Statistical Shape Analysis

Ian Dryden, University of Nottingham, UK.Kanti Mardia, University of Leeds and University of Oxford, UK.

• Preface xixPreface to first edition xxiAcknowledgements for the first edition xxv1 Introduction 11.1 Definition and motivation 11.2 Landmarks 31.3 The shapes package in R 71.4 Practical applications 82 Size measures and shape coordinates 312.1 History 312.2 Size 332.3 Traditional shape coordinates 392.4 Bookstein shape coordinates 412.5 Kendall’s shape coordinates 492.6 Triangle shape coordinates 523 Manifolds, shape and size-and-shape 593.1 Riemannian manifolds 593.2 Shape 613.3 Size-and-shape 663.4 Reflection invariance 663.5 Discussion 674 Shape space 694.1 Shape space distances 694.2 Comparing shape distances 774.3 Planar case 824.4 Tangent space coordinates 885 Size-and-shape space 995.1 Introduction 995.2 Root mean square deviation measures 995.3 Geometry 1015.4 Tangent coordinates for size-and-shape space 1035.5 Geodesics 1045.6 Size-and-shape coordinates 1045.7 Allometry 1076 Manifold means 1116.1 Intrinsic and extrinsic means 1116.2 Population mean shapes 1126.3 Sample mean shape 1136.4 Comparing mean shapes 1156.5 Calculation of mean shapes in R 1186.6 Shape of the means 1206.7 Means in size-and-shape space 1216.8 Principal geodesic mean 1226.9 Riemannian barycentres 1227 Procrustes analysis 1257.1 Introduction 1257.2 Ordinary Procrustes analysis 1267.3 Generalized Procrustes analysis 1347.4 Generalized Procrustes algorithms for shape analysis 1367.5 Generalized Procrustes algorithms for size-and-shape analysis 1437.6 Variants of generalized Procrustes analysis 1457.7 Shape variability: principal component analysis 1507.8 Principal component analysis for size-and-shape 1667.9 Canonical variate analysis 1667.10 Discriminant analysis 1687.11 Independent component analysis 1697.12 Bilateral symmetry 1718 2D Procrustes analysis using complex arithmetic 1758.1 Introduction 1758.2 Shape distance and Procrustes matching 1758.3 Estimation of mean shape 1788.4 Planar shape analysis in R 1818.5 Shape variability 1829 Tangent space inference 1859.1 Tangent space small variability inference for mean shapes 1859.2 Inference using Procrustes statistics under isotropy 1979.3 Size-and-shape tests 2069.4 Edge-based shape coordinates 2129.5 Investigating allometry 21210 Shape and size-and-shape distributions 21710.1 The uniform distribution 21710.2 Complex Bingham distribution 21910.3 Complex Watson distribution 22710.4 Complex angular central Gaussian distribution 23110.5 Complex Bingham quartic distribution 23110.6 A rotationally symmetric shape family 23210.7 Other distributions 23310.8 Bayesian inference 23310.9 Size-and-shape distributions 23710.10 Size-and-shape versus shape 23711 Offset normal shape distributions 23911.1 Introduction 23911.2 Offset normal shape distributions with general covariances 25211.3 Inference for offset normal distributions 25511.4 Practical inference 25811.5 Offset normal size-and-shape distributions 25911.6 Distributions for higher dimensions 26412 Deformations for size and shape change 26912.1 Deformations 26912.2 Affine transformations 27212.3 Pairs of thin-plate splines 27912.4 Alternative approaches and history 30312.5 Kriging 30712.6 Diffeomorphic transformations 31413 Non-parametric inference and regression 31713.1 Consistency 31713.2 Uniqueness of intrinsic means 31813.3 Non-parametric inference 32213.4 Principal geodesics and shape curves 32413.5 Statistical shape change 33213.6 Robustness 33613.7 Incomplete data 33814 Unlabelled size-and-shape and shape analysis 34114.1 The Green–Mardia model 34214.2 Procrustes model 34514.3 Related methods 34814.4 Unlabelled points 34915 Euclidean methods 35315.1 Distance-based methods 35315.2 Multidimensional scaling 35315.3 Multidimensional scaling shape means 35415.4 Euclidean distance matrix analysis for size-and-shape analysis 35715.5 Log-distances and multivariate analysis 36015.6 Euclidean shape tensor analysis 36115.7 Distance methods versus geometrical methods 36216 Curves, surfaces and volumes 36316.1 Shape factors and random sets 36316.2 Outline data 36416.3 Semi-landmarks 36816.4 Square root velocity function 36916.5 Curvature and torsion 37416.6 Surfaces 37416.7 Curvature, ridges and solid shape 37517 Shape in images 37717.1 Introduction 37717.2 High-level Bayesian image analysis 37817.3 Prior models for objects 38017.4 Warping and image averaging 38218 Object data and manifolds 39118.1 Object oriented data analysis 39118.2 Trees 39218.3 Topological data analysis 39318.4 General shape spaces and generalized Procrustes methods 39318.5 Other types of shape 39518.6 Manifolds 39618.7 Reviews 396Exercises 399Appendix 403References 407Index 449

"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)