Statistical Shape Analysis

Transcription

1 Statistical Shape Analysis I. L. Dryden and K. V. Mardia University ofleeds, UK JOHN WILEY& SONS Chichester New York Weinheim Brisbane Singapore Toronto

2 Contents Preface Acknowledgements xv xix 1 Introduction Definition and Motivation Landmarks ' Traditional methods Geometrical methods Practical Applications Biology: Mouse vertebrae Biology: Gorilla skulls Medicine: Brain MR scans of Schizophrenie patients Image analysis: Postcode recognition Archaeology: Alignments of Standing stones Geography: Central Place Theory Geology: Microfossils Biology: Macaque skulls Biology: Sooty mangabeys Agriculture: Fish recognition Agriculture: Robotic harvesting of mushrooms Genetics: Electrophoretic gels 17 2 Preliminaries: Size Measures and Shape Coordinates Configuration Space Size Some Shape Coordinate Systems Angles and ratios of lengths Bookstein coordinates: Planar case Trianglecase Kendali coordinates: Planar case Kendall's spherical coordinates for triangles Watson's triangle coordinates 38

3 x CONTENTS 3 Preliminaries: Planar Procrustes Analysis Introduction Shape Distance and Procrustes Matching Estimation of Mean Shape Shape Variability 47 4 Shape Space and Distances Shape Space Introduction Filtering translation Pre-shape Shape Size-and-shape: Removing location and rotation Reflection shape Alternative standardizations Over-dimensioned case Planar case Distances Procrustes distances Alternative distances Planar case Trianglecase Advanced Shape Coordinate Systems Tangent space coordinates Kent's polar coordinates Bookstein coordinates for three dimensional data Goodall-Mardia QR shape coordinates Goodall-Mardia polar shape coordinates 82 5 General Procrustes Methods Introduction Ordinary Procrustes Analysis Füll ordinary Procrustes analysis Generalized Procrustes Analysis Introduction Algorithm for higher dimensions Variants of Procrustes Analysis Ordinary partial Procrustes Generalized partial Procrustes Reflection Procrustes Shape Variability: Principal Components Analysis Two dimensional data Point distribution modeis PCA in shape analysis and multivariate analysis 107

4 CONTENTS xi 6 Shape Models for Two Dimensional Data Uniform Distribution Complex Bingham Distribution The density The normalizing constant Properties Inference Relationship with the Fisher distribution Complex Watson Distribution The density Calculation of the integrating constant Inference Large concentrations Complex Angular Central Gaussian Distribution A Rotationally Symmetrie Shape Family Offset Normal Shape Distributions Equal mean case in two dimensions The isotropic case in two dimensions The triangle case Approximations: Large and small variations Moments Isotropy Offset Normal Shape Distributions with General Covariances The complex normal case The equal means case Properties Inference for Offset Normal Distributions A Bayesian Approach Practical Inference Tangent Space Inference Tangent Space Inference One sample Hotelling's T 2 test Two independent sample Hotelling's T 2 test Permutation test Extensions Dimension reduetion in inference Inference Using Procrustes Statistics Under Isotropy One sample Goodall's F test Two independent sample Goodall's F test One way analysis of variance Further inference for Procrustes statistics Edge Superimposition Shape Coordinates Bookstein coordinates 168

5 xii CONTENTS Hotelling's T 2 two sample test using Bookstein coordinates Advantages and disadvantages Extensions Size-and-Shape Introduction Allometry Geometry Offset Normal Size-and-Shape Distributions The size-and-shape density Particular Cases The complex normal case The equal means case The isotropic case Inference Using the Offset Normal Model Alternative Distributions Size-and-Shape versus Shape Distributions for Higher Dimensions Introduction QR Decomposition Size-and-Shape Distributions Coincident means Collinear means Planar means Higher rank case Shape Densities and Bartlett's Decomposition Coincident means Collinear means Multivariate Approach Approximations A Rotationally Symmetrie Shape Family Deformations and Describing Shape Change Deformations Introduction Definition and desirable properties D'Arcy Thompson's transformation grids The Affine Deformation The triangle case: Bookstein's hyperbolic shape space PairsofThin-plateSplines Thin-plate splines Transformation grids Principal and partial warp decompositions 214

6 CONTENTS xiü Principal component analysis with non-euclidean metrics Relative warps Alternative Approaches and History Early transformation grids Finite dement analysis Biorthogonal grids Other deformations Kriging Universal kriging Deformations Intrinsic kriging Kriging with derivative constraints Statistical Shape Change Tangent space methods Growth curve modeis for triangle shapes Geometrie components of shape change Paired shape distributions Shape in Images Introduction High-level Bayesian Image Analysis Prior Models for Objects Geometrie parameter approach Landmarks: Shape distributions and point distribution modeis Graphical templates Thin-plate splines Outlines Inference Multiple Objects and Occlusions Classical Hough transform Morphological Operations A Markovian objeet process Warping and Image Averaging Warping Image averaging Merging images Discussion Additional Topics Consistency Distance-based Methods Multidimensional scaling EDMA Tests for shape difference 283

7 xiv CONTENTS Log-distances and multivariate analysis Distance methods versus geometrical methods Euclidean Shape Tensor Analysis Angular shape analysis Incomplete Data General Shape Spaces Definitions Two object matching Generalized matching Affine Shape Least Squares matching: Two objects Least Squares matching: Multiple objects Robust Superimposition Methods Resistance to landmark outliers Object outliers Smoothing Smoothed matching Smoothed principal component analysis Distribution-free Methods Unlabelled Points Fiat triangles and alignments Unlabelled shape densities Further probabilistic issues Delaunay triangles Landmark-free Approaches Curvature Postscript 309 Appendix A: Notation 311 Appendix B: Software and Data 313 Mouse vertebrae 313 Gorilla skulls 317 Handwritten digit 3 data 318 Othersources 320 References and Author Index 321 Index 337