.3 Differential geometry 74

4.1.4 Riemannian distance 76

4.1.5 Minimal geodesics in shape space 77

4.1.6 Planar shape 77

4.1.7 Curvature 79

4.2 Comparing shape distances 79

4.2.1 Relationships 79

4.2.2 Shape distances in R 79

4.2.3 Further discussion 82

4.3 Planar case 84

4.3.1 Complex arithmetic 84

4.3.2 Complex projective space 85

4.3.3 Kent’s polar pre-shape coordinates 87

4.3.4 Triangle case 88

4.4 Tangent space co-ordinates 90

4.4.1 Tangent spaces 90

4.4.2 Procrustes tangent co-ordinates 91

4.4.3 Planar Procrustes tangent co-ordinates 93

4.4.4 Higher dimensional Procrustes tangent co-ordinates 97

4.4.5 Inverse exponential map tangent-coordinates 98

4.4.6 Procrustes residuals 98

4.4.7 Other tangent co-ordinates 99

4.4.8 Tangent space coordinates in R 99

5 Size-and-shape space 101

5.1 Introduction 101

5.2 RMSD measures 101

5.3 Geometry 102

5.4 Tangent co-ordinates for size-and-shape space 105

5.5 Geodesics 105

5.6 Size-and-shape co-ordinates 106

5.6.1 Bookstein-type coordinates for size-and-shape analysis 106

5.6.2 Goodall–Mardia QR size-and-shape co-ordinates 107

5.7 Allometry 108

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 117

6.6 Shape of the means 120

6.7 Means in size-and-shape space 120

6.7.1 Fr´echet and Karcher means 120

6.7.2 Size-and-shape of the means 121

6.8 Principal geodesic mean 121

6.9 Riemannian barycentres 122

7 Procrustes analysis 123

7.1 Introduction 123

7.2 Ordinary Procrustes analysis 124

7.2.1 Full ordinary Procrustes analysis 124

7.2.2 Ordinary Procrustes analysis in R 127

7.2.3 Ordinary partial Procrustes 129

7.2.4 Reflection Procrustes 130

7.3 Generalized Procrustes analysis 131

7.3.1 Introduction 131

7.4 Generalized Procrustes algorithms for shape analysis 135

7.4.1 Algorithm: GPA-Shape-1 135

7.4.2 Algorithm: GPA-Shape-2 137

7.4.3 GPA in R 137

7.5 Generalized Procrustes algorithms for size-and-shape analysis 140

7.5.1 Algorithm: GPA-Size-and-Shape-1 140

7.5.2 Algorithm: GPA-Size-and-Shape-2 141

7.5.3 Partial generalized Procrustes analysis in R 141

7.5.4 Reflection generalized Procrustes analysis in R 141

7.6 Variants of generalized Procrustes Analysis 142

7.6.1 Summary 142

7.6.2 Unit size partial Procrustes 142

7.6.3 Weighted Procrustes analysis 143

7.7 Shape variability: principal components analysis 147

7.7.1 Shape PCA 147

7.7.2 Kent’s shape PCA 149

7.7.3 Shape PCA in R 149

7.7.4 Point distribution models 162

7.7.5 PCA in shape analysis and multivariate analysis 164

7.8 PCA for size-and-shape 164

7.9 Canonical variate analysis 165

7.10 Discriminant analysis 167

7.11 Independent components analysis 168

7.12 Bilateral symmetry 170

8 2D Procrustes analysis using complex arithmetic 173

8.1 Introduction 173

8.2 Shape distance and Procrustes matching 173

8.3 Estimation of mean shape 176

8.4 Planar shape analysis in R 178

8.5 Shape variability 179

9 Tangent space inference 185

9.1 Tangent space small variability inference for mean shapes 185

9.1.1 One sample Hotelling’s T 2 test 185

9.1.2 Two independent sample Hotelling’s T 2 test 188

9.1.3 Permutation and bootstrap tests 193

9.1.4 Fast permutation and bootstrap tests 194

9.1.5 Extensions and regularization 196

9.2 Inference using Procrustes statistics under isotropy 196

9.2.1 One sample Goodall’s F test 197

9.2.2 Two independent sample Goodall’s F test 199

9.2.3 Further two sample tests 203

9.2.4 One way analysis of variance 204

9.3 Size-and-shape tests 205

9.3.1 Tests using Procrustes size-and-shape tangent space 205

9.3.2 Case-study: Size-and-shape analysis and mutation 207

9.4 Edge-based shape coordinates 210

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

10.2.1 The density 219

10.2.2 Relation to the complex normal distribution 220

10.2.3 Relation to real Bingham distribution 220

10.2.4 The normalizing constant 221

10.2.5 Properties 221

10.2.6 Inference 223

10.2.7 Approximations and computation 224

10.2.8 Relationship with the Fisher-von Mises distribution 225

10.2.9 Simulation 226

10.3 ComplexWatson distribution 226

10.3.1 The density 226

10.3.2 Inference 227

10.3.3 Large concentrations 228

10.4 Complex Angular central Gaussian distribution 230

10.5 Complex Bingham quartic distribution 230

10.6 A rotationally symmetric shape family 230

10.7 Other distributions 231

10.8 Bayesian inference 232

10.9 Size-and-shape distributions 234

10.9.1 Rotationally symmetric size-and-shape family 234

10.9.2 Central complex Gaussian distribution 236

10.10Size-and-shape versus shape 236

11 Offset normal shape distributions 237

11.1 Introduction 237

11.1.1 Equal mean case in two dimensions 237

11.1.2 The isotropic case in two dimensions 242

11.1.3 The triangle case 246

11.1.4 Approximations: Large and small variations 247

11.1.5 Exact Moments 249

11.1.6 Isotropy 249

11.2 Offset normal shape distributions with general covariances 250

11.2.1 The complex normal case 251

11.2.2 General covariances: small variations 251

11.3 Inference for offset normal distributions 253

11.3.1 General MLE 253

11.3.2 Isotropic case 253

11.3.3 Exact istropic MLE in R 256

11.3.4 EM algorithm and extensions 256

11.4 Practical Inference 257

11.5 Offset normal size-and-shape distributions 257

11.5.1 The isotropic case 258

11.5.2 Inference using the offset normal size-and-shape model 260

11.6 Distributions for higher dimensions 262

11.6.1 Introduction 262

11.6.2 QR Decomposition 262

11.6.3 Size-and-shape distributions 263

11.6.4 Multivariate approach 264

11.6.5 Approximations 265

12 Deformations for size and shape change 267

12.1 Deformations 267

12.1.1 Introduction 267

12.1.2 Definition and desirable properties 268

12.1.3 D’Arcy Thompson’s transformation grids 268

12.2 Affine transformations 270

12.2.1 Exact match 270

12.2.2 Least squares matching: Two objects 270

12.2.3 Least squares matching: Multiple objects 272

12.2.4 The triangle case: Bookstein’s hyperbolic shape space 275

12.3 Pairs of Thin-plate Splines 277

12.3.1 Thin-plate splines 277

12.3.2 Transformation grids 279

12.3.3 Thin-plate splines in R 282

12.3.4 Principal and partial warp decompositions 287

12.3.5 Principal component analysis with non-Euclidean metrics 296

12.3.6 Relative warps 299

12.4 Alternative approaches and history 303

12.4.1 Early transformation grids 303

12.4.2 Finite element analysis 306

12.4.3 Biorthogonal grids 309

12.5 Kriging 309

12.5.1 Universal kriging 309

12.5.2 Deformations 311

12.5.3 Intrinsic kriging 311

12.5.4 Kriging with derivative constraints 313

12.5.5 Smoothed matching 313

12.6 Diffeomorphic transformations 315

13 Non-parametric inference and regression 317

13.1 Consistency 317

13.2 Uniqueness of intrinsic means 318

13.3 Non-parametric inference 321

13.3.1 Central limit theorems and non-parametric tests 321

13.3.2 M-estimators 323

13.4 Principal geodesics and shape curves 323

13.4.1 Tangent space methods and longitudinal data 323

13.4.2 Growth curve models for triangle shapes 325

13.4.3 Geodesic hypothesis 325

13.4.4 Principal geodesic analysis 326

13.4.5 Principal nested spheres and shape spaces 327

13.4.6 Unrolling and unwrapping 328

13.4.7 Manifold splines 331

13.5 Statistical shape change 333

13.5.1 Geometric components of shape change 334

13.5.2 Paired shape distributions 336

13.6 Robustness 336

13.7 Incomplete Data 340

14 Unlabelled size-and-shape and shape analysis 341

14.1 The Green-Mardia model 342

14.1.1 Likelihood 342

14.1.2 Prior and posterior distributions 343

14.1.3 MCMC simulation 344

14.2 Procrustes model 346

14.2.1 Prior and posterior distributions 347

14.2.2 MCMC Inference 347

14.3 Related methods 349

14.4 Unlabelled Points 350

14.4.1 Flat triangles and alignments 350

14.4.2 Unlabelled shape densities 351

14.4.3 Further probabilistic issues 351

14.4.4 Delaunay triangles 352

15 Euclidean methods 355

15.1 Distance-based methods 355

15.2 Multidimensional scaling 355

15.2.1 Classical MDS 355

15.2.2 MDS for size-and-shape 356

15.3 MDS shape means 356

15.4 EDMA for size-and-shape analysis 359

15.4.1 Mean shape 359

15.4.2 Tests for shape difference 360

15.5 Log-distances and multivariate analysis 362

15.6 Euclidean shape tensor analysis 363

15.7 Distance methods versus geometrical methods 363

16 Curves, surfaces and volumes 365

16.1 Shape factors and random sets 365

16.2 Outline data 366

16.2.1 Fourier series 366

16.2.2 Deformable template outlines 367

16.2.3 Star-shaped objects 368

16.2.4 Featureless outlines 369

16.3 Semi-landmarks 370

16.4 Square root velocity function 371

16.4.1 SRVF and quotient space for size-and-shape 371

16.4.2 Quotient space inference 372

16.4.3 Ambient space inference 373

16.5 Curvature and torsion 375

16.6 Surfaces 376

16.7 Curvature, ridges and solid shape 376

17 Shape in images 379

17.1 Introduction 379

17.2 High-level Bayesian image analysis 380

17.3 Prior models for objects 381

17.3.1 Geometric parameter approach 382

17.3.2 Active shape models and active appearance models 382

17.3.3 Graphical templates 383

17.3.4 Thin-plate splines 383

17.3.5 Snake 384

17.3.6 Inference 384

17.4 Warping and image averaging 384

17.4.1 Warping 384

17.4.2 Image averaging 385

17.4.3 Merging images 386

17.4.4 Consistency of deformable models 392

17.4.5 Discussion 392

18 Object data and manifolds 395

18.1 Object oriented data analysis 395

18.2 Trees 396

18.3 Topological data analysis 397

18.4 General shape spaces and generalized Procrustes methods 397

18.4.1 Definitions 397

18.4.2 Two object matching 398

18.4.3 Generalized matching 399

18.5 Other types of shape 399

18.6 Manifolds 400

18.7 Reviews 400

19 Exercises 403

20 Bibliography 409

References 409

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