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 Kendalls shape coordinates 51
2.6 Triangle shape co-ordinates 53
2.6.1 Bookstein co-ordinates for triangles 53
2.6.2 Kendalls spherical coordinates for triangles 56
2.6.3 Spherical projections 58
2.6.4 Watsons 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 Kents 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 Kents 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 Hotellings T 2 test 185
9.1.2 Two independent sample Hotellings 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 Goodalls F test 197
9.2.2 Two independent sample Goodalls 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 DArcy Thompsons 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: Booksteins 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
