Details

Statistical Shape Analysis


Statistical Shape Analysis

With Applications in R
Wiley Series in Probability and Statistics, Band 995 2. Aufl.

von: Ian L. Dryden, Kanti V. Mardia

77,99 €

Verlag: Wiley
Format: PDF
Veröffentl.: 28.06.2016
ISBN/EAN: 9781119072508
Sprache: englisch
Anzahl Seiten: 496

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Beschreibungen

<p><b>A thoroughly revised and updated edition of this introduction to modern statistical methods for shape analysis</b></p> <p>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.</p> <p>This book is a significant update of the highly-regarded <i>Statistical Shape Analysis</i> 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.</p> <p>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.</p> <ul> <li>Offers a detailed yet accessible treatment of statistical methods for shape analysis</li> <li>Includes numerous examples and applications from many disciplines</li> <li>Provides R code for implementing the examples</li> <li>Covers a wide variety of recent developments in shape analysis</li> </ul> <p><i>Shape Analysis, with Applications in R</i> 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.</p>
<p><b>1 Introduction 1</b></p> <p>1.1 Definition and Motivation 1</p> <p>1.2 Landmarks 3</p> <p>1.3 The shapes package in R 6</p> <p>1.4 Practical Applications 8</p> <p>1.4.1 Biology: Mouse vertebrae 8</p> <p>1.4.2 Image analysis: Postcode recognition 11</p> <p>1.4.3 Biology: Macaque skulls 12</p> <p>1.4.4 Chemistry: Steroid molecules 15</p> <p>1.4.5 Medicine: SchizophreniaMR images 16</p> <p>1.4.6 Medicine and law: Fetal Alcohol Spectrum Disorder 16</p> <p>1.4.7 Pharmacy: DNA molecules 18</p> <p>1.4.8 Biology: Great ape skulls 19</p> <p>1.4.9 Bioinformatics: Protein matching 22</p> <p>1.4.10 Particle science: Sand grains 22</p> <p>1.4.11 Biology: Rat skull growth 24</p> <p>1.4.12 Biology: Sooty mangabeys 25</p> <p>1.4.13 Physiotherapy: Human movement data 25</p> <p>1.4.14 Genetics: Electrophoretic gels 26</p> <p>1.4.15 Medicine: Cortical surface shape 26</p> <p>1.4.16 Geology:Microfossils 28</p> <p>1.4.17 Geography: Central Place Theory 29</p> <p>1.4.18 Archaeology: Alignments of standing stones 32</p> <p><b>2 Size measures and shape coordinates 33</b></p> <p>2.1 History 33</p> <p>2.2 Size 35</p> <p>2.2.1 Configuration space 35</p> <p>2.2.2 Centroid size 35</p> <p>2.2.3 Other size measures 38</p> <p>2.3 Traditional shape coordinates 41</p> <p>2.3.1 Angles 41</p> <p>2.3.2 Ratios of lengths 42</p> <p>2.3.3 Penrose coefficent 43</p> <p>2.4 Bookstein shape coordinates 44</p> <p>2.4.1 Planar landmarks 44</p> <p>2.4.2 Bookstein-type coordinates for three dimensional data 49</p> <p>2.5 Kendall’s shape coordinates 51</p> <p>2.6 Triangle shape co-ordinates 53</p> <p>2.6.1 Bookstein co-ordinates for triangles 53</p> <p>2.6.2 Kendall’s spherical coordinates for triangles 56</p> <p>2.6.3 Spherical projections 58</p> <p>2.6.4 Watson’s triangle coordinates 58</p> <p><b>3 Manifolds, shape and size-and-shape 61</b></p> <p>3.1 Riemannian Manifolds 61</p> <p>3.2 Shape 63</p> <p>3.2.1 Ambient and quotient space 63</p> <p>3.2.2 Rotation 63</p> <p>3.2.3 Coincident and collinear points 65</p> <p>3.2.4 Filtering translation 65</p> <p>3.2.5 Pre-shape 65</p> <p>3.2.6 Shape 66</p> <p>3.3 Size-and-shape 67</p> <p>3.4 Reflection invariance 68</p> <p>3.5 Discussion 69</p> <p>3.5.1 Standardizations 69</p> <p>3.5.2 Over-dimensioned case 69</p> <p>3.5.3 Hierarchies 70</p> <p><b>4 Shape space 71</b></p> <p>4.1 Shape space distances 71</p> <p>4.1.1 Procrustes distances 71</p> <p>4.1.2 Procrustes 74</p> <p>4.1.3 Differential geometry 74</p> <p>4.1.4 Riemannian distance 76</p> <p>4.1.5 Minimal geodesics in shape space 77</p> <p>4.1.6 Planar shape 77</p> <p>4.1.7 Curvature 79</p> <p>4.2 Comparing shape distances 79</p> <p>4.2.1 Relationships 79</p> <p>4.2.2 Shape distances in R 79</p> <p>4.2.3 Further discussion 82</p> <p>4.3 Planar case 84</p> <p>4.3.1 Complex arithmetic 84</p> <p>4.3.2 Complex projective space 85</p> <p>4.3.3 Kent’s polar pre-shape coordinates 87</p> <p>4.3.4 Triangle case 88</p> <p>4.4 Tangent space co-ordinates 90</p> <p>4.4.1 Tangent spaces 90</p> <p>4.4.2 Procrustes tangent co-ordinates 91</p> <p>4.4.3 Planar Procrustes tangent co-ordinates 93</p> <p>4.4.4 Higher dimensional Procrustes tangent co-ordinates 97</p> <p>4.4.5 Inverse exponential map tangent-coordinates 98</p> <p>4.4.6 Procrustes residuals 98</p> <p>4.4.7 Other tangent co-ordinates 99</p> <p>4.4.8 Tangent space coordinates in R 99</p> <p><b>5 Size-and-shape space 101</b></p> <p>5.1 Introduction 101</p> <p>5.2 RMSD measures 101</p> <p>5.3 Geometry 102</p> <p>5.4 Tangent co-ordinates for size-and-shape space 105</p> <p>5.5 Geodesics 105</p> <p>5.6 Size-and-shape co-ordinates 106</p> <p>5.6.1 Bookstein-type coordinates for size-and-shape analysis 106</p> <p>5.6.2 Goodall–Mardia QR size-and-shape co-ordinates 107</p> <p>5.7 Allometry 108</p> <p><b>6 Manifold means 111</b></p> <p>6.1 Intrinsic and extrinsic means 111</p> <p>6.2 Population mean shapes 112</p> <p>6.3 Sample mean shape 113</p> <p>6.4 Comparing mean shapes 115</p> <p>6.5 Calculation of mean shapes in R 117</p> <p>6.6 Shape of the means 120</p> <p>6.7 Means in size-and-shape space 120</p> <p>6.7.1 Fr´echet and Karcher means 120</p> <p>6.7.2 Size-and-shape of the means 121</p> <p>6.8 Principal geodesic mean 121</p> <p>6.9 Riemannian barycentres 122</p> <p><b>7 Procrustes analysis 123</b></p> <p>7.1 Introduction 123</p> <p>7.2 Ordinary Procrustes analysis 124</p> <p>7.2.1 Full ordinary Procrustes analysis 124</p> <p>7.2.2 Ordinary Procrustes analysis in R 127</p> <p>7.2.3 Ordinary partial Procrustes 129</p> <p>7.2.4 Reflection Procrustes 130</p> <p>7.3 Generalized Procrustes analysis 131</p> <p>7.3.1 Introduction 131</p> <p>7.4 Generalized Procrustes algorithms for shape analysis 135</p> <p>7.4.1 Algorithm: GPA-Shape-1 135</p> <p>7.4.2 Algorithm: GPA-Shape-2 137</p> <p>7.4.3 GPA in R 137</p> <p>7.5 Generalized Procrustes algorithms for size-and-shape analysis 140</p> <p>7.5.1 Algorithm: GPA-Size-and-Shape-1 140</p> <p>7.5.2 Algorithm: GPA-Size-and-Shape-2 141</p> <p>7.5.3 Partial generalized Procrustes analysis in R 141</p> <p>7.5.4 Reflection generalized Procrustes analysis in R 141</p> <p>7.6 Variants of generalized Procrustes Analysis 142</p> <p>7.6.1 Summary 142</p> <p>7.6.2 Unit size partial Procrustes 142</p> <p>7.6.3 Weighted Procrustes analysis 143</p> <p>7.7 Shape variability: principal components analysis 147</p> <p>7.7.1 Shape PCA 147</p> <p>7.7.2 Kent’s shape PCA 149</p> <p>7.7.3 Shape PCA in R 149</p> <p>7.7.4 Point distribution models 162</p> <p>7.7.5 PCA in shape analysis and multivariate analysis 164</p> <p>7.8 PCA for size-and-shape 164</p> <p>7.9 Canonical variate analysis 165</p> <p>7.10 Discriminant analysis 167</p> <p>7.11 Independent components analysis 168</p> <p>7.12 Bilateral symmetry 170</p> <p><b>8 2D Procrustes analysis using complex arithmetic 173</b></p> <p>8.1 Introduction 173</p> <p>8.2 Shape distance and Procrustes matching 173</p> <p>8.3 Estimation of mean shape 176</p> <p>8.4 Planar shape analysis in R 178</p> <p>8.5 Shape variability 179</p> <p><b>9 Tangent space inference 185</b></p> <p>9.1 Tangent space small variability inference for mean shapes 185</p> <p>9.1.1 One sample Hotelling’s T 2 test 185</p> <p>9.1.2 Two independent sample Hotelling’s T 2 test 188</p> <p>9.1.3 Permutation and bootstrap tests 193</p> <p>9.1.4 Fast permutation and bootstrap tests 194</p> <p>9.1.5 Extensions and regularization 196</p> <p>9.2 Inference using Procrustes statistics under isotropy 196</p> <p>9.2.1 One sample Goodall’s F test 197</p> <p>9.2.2 Two independent sample Goodall’s F test 199</p> <p>9.2.3 Further two sample tests 203</p> <p>9.2.4 One way analysis of variance 204</p> <p>9.3 Size-and-shape tests 205</p> <p>9.3.1 Tests using Procrustes size-and-shape tangent space 205</p> <p>9.3.2 Case-study: Size-and-shape analysis and mutation 207</p> <p>9.4 Edge-based shape coordinates 210</p> <p>9.5 Investigating allometry 212</p> <p><b>10 Shape and size-and-shape distributions 217</b></p> <p>10.1 The Uniform distribution 217</p> <p>10.2 Complex Bingham distribution 219</p> <p>10.2.1 The density 219</p> <p>10.2.2 Relation to the complex normal distribution 220</p> <p>10.2.3 Relation to real Bingham distribution 220</p> <p>10.2.4 The normalizing constant 221</p> <p>10.2.5 Properties 221</p> <p>10.2.6 Inference 223</p> <p>10.2.7 Approximations and computation 224</p> <p>10.2.8 Relationship with the Fisher-von Mises distribution 225</p> <p>10.2.9 Simulation 226</p> <p>10.3 ComplexWatson distribution 226</p> <p>10.3.1 The density 226</p> <p>10.3.2 Inference 227</p> <p>10.3.3 Large concentrations 228</p> <p>10.4 Complex Angular central Gaussian distribution 230</p> <p>10.5 Complex Bingham quartic distribution 230</p> <p>10.6 A rotationally symmetric shape family 230</p> <p>10.7 Other distributions 231</p> <p>10.8 Bayesian inference 232</p> <p>10.9 Size-and-shape distributions 234</p> <p>10.9.1 Rotationally symmetric size-and-shape family 234</p> <p>10.9.2 Central complex Gaussian distribution 236</p> <p>10.10Size-and-shape versus shape 236</p> <p><b>11 Offset normal shape distributions 237</b></p> <p>11.1 Introduction 237</p> <p>11.1.1 Equal mean case in two dimensions 237</p> <p>11.1.2 The isotropic case in two dimensions 242</p> <p>11.1.3 The triangle case 246</p> <p>11.1.4 Approximations: Large and small variations 247</p> <p>11.1.5 Exact Moments 249</p> <p>11.1.6 Isotropy 249</p> <p>11.2 Offset normal shape distributions with general covariances 250</p> <p>11.2.1 The complex normal case 251</p> <p>11.2.2 General covariances: small variations 251</p> <p>11.3 Inference for offset normal distributions 253</p> <p>11.3.1 General MLE 253</p> <p>11.3.2 Isotropic case 253</p> <p>11.3.3 Exact istropic MLE in R 256</p> <p>11.3.4 EM algorithm and extensions 256</p> <p>11.4 Practical Inference 257</p> <p>11.5 Offset normal size-and-shape distributions 257</p> <p>11.5.1 The isotropic case 258</p> <p>11.5.2 Inference using the offset normal size-and-shape model 260</p> <p>11.6 Distributions for higher dimensions 262</p> <p>11.6.1 Introduction 262</p> <p>11.6.2 QR Decomposition 262</p> <p>11.6.3 Size-and-shape distributions 263</p> <p>11.6.4 Multivariate approach 264</p> <p>11.6.5 Approximations 265</p> <p><b>12 Deformations for size and shape change 267</b></p> <p>12.1 Deformations 267</p> <p>12.1.1 Introduction 267</p> <p>12.1.2 Definition and desirable properties 268</p> <p>12.1.3 D’Arcy Thompson’s transformation grids 268</p> <p>12.2 Affine transformations 270</p> <p>12.2.1 Exact match 270</p> <p>12.2.2 Least squares matching: Two objects 270</p> <p>12.2.3 Least squares matching: Multiple objects 272</p> <p>12.2.4 The triangle case: Bookstein’s hyperbolic shape space 275</p> <p>12.3 Pairs of Thin-plate Splines 277</p> <p>12.3.1 Thin-plate splines 277</p> <p>12.3.2 Transformation grids 279</p> <p>12.3.3 Thin-plate splines in R 282</p> <p>12.3.4 Principal and partial warp decompositions 287</p> <p>12.3.5 Principal component analysis with non-Euclidean metrics 296</p> <p>12.3.6 Relative warps 299</p> <p>12.4 Alternative approaches and history 303</p> <p>12.4.1 Early transformation grids 303</p> <p>12.4.2 Finite element analysis 306</p> <p>12.4.3 Biorthogonal grids 309</p> <p>12.5 Kriging 309</p> <p>12.5.1 Universal kriging 309</p> <p>12.5.2 Deformations 311</p> <p>12.5.3 Intrinsic kriging 311</p> <p>12.5.4 Kriging with derivative constraints 313</p> <p>12.5.5 Smoothed matching 313</p> <p>12.6 Diffeomorphic transformations 315</p> <p><b>13 Non-parametric inference and regression 317</b></p> <p>13.1 Consistency 317</p> <p>13.2 Uniqueness of intrinsic means 318</p> <p>13.3 Non-parametric inference 321</p> <p>13.3.1 Central limit theorems and non-parametric tests 321</p> <p>13.3.2 M-estimators 323</p> <p>13.4 Principal geodesics and shape curves 323</p> <p>13.4.1 Tangent space methods and longitudinal data 323</p> <p>13.4.2 Growth curve models for triangle shapes 325</p> <p>13.4.3 Geodesic hypothesis 325</p> <p>13.4.4 Principal geodesic analysis 326</p> <p>13.4.5 Principal nested spheres and shape spaces 327</p> <p>13.4.6 Unrolling and unwrapping 328</p> <p>13.4.7 Manifold splines 331</p> <p>13.5 Statistical shape change 333</p> <p>13.5.1 Geometric components of shape change 334</p> <p>13.5.2 Paired shape distributions 336</p> <p>13.6 Robustness 336</p> <p>13.7 Incomplete Data 340</p> <p><b>14 Unlabelled size-and-shape and shape analysis 341</b></p> <p>14.1 The Green-Mardia model 342</p> <p>14.1.1 Likelihood 342</p> <p>14.1.2 Prior and posterior distributions 343</p> <p>14.1.3 MCMC simulation 344</p> <p>14.2 Procrustes model 346</p> <p>14.2.1 Prior and posterior distributions 347</p> <p>14.2.2 MCMC Inference 347</p> <p>14.3 Related methods 349</p> <p>14.4 Unlabelled Points 350</p> <p>14.4.1 Flat triangles and alignments 350</p> <p>14.4.2 Unlabelled shape densities 351</p> <p>14.4.3 Further probabilistic issues 351</p> <p>14.4.4 Delaunay triangles 352</p> <p><b>15 Euclidean methods 355</b></p> <p>15.1 Distance-based methods 355</p> <p>15.2 Multidimensional scaling 355</p> <p>15.2.1 Classical MDS 355</p> <p>15.2.2 MDS for size-and-shape 356</p> <p>15.3 MDS shape means 356</p> <p>15.4 EDMA for size-and-shape analysis 359</p> <p>15.4.1 Mean shape 359</p> <p>15.4.2 Tests for shape difference 360</p> <p>15.5 Log-distances and multivariate analysis 362</p> <p>15.6 Euclidean shape tensor analysis 363</p> <p>15.7 Distance methods versus geometrical methods 363</p> <p><b>16 Curves, surfaces and volumes 365</b></p> <p>16.1 Shape factors and random sets 365</p> <p>16.2 Outline data 366</p> <p>16.2.1 Fourier series 366</p> <p>16.2.2 Deformable template outlines 367</p> <p>16.2.3 Star-shaped objects 368</p> <p>16.2.4 Featureless outlines 369</p> <p>16.3 Semi-landmarks 370</p> <p>16.4 Square root velocity function 371</p> <p>16.4.1 SRVF and quotient space for size-and-shape 371</p> <p>16.4.2 Quotient space inference 372</p> <p>16.4.3 Ambient space inference 373</p> <p>16.5 Curvature and torsion 375</p> <p>16.6 Surfaces 376</p> <p>16.7 Curvature, ridges and solid shape 376</p> <p><b>17 Shape in images 379</b></p> <p>17.1 Introduction 379</p> <p>17.2 High-level Bayesian image analysis 380</p> <p>17.3 Prior models for objects 381</p> <p>17.3.1 Geometric parameter approach 382</p> <p>17.3.2 Active shape models and active appearance models 382</p> <p>17.3.3 Graphical templates 383</p> <p>17.3.4 Thin-plate splines 383</p> <p>17.3.5 Snake 384</p> <p>17.3.6 Inference 384</p> <p>17.4 Warping and image averaging 384</p> <p>17.4.1 Warping 384</p> <p>17.4.2 Image averaging 385</p> <p>17.4.3 Merging images 386</p> <p>17.4.4 Consistency of deformable models 392</p> <p>17.4.5 Discussion 392</p> <p><b>18 Object data and manifolds 395</b></p> <p>18.1 Object oriented data analysis 395</p> <p>18.2 Trees 396</p> <p>18.3 Topological data analysis 397</p> <p>18.4 General shape spaces and generalized Procrustes methods 397</p> <p>18.4.1 Definitions 397</p> <p>18.4.2 Two object matching 398</p> <p>18.4.3 Generalized matching 399</p> <p>18.5 Other types of shape 399</p> <p>18.6 Manifolds 400</p> <p>18.7 Reviews 400</p> <p>19 Exercises 403</p> <p>20 Bibliography 409</p> <p>References 409</p>
<p>"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." (<i>Mathematical Reviews/MathSciNet</i>, July 2017)</p><p>"Statistical methods applied to shape analysis. Great for biologists, but strong mathematical treatment and accompanying code expands possible applications." (<i>Raspberry Pi,</i> March 2017)</p>
<p><b>Ian Dryden</b>, University of Nottingham, UK.</p> <p><b>Kanti Mardia</b>, University of Leeds and University of Oxford, UK.</p>
<p><b>A thoroughly revised and updated edition of this introduction to modern statistical methods for shape analysis</b></p> <p>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.</p> <p>This book is a significant update of the highly-regarded <i>Statistical Shape Analysis</i> 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.</p> <p>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.</p> <ul> <li>Offers a detailed yet accessible treatment of statistical methods for shape analysis</li> <li>Includes numerous examples and applications from many disciplines</li> <li>Provides R code for implementing the examples</li> <li>Covers a wide variety of recent developments in shape analysis</li> </ul> <p><i>Shape Analysis, with Applications in R</i> 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.</p>

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