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