Statistical Shape Analysis

774.1.6 Planar shape 774.1.7 Curvature 794.2 Comparing shape distances 794.2.1 Relationships 794.2.2 Shape distances in R 794.2.3 Further discussion 824.3 Planar case 844.3.1 Complex arithmetic 844.3.2 Complex projective space 854.3.3 Kent's polar pre-shape coordinates 874.3.4 Triangle case 884.4 Tangent space co-ordinates 904.4.1 Tangent spaces 904.4.2 Procrustes tangent co-ordinates 914.4.3 Planar Procrustes tangent co-ordinates 934.4.4 Higher dimensional Procrustes tangent co-ordinates 974.4.5 Inverse exponential map tangent-coordinates 984.4.6 Procrustes residuals 984.4.7 Other tangent co-ordinates 994.4.8 Tangent space coordinates in R 995 Size-and-shape space 1015.1 Introduction 1015.2 RMSD measures 1015.3 Geometry 1025.4 Tangent co-ordinates for size-and-shape space 1055.5 Geodesics 1055.6 Size-and-shape co-ordinates 1065.6.1 Bookstein-type coordinates for size-and-shape analysis 1065.6.2 Goodall-Mardia QR size-and-shape co-ordinates 1075.7 Allometry 1086 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 1176.6 Shape of the means 1206.7 Means in size-and-shape space 1206.7.1 Fr¿echet and Karcher means 1206.7.2 Size-and-shape of the means 1216.8 Principal geodesic mean 1216.9 Riemannian barycentres 1227 Procrustes analysis 1237.1 Introduction 1237.2 Ordinary Procrustes analysis 1247.2.1 Full ordinary Procrustes analysis 1247.2.2 Ordinary Procrustes analysis in R 1277.2.3 Ordinary partial Procrustes 1297.2.4 Reflection Procrustes 1307.3 Generalized Procrustes analysis 1317.3.1 Introduction 1317.4 Generalized Procrustes algorithms for shape analysis 1357.4.1 Algorithm: GPA-Shape-1 1357.4.2 Algorithm: GPA-Shape-2 1377.4.3 GPA in R 1377.5 Generalized Procrustes algorithms for size-and-shape analysis 1407.5.1 Algorithm: GPA-Size-and-Shape-1 1407.5.2 Algorithm: GPA-Size-and-Shape-2 1417.5.3 Partial generalized Procrustes analysis in R 1417.5.4 Reflection generalized Procrustes analysis in R 1417.6 Variants of generalized Procrustes Analysis 1427.6.1 Summary 1427.6.2 Unit size partial Procrustes 1427.6.3 Weighted Procrustes analysis 1437.7 Shape variability: principal components analysis 1477.7.1 Shape PCA 1477.7.2 Kent's shape PCA 1497.7.3 Shape PCA in R 1497.7.4 Point distribution models 1627.7.5 PCA in shape analysis and multivariate analysis 1647.8 PCA for size-and-shape 1647.9 Canonical variate analysis 1657.10 Discriminant analysis 1677.11 Independent components analysis 1687.12 Bilateral symmetry 1708 2D Procrustes analysis using complex arithmetic 1738.1 Introduction 1738.2 Shape distance and Procrustes matching 1738.3 Estimation of mean shape 1768.4 Planar shape analysis in R 1788.5 Shape variability 1799 Tangent space inference 1859.1 Tangent space small variability inference for mean shapes 1859.1.1 One sample Hotelling's T 2 test 1859.1.2 Two independent sample Hotelling's T 2 test 1889.1.3 Permutation and bootstrap tests 1939.1.4 Fast permutation and bootstrap tests 1949.1.5 Extensions and regularization 1969.2 Inference using Procrustes statistics under isotropy 1969.2.1 One sample Goodall's F test 1979.2.2 Two independent sample Goodall's F test 1999.2.3 Further two sample tests 2039.2.4 One way analysis of variance 2049.3 Size-and-shape tests 2059.3.1 Tests using Procrustes size-and-shape tangent space 2059.3.2 Case-study: Size-and-shape analysis and mutation 2079.4 Edge-based shape coordinates 2109.5 Investigating allometry 21210 Shape and size-and-shape distributions 21710.1 The Uniform distribution 21710.2 Complex Bingham distribution 21910.2.1 The density 21910.2.2 Relation to the complex normal distribution 22010.2.3 Relation to real Bingham distribution 22010.2.4 The normalizing constant 22110.2.5 Properties 22110.2.6 Inference 22310.2.7 Approximations and computation 22410.2.8 Relationship with the Fisher-von Mises distribution 22510.2.9 Simulation 22610.3 ComplexWatson distribution 22610.3.1 The density 22610.3.2 Inference 22710.3.3 Large concentrations 22810.4 Complex Angular central Gaussian distribution 23010.5 Complex Bingham quartic distribution 23010.6 A rotationally symmetric shape family 23010.7 Other distributions 23110.8 Bayesian inference 23210.9 Size-and-shape distributions 23410.9.1 Rotationally symmetric size-and-shape family 23410.9.2 Central complex Gaussian distribution 23610.10Size-and-shape versus shape 23611 Offset normal shape distributions 23711.1 Introduction 23711.1.1 Equal mean case in two dimensions 23711.1.2 The isotropic case in two dimensions 24211.1.3 The triangle case 24611.1.4 Approximations: Large and small variations 24711.1.5 Exact Moments 24911.1.6 Isotropy 24911.2 Offset normal shape distributions with general covariances 25011.2.1 The complex normal case 25111.2.2 General covariances: small variations 25111.3 Inference for offset normal distributions 25311.3.1 General MLE 25311.3.2 Isotropic case 25311.3.3 Exact istropic MLE in R 25611.3.4 EM algorithm and extensions 25611.4 Practical Inference 25711.5 Offset normal size-and-shape distributions 25711.5.1 The isotropic case 25811.5.2 Inference using the offset normal size-and-shape model 26011.6 Distributions for higher dimensions 26211.6.1 Introduction 26211.6.2 QR Decomposition 26211.6.3 Size-and-shape distributions 26311.6.4 Multivariate approach 26411.6.5 Approximations 26512 Deformations for size and shape change 26712.1 Deformations 26712.1.1 Introduction 26712.1.2 Definition and desirable properties 26812.1.3 D'Arcy Thompson's transformation grids 26812.2 Affine transformations 27012.2.1 Exact match 27012.2.2 Least squares matching: Two objects 27012.2.3 Least squares matching: Multiple objects 27212.2.4 The triangle case: Bookstein's hyperbolic shape space 27512.3 Pairs of Thin-plate Splines 27712.3.1 Thin-plate splines 27712.3.2 Transformation grids 27912.3.3 Thin-plate splines in R 28212.3.4 Principal and partial warp decompositions 28712.3.5 Principal component analysis with non-Euclidean metrics 29612.3.6 Relative warps 29912.4 Alternative approaches and history 30312.4.1 Early transformation grids 30312.4.2 Finite element analysis 30612.4.3 Biorthogonal grids 30912.5 Kriging 30912.5.1 Universal kriging 30912.5.2 Deformations 31112.5.3 Intrinsic kriging 31112.5.4 Kriging with derivative constraints 31312.5.5 Smoothed matching 31312.6 Diffeomorphic transformations 31513 Non-parametric inference and regression 31713.1 Consistency 31713.2 Uniqueness of intrinsic means 31813.3 Non-parametric inference 32113.3.1 Central limit theorems and non-parametric tests 32113.3.2 M-estimators 32313.4 Principal geodesics and shape curves 32313.4.1 Tangent space methods and longitudinal data 32313.4.2 Growth curve models for triangle shapes 32513.4.3 Geodesic hypothesis 32513.4.4 Principal geodesic analysis 32613.4.5 Principal nested spheres and shape spaces 32713.4.6 Unrolling and unwrapping 32813.4.7 Manifold splines 33113.5 Statistical shape change 33313.5.1 Geometric components of shape change 33413.5.2 Paired shape distributions 33613.6 Robustness 33613.7 Incomplete Data 34014 Unlabelled size-and-shape and shape analysis 34114.1 The Green-Mardia model 34214.1.1 Likelihood 34214.1.2 Prior and posterior distributions 34314.1.3 MCMC simulation 34414.2 Procrustes model 34614.2.1 Prior and posterior distributions 34714.2.2 MCMC Inference 34714.3 Related methods 34914.4 Unlabelled Points 35014.4.1 Flat triangles and alignments 35014.4.2 Unlabelled shape densities 35114.4.3 Further probabilistic issues 35114.4.4 Delaunay triangles 35215 Euclidean methods 35515.1 Distance-based methods 35515.2 Multidimensional scaling 35515.2.1 Classical MDS 35515.2.2 MDS for size-and-shape 35615.3 MDS shape means 35615.4 EDMA for size-and-shape analysis 35915.4.1 Mean shape 35915.4.2 Tests for shape difference 36015.5 Log-distances and multivariate analysis 36215.6 Euclidean shape tensor analysis 36315.7 Distance methods versus geometrical methods 36316 Curves, surfaces and volumes 36516.1 Shape factors and random sets 36516.2 Outline data 36616.2.1 Fourier series 36616.2.2 Deformable template outlines 36716.2.3 Star-shaped objects 36816.2.4 Featureless outlines 36916.3 Semi-landmarks 37016.4 Square root velocity function 37116.4.1 SRVF and quotient space for size-and-shape 37116.4.2 Quotient space inference 37216.4.3 Ambient space inference 37316.5 Curvature and torsion 37516.6 Surfaces 37616.7 Curvature, ridges and solid shape 37617 Shape in images 37917.1 Introduction 37917.2 High-level Bayesian image analysis 38017.3 Prior models for objects 38117.3.1 Geometric parameter approach 38217.3.2 Active shape models and active appearance models 38217.3.3 Graphical templates 38317.3.4 Thin-plate splines 38317.3.5 Snake 38417.3.6 Inference 38417.4 Warping and image averaging 38417.4.1 Warping 38417.4.2 Image averaging 38517.4.3 Merging images 38617.4.4 Consistency of deformable models 39217.4.5 Discussion 39218 Object data and manifolds 39518.1 Object oriented data analysis 39518.2 Trees 39618.3 Topological data analysis 39718.4 General shape spaces and generalized Procrustes methods 39718.4.1 Definitions 39718.4.2 Two object matching 39818.4.3 Generalized matching 39918.5 Other types of shape 39918.6 Manifolds 40018.7 Reviews 40019 Exercises 40320 Bibliography 409References 409