Statistical Shape Analysis

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