This edition first published 2016
© 2016 John Wiley and Sons Ltd

Registered office
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the authors to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book.

Limit of Liability/Disclaimer of Warranty: While the publisher and authors have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the authors shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Names: Dryden, I. L. (Ian L.), author. | Mardia, K. V., author.
Title: Statistical shape analysis with applications in R / Ian L. Dryden and Kanti V. Mardia.
Other titles: Statistical shape analysis | Wiley series in probability and statistics.
Description: Second edition. | Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2016. | Series: Wiley
   series in probability and statistics | Originally published as: Statistical shape analysis, 1998 |
   Includes bibliographical references and index.
Identifiers: LCCN 2016011608 (print) | LCCN 2016016288 (ebook) | ISBN 9780470699621 (cloth) |
   ISBN 9781119072508 (pdf) | ISBN 9781119072515 (epub)
Subjects: LCSH: (1) Shape theory (Topology)--Statistical methods (2) Statistics, multivariate analysis.
Classification: LCC QA612.7 .D79 2016 (print) | LCC QA612.7 (ebook) | DDC 514/.24--dc23
LC record available at https://lccn.loc.gov/2016011608

A catalogue record for this book is available from the British Library.

Cover image: Shape analysis of two proteins 1cyd (left) and 1a27 (right) from the Protein Data Bank

ISBN: 9780470699621

CONTENTS

Preface

Preface to the first edition

Acknowledgements for the first edition

1 Introduction

1.1 Definition and motivation
1.2 Landmarks
1.3 The shapes package in R
1.4 Practical applications

2 Size measures and shape coordinates

2.1 History
2.2 Size
2.3 Traditional shape coordinates
2.4 Bookstein shape coordinates
2.5 Kendall’s shape coordinates
2.6 Triangle shape coordinates

3 Manifolds, shape and size-and-shape

3.1 Riemannian manifolds
3.2 Shape
3.3 Size-and-shape
3.4 Reflection invariance
3.5 Discussion

4 Shape space

4.1 Shape space distances
4.2 Comparing shape distances
4.3 Planar case
4.4 Tangent space coordinates

5 Size-and-shape space

5.1 Introduction
5.2 Root mean square deviation measures
5.3 Geometry
5.4 Tangent coordinates for size-and-shape space
5.5 Geodesics
5.6 Size-and-shape coordinates
5.7 Allometry

6 Manifold means

6.1 Intrinsic and extrinsic means
6.2 Population mean shapes
6.3 Sample mean shape
6.4 Comparing mean shapes
6.5 Calculation of mean shapes in R
6.6 Shape of the means
6.7 Means in size-and-shape space
6.8 Principal geodesic mean
6.9 Riemannian barycentres

7 Procrustes analysis

7.1 Introduction
7.2 Ordinary Procrustes analysis
7.3 Generalized Procrustes analysis
7.4 Generalized Procrustes algorithms for shape analysis
7.5 Generalized Procrustes algorithms for size-and-shape analysis
7.6 Variants of generalized Procrustes analysis
7.7 Shape variability: principal component analysis
7.8 Principal component analysis for size-and-shape
7.9 Canonical variate analysis
7.10 Discriminant analysis
7.11 Independent component analysis
7.12 Bilateral symmetry

8 2D Procrustes analysis using complex arithmetic

8.1 Introduction
8.2 Shape distance and Procrustes matching
8.3 Estimation of mean shape
8.4 Planar shape analysis in R
8.5 Shape variability

9 Tangent space inference

9.1 Tangent space small variability inference for mean shapes
9.2 Inference using Procrustes statistics under isotropy
9.3 Size-and-shape tests
9.4 Edge-based shape coordinates
9.5 Investigating allometry

10 Shape and size-and-shape distributions

10.1 The uniform distribution
10.2 Complex Bingham distribution
10.3 Complex Watson distribution
10.4 Complex angular central Gaussian distribution
10.5 Complex Bingham quartic distribution
10.6 A rotationally symmetric shape family
10.7 Other distributions
10.8 Bayesian inference
10.9 Size-and-shape distributions
10.10 Size-and-shape versus shape

11 Offset normal shape distributions

11.1 Introduction
11.2 Offset normal shape distributions with general covariances
11.3 Inference for offset normal distributions
11.4 Practical inference
11.5 Offset normal size-and-shape distributions
11.6 Distributions for higher dimensions

12 Deformations for size and shape change

12.1 Deformations
12.2 Affine transformations
12.3 Pairs of thin-plate splines
12.4 Alternative approaches and history
12.5 Kriging
12.6 Diffeomorphic transformations

13 Non-parametric inference and regression

13.1 Consistency
13.2 Uniqueness of intrinsic means
13.3 Non-parametric inference
13.4 Principal geodesics and shape curves
13.5 Statistical shape change
13.6 Robustness
13.7 Incomplete data

14 Unlabelled size-and-shape and shape analysis

14.1 The Green–Mardia model
14.2 Procrustes model
14.3 Related methods
14.4 Unlabelled points

15 Euclidean methods

15.1 Distance-based methods
15.2 Multidimensional scaling
15.3 Multidimensional scaling shape means
15.4 Euclidean distance matrix analysis for size-and-shape analysis
15.5 Log-distances and multivariate analysis
15.6 Euclidean shape tensor analysis
15.7 Distance methods versus geometrical methods

16 Curves, surfaces and volumes

16.1 Shape factors and random sets
16.2 Outline data
16.3 Semi-landmarks
16.4 Square root velocity function
16.5 Curvature and torsion
16.6 Surfaces
16.7 Curvature, ridges and solid shape

17 Shape in images

17.1 Introduction
17.2 High-level Bayesian image analysis
17.3 Prior models for objects
17.4 Warping and image averaging

18 Object data and manifolds

18.1 Object oriented data analysis
18.2 Trees
18.3 Topological data analysis
18.4 General shape spaces and generalized Procrustes methods
18.5 Other types of shape
18.6 Manifolds
18.7 Reviews

Exercises

Appendix

References

Index

WILEY SERIES IN PROBABILITY AND STATISTICS

EULA

List of Illustrations

Chapter 1

Figure 1.1 Two outlines of the same second thoracic (T2) vertebra of a mouse, which have different locations, rotations and scales but the same shape.
Figure 1.2 From Galileo (1638) illustrating the differences in shapes of the bones of small and large animals.
Figure 1.3 Anatomical landmarks located on the side view of a macaque monkey skull.
Figure 1.4 Image of a T2 mouse vertebra with six mathematical landmarks on the outline joined by lines (dark +) and 42 pseudo-landmarks (light +). Source: Dryden & Mardia 1998. Reproduced with permission from John Wiley & Sons.
Figure 1.5 Six labelled triangles: A and B have the same size and labelled shape; C has the same labelled shape as A and B (but larger size); D has a different labelled shape but its labels can be permuted to give the same unlabelled shape as A, B and C; triangle E can be reflected to have the same labelled shape as D; triangle F has a different shape from A, B, C, D and E.
Figure 1.6 Six mathematical landmarks (+) on a second thoracic mouse vertebra, together with 54 pseudo-landmarks around the outline, approximately equally spaced between pairs of landmarks. The landmarks are 1 and 2 at maximum points of approximate curvature function (usually at the widest part of the vertebra rather than on the tips), 3 and 5 at the extreme points of negative curvature at the base of the spinous process, 4 at the tip of the spinous process, and 6 at the maximal curvature point on the opposite side of the bone from 4.
Figure 1.7 The three groups of T2 mouse landmarks, with k = 6 landmarks per bone: (a) 30 Control; (b) 23 Large; and (c) 23 Small mice.
Figure 1.8 The three groups of T2 vertebra outlines, with 60 points per bone: (a) 30 Control; (b) 23 Large; and (c) 23 Small mice.
Figure 1.9 A handwritten digit ‘3’ from the postcode dataset, with 13 labelled mathematical landmarks. Landmark 1 is at the extreme bottom left, 4 is at the maximum curvature of the bottom arc, 7 is at the extreme end of the central protrusion, 10 is at the maximum curvature of the top arc and 13 is the extreme top left point. Landmarks 2, 3, 5, 6, 8, 9, 11 and 12 are pseudo-landmarks at approximately equal intervals between the mathematical landmarks.
Figure 1.10 The thirty digit 3 configurations, each with 13 landmarks.
Figure 1.11 A 3D macaque skull: (a) side view; (b) frontal view; and (c) bottom view. A total of 26 landmarks are displayed on the skull and a subset of 7 was taken for the analysis. The seven chosen landmarks are: 1, prosthion; 7, opisthion; 10, bregma; 12, nasion; 15, asterion; 16, midpoint of zyg/temp suture; and 17, interfrontomalare.
Figure 1.12 The macaque skull data with seven landmarks from 18 individuals, with each landmark displayed by a different colour.
Figure 1.13 The first 17 carbon atoms in the 31 steroid molecules.
Figure 1.14 The 13 landmarks on a near midsagittal section from a brain scan of a schizophrenia patient. The landmark positions are approximately located at each cross (+). Source: Adapted from Bookstein 1996b. Reproduced with permission from Springer Science+Business Media.
Figure 1.15 The dataset of 13 landmarks per individual from the schizophrenia study, with circles for controls and triangles for patients.
Figure 1.16 A landmark and 39 semi-landmarks on the outline of the corpus callosum from an MR image of a prisoner. Source: Mardia et al. 2013a. Reproduced with permission from John Wiley & Sons.
Figure 1.17 A small dataset of 22 phosphorous atoms from a DNA molecule at n = 30 time points.
Figure 1.18 Eight landmarks on the midline section of the ape cranium. The face region is taken to be comprised of landmarks: 7, nasion (n); 4, basion (ba); 5, staphylion (st); 1, prosthion (pr); and 6, nariale (na). The braincase region is taken to be comprised of landmarks: 7, 4 and 8, bregma (b); 2, lambda (l); and 3, opisthion (o).
Figure 1.19 The six groups of great ape skull landmarks: (left column) female and male gorillas; (middle column) female and male chimpanzees; and (right column) female and male orangutans.
Figure 1.20 The proteins 1a27 (a) and 1cyd (b) from the PDB databank (Tanaka et al. 1996; Mazza 1997; Berman et al. 2000).
Figure 1.21 The sand particle outlines: (a) sea sand; and (b) river sand.
Figure 1.22 The eight landmarks on the 18 rat skulls, observed at eight time points.
Figure 1.23 The 12 landmarks on the midline of the skull of a juvenile sooty mangabey. The chosen landmarks are nasion (n), rhinion (r), nariale (na), prosthion (pr), incisive canal (i), palatine junction (p), posterior nasal spine (pns), basisphenoid (bs), basion (ba), opisthion (o), lambda (l) and bregma (b).
Figure 1.24 The electrophoretic gel images from (a) gel A and (b) gel B. The invariant spots are marked with a ‘+’ in both images. Source: Adapted from Horgan et al. 1992.
Figure 1.25 The five series of configurations projected into the plane of the table. Each series consists of 10 quadrilaterals observed at equal fractions of the time taken to carry out the pointing movement. Source: Kume et al. 2007. Reproduced with permission of Oxford University Press.
Figure 1.26 A set of 62 501 cortical surface points in three dimensions. The colouring indicates the ordering of the points.
Figure 1.27 Landmarks from 21 mean outlines of microfossils.
Figure 1.28 The Voronoi polygons (unbroken lines) and Delaunay triangulation (broken lines) for a completely regular configuration, that is ideal central places.
Figure 1.29 The Voronoi polygons (unbroken lines) and Delaunay triangles (broken lines) for the Iowa towns. The Voronoi polygons at the edges are not shown fully.
Figure 1.30 The map of 52 megalithic sites (+) that form the ‘Old Stones of Land’s End’ in Cornwall. Source: Stoyan, Kendall & Mecke 1995. Reproduced with permission from John Wiley & Sons.

Chapter 2

Figure 2.1 Thompson (1917)’s example of a species of fish Diodon being geometrically transformed into another species Orthagoriscus. Source: Thompson 1917. Reproduced with permission of Cambridge University Press.
Figure 2.2 Boxplots of the centroid sizes for the apes' data by each group.
Figure 2.3 Four example triangles. The ranking of the triangles in terms of size differs when different choices of size measure are considered.
Figure 2.4 A labelled triangle with three internal angles marked. Two of the internal angles could be used to measure the shape of the triangle.
Figure 2.5 Examples of pathological cases where angles are an inappropriate shape measure. The landmarks are at the centre of the discs.
Figure 2.6 The geometrical interpretation of Bookstein coordinates. The original triangle in (a) is transformed by (b) translation, (c) rotation and finally (d) rescaling to give the Bookstein coordinates as the coordinates of the point labelled 3 in plot (d).
Figure 2.7 Scatter plots of the Bookstein coordinates of the T2 Small vertebrae registered on the baseline 1, 2.
Figure 2.8 Pairwise scatter plots of centroid size S and Bookstein coordinates u^B₃, …, u₆^B, v^B₃, …, v₆^B for the T2 Small vertebrae.
Figure 2.9 The 24 landmarks located on a macaque monkey skull where the page is regarded as the x–y plane and into the page is the z-axis. The skull is translated, rotated and rescaled so that landmarks 1 and 6 (see Section 1.2.8) lie in the x–y plane of the page, at unit length apart, and landmark 10 is also rotated to be in the x–y plane.
Figure 2.10 The shape space of triangles, using the Bookstein coordinates (U^B, V^B). Each triangle is plotted with its centre at the shape coordinates (U^B, V^B). The equilateral triangles are located at the points marked E and F. The isosceles triangles are located on the unbroken lines and circles (——) and the right-angled triangles are located on the broken lines and circles (- - - -). The flat (collinear) triangles are located on the V^B = 0 line (the U^B-axis). All triangles could be relabelled and reflected to lie in the region AOE, bounded by the arc of isosceles triangles AE, the line of isosceles triangles OE and the line of flat triangles AO.
Figure 2.11 Kendall’s spherical shape space for triangles in m = 2 dimensions. The shape coordinates are the latitude θ (with zero at the North pole) and the longitude ϕ.
Figure 2.12 Part of the shape space of triangles projected onto the equal-area projection Schmidt net. If relabelling and reflection was not important, then all triangles could be projected into this sector.

Chapter 3

Figure 3.1 The hierarchies of the various spaces. Source: Adapted from Goodall & Mardia 1992.

Chapter 4

Figure 4.1 A diagrammatic schematic view of two fibres [X₁] and [X₂] on the pre-shape sphere, which correspond to the shapes of the original configuration matrices X₁ and X₂ which have pre-shapes Z₁ and Z₂. Also displayed are the smallest great circle ρ and chordal distance d_P between the fibres.
Figure 4.2 Section of the pre-shape sphere, illustrating schematically the relationship between the distances d_F, d_P and ρ.
Figure 4.3 Multidimensional scaling plots using (a) Riemannian distance and (b) full Procrustes distance.
Figure 4.4 The geometry of Procrustes fits in calculating Procrustes distances. The first two rows show each configuration (left), centred (middle) and scaled (right). The last row shows the partial Procrustes fit (left) and full Procrustes fit (middle) of the centred pre-shape from configuration 1 (- - - -) onto the centred pre-shape of configuration 2 (——–). Source: Adapted from Bookstein 1996b.
Figure 4.5 (a) Six landmarks from a T2 mouse vertebra viewed at an observed location, scale and rotation. (b) An icon for the T2 vertebra, which is the centred pre-shape [with the same rotation as in (a)].
Figure 4.6 Section of the shape sphere for triangles, illustrating schematically the relationship between the Procrustes distances d_P, ρ and d_F.
Figure 4.7 A geometrical view of the tangent coordinates on the real sphere.
Figure 4.8 A diagrammatic view of a section of the pre-shape sphere, showing the partial tangent plane coordinates v_P and the full Procrustes tangent plane coordinates v_F discussed in Section 4.26.
Figure 4.9 Icons for partial Procrustes tangent coordinates for the T2 vertebral data (Small group).
Figure 4.10 Pairwise scatter plots for centroid size (S) and the (x, y) coordinates of icons for the partial Procrustes tangent coordinates for the T2 vertebral data (Small group).

Chapter 5

Figure 5.1 (a) Multidimensional scaling plot using Riemannian size-and-shape distance and (b) boxplots of centroid size for controls and schizophrenia patients.
Figure 5.2 A scatter plot of the Bookstein coordinates for the microfossil data. The square root of the sample centroid size has been plotted at each point, and it is clear that size increases with V.
Figure 5.3 Pairwise scatter plots of log S versus the Bookstein coordinates U and V for the microfossil data.
Figure 5.4 Plot of residuals versus fitted values for the regression of (a) V on √S and (b) V on log S.

Chapter 6

Figure 6.1 Multidimensional scaling plot of the male gorilla data and 13 means using Riemannian distance to form the dissimilarity matrix. (a) The principal coordinates of the 29 skulls (circles) and the 13 means (numbers). (b) Zoomed in version of (a) near the origin. (c) Another zoomed in version of (a) near the origin. In each of (a)–(c) the mean shapes are indicated by their number label, and the first ten means are joined by straight lines. (d) A plot of the observed ratios and expected approximate ratios of Riemannian distances for different h.
Figure 6.2 (a) Multidimensional scaling plot of the digit 3 data and ten means using Riemannian distance to form the dissimilarity matrix. The ten sample mean shapes are: 1, h = 0.001; 2, intrinsic; 3, partial Procrustes; 4, isotropic MLE; 5, full Procrustes; 6, h = 2; 7, h = 3; 8, h = 5; 9, h = 10; and 10, h = 20. (b) A zoomed in view showing the ten means and lines joining them in order of resistance to outlier shapes, with 10 being the most resistant.

Chapter 7

Figure 7.1 Unregistered sooty mangabeys: juvenile (—–); and adult (- - -).
Figure 7.2 The Procrustes fit of (a) the adult sooty mangabey (- - -) onto the juvenile (—–) and (b) the juvenile onto the adult.
Figure 7.3 The male (a) and female (b) macaque skulls registered by full GPA.
Figure 7.4 The male (red) mean shape registered to the female (blue) mean shape of the macaque skulls registered by OPA.
Figure 7.5 Plots of the first three PCs. In the jth row: mean − 3sd PCj, mean, mean + 3sd PCj (where j = 1, 2, 3).
Figure 7.6 (a) Plots of the mean (red spheres) with vectors to figures along the first three PCs: (black) mean + 3sd PC1; (red) mean + 3sd PC2; and (green) mean + 3sd PC3. (b) The mean (red) and a figure at mean + 3sd PC1 (blue) with a deformed grid on the blue figure at z = −1, which was deformed from being a square grid on the red figure at z = −1.
Figure 7.7 Procrustes rotated outlines of T2 Small mouse vertebrae.
Figure 7.8 Two rows of series of T2 vertebral shapes evaluated along the first two PCs – the ith row shows the shapes at c { − 6, 0, 6} standard deviations along the ith PC. Note that in each row the middle plot (c = 0) is the full Procrustes mean shape. By magnifying the usual range of c by 2 the structure of each PC is more clearly illustrated.
Figure 7.9 The first (a) and second (b) PCs for the T2 Small vertebral data. The plot shows the icons overlaid on the same picture. Each plot shows the shapes at c { − 6, −4, −2} (---*---), the mean shape at c = 0 (circled +) and the shapes at c { + 6, +4, +2} (…+…) standard deviations along each PC.
Figure 7.10 The first (a) and second (b) PCs for the T2 Small vertebral outline data. Each plot shows the full Procustes mean shape with vectors drawn from the mean (+) to an icon which is c = +6 standard deviations along each PC from the mean shape.
Figure 7.11 The first (a) and second (b) PCs for the T2 Small vertebral outline data. A square grid is drawn on the mean shape and deformed using a pair of thin-plate splines (see Chapter 12) to an icon c = 6 standard deviations along each PC (indicated by a vector from the mean to the icon). The plots just show the deformed grid at c = 6 for each PC and not the starting grids on the mean.
Figure 7.12 Pairwise plots of (s_i, ρ_i, c_i1, c_i2, c_i3)^T, i = 1, …, n, centroid size, Riemannian distance to the mean shape and the first three standardized PC scores, for the T2 Small vertebral outline data.
Figure 7.13 The full Procrustes coordinates for all 30 handwritten digits.
Figure 7.14 Pairwise plots of (s_i, ρ_i, c_i1, c_i2, c_i3)^T, i = 1, …, n, the centroid size, Riemannian distance ρ to the mean shape and the first three PC scores, for the digit 3 data. There appears to be an outlier with a particularly large value of ρ. Closer inspection indicates that the first digit may have poorly identified landmarks.
Figure 7.15 Principal component analysis of the digit number 3s. The ith row represents the ith PC, with configurations evaluated at − 3, 0, 3 standard deviations along each PC from the Procrustes mean. The central figure on each row (c = 0) is the Procrustes mean shape.
Figure 7.16 The first three PCs for the digit 3 data from the first (a) and second (b) PCs. Each plot shows the shapes at c { − 3, −2, −1} (---*---), the mean shape at c = 0 (circled +) and the shapes at c { + 3, +2, +1} (…+…) standard deviations along each PC.
Figure 7.17 Varying hands: the first three PCs with values of c { − 2, −1, 0, 1, 2} here. (Reproduced by permission of Carfax Publishing Ltd.) Source: Cootes et al. 1994.
Figure 7.18 Plot of size-and-shape PC1 scores versus centroid size, for the gorilla data. Note the females all have centroid size less than 247 and the males are all greater than 261.
Figure 7.19 Plots of the first two canonical variates for the T2 mouse vertebral data. The three groups are Large (l), Small (s) and Control (c).
Figure 7.20 Plots of the first two canonical variates for the great ape data. The six groups are male gorillas (gorm), female gorillas (gorf), male chimpanzees (panm), female chimpanzees (panf), male orang utans (pongom) and female orang utans (pongof).
Figure 7.21 Plot of the first three independent components scores (top row), and the first three PC scores (bottom row) for the mouse vertebral outline data. The observations are labelled by group: Control (c); Large (l); and Small (s). The ordering of the ICs is arbitrary.
Figure 7.22 The T2 Small landmarks (a) and the reflected relabelled landmarks (b).
Figure 7.23 The symmetric and asymmetric PCs from the T2 Small vertebrae, with the bilateral symmetric mean and PC vectors magnified three times.

Chapter 8

Figure 8.1 (a) The 30 female gorilla skull landmarks registered in the coordinate system as recorded by a digitizer. (b) The original 29 male gorilla skull landmarks.
Figure 8.2 The full Procrustes fits of the female gorilla skulls.
Figure 8.3 The full Procrustes fits of the male gorilla skulls.
Figure 8.4 The male (—) and female (- - -) full Procrustes mean shapes registered by GPA.
Figure 8.5 The Procrustes mean shape of the T2 vertebal data (landmarks at the dots) and vectors to 6 standard deviations along the first PC (a) and second PC (b).
Figure 8.6 The mean shape with vectors to a figure at 6 standard deviations along the first PC (a) and second PC (b), both the same PCs as in Figure 8.5 but the icons are registered on a common baseline 1, 2.

Chapter 9

Figure 9.1 A piecewise linear template number 3 digit, with two equal sized arcs, and with 12 landmarks lying on two regular octagons.
Figure 9.2 The first PC for the gorilla females (a) and males (b). The mean shape is drawn with vectors to an icon +3 (—–) standard deviations along the first PC away from the mean.
Figure 9.3 Principal components for the gorilla data using a pooled within group covariance matrix. The first row displays PCs 1, 2 and 3 (from left to right) and the second row displays PCs 9, 11 and 12 (from left to right). In each plot the mean shape is drawn with vectors to an icon +3 standard deviations along the PC away from the mean. The vectors on the top row are magnified 3 times and the vectors on the bottom row are magnified 10 times.
Figure 9.4 Pairwise scatter plots of the centroid sizes, the full Procrustes distances to the pooled mean and PC scores 9, 11, 2, 12, 1 (s_i, d_Fi, c_i9, c_i11, c_i2, c_i12, c_i1)^T for the gorilla data: males (m); and females (f). These particular PC scores c_ij have the highest correlation with the observed group shape difference.
Figure 9.5 The Procrustes rotated brain landmark data for the 14 controls (a) and 14 schizophrenia patients (b).
Figure 9.6 The full Procrustes mean shapes of the normal subjects (x) and schizophrenia patients (+) for the brain landmark data, rotated to each other by GPA.
Figure 9.7 Dendrograms for the five fragments based on the RMSD for (a) Set 1 and (b) Set 2: 1, AG; 2, AG + Q; 3, AG + R; 4, FAR; and 5, FAR − Q. Source: Mardia 2013. Reproduced with permission from Taylor & Francis.
Figure 9.8 Pairwise scatter plots for centroid size and the first three PC scores of shape. There is a positive correlation between the first PC score and size.
Figure 9.9 Regularization path for the LASSO (a) and cross-validation fitting (b).

Chapter 10

Figure 10.1 The smoothed Procrustes mean of the T2 Small data: (a) λ/κ = 0; (b) λ/κ = 0.1; (c) λ/κ = 1.0; and (d) λ/κ = 100.

Chapter 11

Figure 11.1 The density (f) of the Riemannian distance ρ from any fixed shape for different numbers of points k, in the uniform case. The plot shows the densities for values of k = 3, 4, 6, 10, and the mode of the distribution increases as k increases.
Figure 11.2 The isotropic model is appropriate for independent isotropically perturbed landmarks (a). The resulting shape distribution can be thought of as the distribution of the landmarks after translating, rotating and rescaling so that the baseline is sent to a fixed position (b).
Figure 11.3 The exact isotropic MLE mean shapes for the schizophrenia patients (S) and control group (C), pictured in Bookstein coordinates with baseline 2, 1 (x).
Figure 11.4 The three labelled landmarks taken on each T1 mouse vertebra. The baseline was taken as 1–2 in the calculations described in the text. Source: Dryden & Mardia 1992. Reproduced with permission of Oxford University Press.

Chapter 12

Figure 12.1 An initial square grid placed on a human skull. Source: Thompson 1917. Reproduced with permission of Cambridge University Press.
Figure 12.2 Cartesian transformation grids from the human skull in Figure 12.1 to a chimpanzee (a) and a baboon (b). Source: Thompson 1917. Reproduced with permission of Cambridge University Press.
Figure 12.3 Cartesian transformation grids from one species of fish to another. The transformation is an affine deformation. Source: Thompson 1917. Reproduced with permission of Cambridge University Press.
Figure 12.4 The superimposed gel A onto gel B using an affine match, using the 10 invariant spots chosen by an expert. The four grey levels indicate the correspondence of dark and light pixels in the registered images. The key used is: black, dark pixel in gel A and a light pixel in gel B; dark grey, matched dark pixels in gels A and B; light grey, matched light pixels in gels A and B; and white, dark pixel in gel B and a light pixel in gel A.
Figure 12.5 Bookstein’s hyperbolic shape space considers the distance between two shapes of triangles A and B as approximately δ/h for small shape differences.
Figure 12.6 Transformation grids for the square (left column) to kite (right column). In the second row the same figures as in the first row have been rotated by 45° and the deformed grid does look different, even though the transformation is the same. Source: Adapted from Bookstein 1989.
Figure 12.7 A thin-plate spline transformation grid between the control mean shape estimate and the schizophrenia mean shape estimate, from the schizophrenia study. The square grid is placed on the estimated mean control shape (a) and the curved grid is pictured on the estimated mean shape of the schizophrenic patients (b), magnified three times, with arrows drawn from the control mean (red) to patient mean (green).
Figure 12.8 The thin-plate spline transformation grid from the patient mean (a) to the control mean (b), with arrows drawn from the patient mean (red) to control mean (green). The shape change has been magnified three times.
Figure 12.9 A series of grids showing the shape changes in the skull of some sooty mangabey monkeys (read across the rows from left to right): (first row) age stage 1, age stages 1 to 2, 1 to 3; (second row) age stages 1 to 4, 1 to 5.
Figure 12.10 The pair of thin-plate splines deformation from a non-symmetric T configuration of k = 5 landmarks (a) deformed into a second figure (b), and the principal warps for the non-symmetric T example drawn as surfaces above the Cartesian plane: (c) the first principal warp; and (d) the second principal warp. Source: Adapted from Bookstein 1989.
Figure 12.11 (a) The initial grid on the non-symmetric T, (b) the first partial warp, (c) the second partial warp, and (d) the affine component. Source: Adapted from Bookstein 1989.
Figure 12.12 A thin-plate spline transformation grid between a female and a male gorilla skull midline. (a) The square grid is placed on the estimated mean female skull and (b) the curved grid is pictured on the estimated mean male skull. The deformation is magnified three times here, for ease of interpretation.
Figure 12.13 The five principal warps for the the pooled mean shape of the gorillas. The top row shows the first three principal warps (first on the left to third on the right) and the bottom row shows the last two principal warps (fourth on the left, fifth on the right).
Figure 12.14 The five principal warps for the pooled mean shape of the gorillas. The top row shows the first three principal warps (first on the left to third on the right) and the bottom row shows the last two principal warps (fourth on the left, fifth on the right). The principal warps are pictured by deforming the full Procrustes mean shape for the pooled dataset of all 59 gorillas to figures which have a partial warp score of 0.15 along each warp.
Figure 12.15 The affine component and the partial warps for the deformation from the female to the male gorilla (all magnified three times). The top row shows the affine component and the first two partial warps (affine on the left, first in the middle and second on the right) and the bottom row shows the last three partial warps (third on the left, fourth in the middle and fifth on the right).
Figure 12.16 The affine scores and the partial warp scores for female (f) and male (m) gorilla skulls. The top row shows the affine scores and the first two partial warp scores (affine on the left, first in the middle and second on the right) and the bottom row shows the last three partial warp scores (third on the left, fourth in the middle and fifth on the right).
Figure 12.17 A plot of the first two relative warp scores with respect to (a) bending energy (α = 1), (b) inverse bending energy (α = − 1) and (c) Procrustes metric (α = 0) for the female (f) and male (m) gorilla skulls.
Figure 12.18 Deformation grids for the first two relative warps from the gorilla data with respect to bending energy (α = 1), emphasizing large scale variation. A square grid is drawn onto the pooled mean shape (not shown) and then deformed (by PTPS) to a shape at +6 standard deviations along the first relative warp (a) and −6 standard deviations along the second relative warp (b).
Figure 12.19 Deformation grids for the first two relative warps with respect to the inverse bending energy (α = − 1), emphasizing the small scale variation. A square grid is drawn onto the pooled mean shape (not shown) and then deformed (by PTPS) to shapes at 6 standard deviations along the relative warps [first in (a), second in (b)].
Figure 12.20 Deformation grids for the first two relative warp scores with respect to the identity matrix (α = 0), for the gorilla data. A square grid is drawn onto the pooled mean shape (not shown) and then deformed (by PTPS) to a shape at +6 standard deviations along the first relative warp (a) and −6 standard deviations along the second relative warp (b).
Figure 12.21 Early transformation grids of human profiles. Source: Bookstein 1996b. Reproduced with permission from Springer Science+Business Media.
Figure 12.22 Early transformation grids modelling six stages through life. Source: Medawar 1944.
Figure 12.23 Transformation grids for four skulls. Source: Sneath 1967. Reproduced with permission from John Wiley & Sons.
Figure 12.24 Transformation grids for a pair of skulls. Source: Cheverud & Richtsmeier 1986. Reproduced with permission of Oxford University Press.
Figure 12.25 One finite element description for the shape change from the average female (shown) to the average male gorilla. The crosses represent the principal axes of the deformation in each triangle – the magnitudes of the largest and least amount of stretching are given, together with the directions of the strains. Source: O'Higgins & Dryden 1993. Reproduced with permission from Elsevier.
Figure 12.26 The deformed cuboid grid from the full Procrustes mean male to an icon which is three standard deviations along the first PC of shape variability. We see four different views of the icon in 3D space. In particular, view (c) is the top view of the skull and view (d) is the side view. We can see that there is more deformation in the grid at the back and top of the skull, showing the region of greatest shape variability. The deformed grid has been calculated using intrinsic kriging.

Chapter 13

Figure 13.1 Multidimensional scaling plots of the rat data, using the pairwise Riemannian distances between all 144 rat skull shapes, with shapes for each individual rat joined by lines.
Figure 13.2 A diagrammatic view of unrolling and unwrapping with respect to a piecewise geodesic curve. Source: Kume et al. 2007. Reproduced with permission of Oxford University Press.
Figure 13.3 The first two PC scores of the unrolling of the human movement data paths with respect to the fitted mean path. In (a) fitted smoothing splines are shown in solid black (λ = 0.00013) with the projected data points joined by dashed lines. In (b) fitted approximate geodesics (λ = 60658.8) are shown in solid black, with the projected data points joined by dashed lines. In both plots the encircled points are knots of the mean path. Source: Kume et al. 2007. Reproduced with permission of Oxford University Press.
Figure 13.4 The original data, the noisy data, and interpolated and smoothed shape sequences using different techniques. Source: Su et al. 2012. Reproduced with permission of Elsevier.
Figure 13.5 An affine deformation from (a) to (b). In (c) we see the change vectors in Bookstein coordinates, which are all parallel with length proportional to the distance from ϕ = 0, and direction depending on the sign of the ϕ coordinate.
Figure 13.6 The fitted gel A registered by affine fitting using the invariant points in the gels, with two points poorly located in gel B. (a) Least squares affine transformation of gel A; (b) LMS affine transformation of gel A.

Chapter 14

Figure 14.1 The active site locations of proteins 1cyd (upper point cloud) and 1a27 (lower point cloud) from the Protein Data Bank. The lines connect the top 35 estimated aligned active sites from Green and Mardia (2006).
Figure 14.2 The Iowa central place data on Kendall’s spherical blackboard. There are some special points marked by symbols representing a full Procrustes mean shape (*), the mean of the uniform distribution (X), the centre of the bell (+), and the circle is the mean of 63 simulated values from the Miles distribution. Source: Mardia 1989b. Reproduced with permission from John Wiley & Sons.

Chapter 16

Figure 16.1 A leaf outline which is a star-shaped object. Various radii are drawn from the centre to the outline.
Figure 16.2 (a) Unregistered curves and (b) registration through . Source: Cheng et al. 2016.
Figure 16.3 (a) Correspondence based on and (b) 95% credibility interval for γ(t). In (a) one of the bones is drawn artificially smaller in order to better illustrate the correspondence. Source: Cheng et al. 2016.
Figure 16.4 The original curves from the Small group, (a) without and (b) with registration. The dashed curve in (b) is the estimated μ_A and grey colour shows the credible region given by 10 000 samples of mean. Source: Cheng et al. 2016.
Figure 16.5 Range data surface of a human face: (a) before surgery; and (b) after surgery to correct a cleft palate.

Chapter 17

Figure 17.1 A simple digital image of the letter H.
Figure 17.2 The display tool of Mardia and Little (1994) for combining information from an X-ray (far left) and a nuclear medicine image (far right). Thin-plate spline transformations are used to register the images, and perhaps the most useful image is the second from the left which displays the highest nuclear medicine grey levels on top of the X-ray image. These bright areas could indicate possible locations of tumours or other abnormalities. The precise location is easier to see in this new image compared with the pure nuclear medicine image. Source: Adapted from Mardia & Little 1994.
Figure 17.3 (a–e) Images of five first thoracic (T1) mouse vertebrae.
Figure 17.4 An average T1 vertebral image obtained from five vertebral images.
Figure 17.5 Composite photographs produced by Galton by taking multiple exposures of several portraits. Source: Galton 1883.
Figure 17.6 Photographs of Einstein at ages (a) 49 and (b) 71. Image (c) is a merged image between (a) and (b). Source (a): Photograph by Keystone/Getty Images. (b): Photograph by Doreen Spooner/Keystone Features/Getty Images.
Figure 17.7 Photographs of (a) Newton and (b) Einstein. Image (c) is a merged image between (a) and (b) – ‘Newstein’. Source (b): Photograph by Fred Stein Archive/Archive Photos/Getty Images.

Exercises

Figure E.1 Some triangles.

Preface

Since the publication of the first edition of this book (Dryden and Mardia 1998) there have been numerous exciting novel developments in the field of statistical shape analysis. Although a book length treatment of the new developments is certainly merited, much of the work that we discussed in the first edition still forms the foundations of new methodology. The shear volume of applications of the methodology has multiplied significantly, and we are frequently amazed by the breadth of applications of the field.

The first edition of the book primarily discussed the topic of landmark shape analysis, which is still the core material of the field. We have updated the material, with a new focus on illustrating the methodology with examples based on the shapes package (Dryden 2015) in R (R Development Core Team 2015). This new focus on R applications and an extension of the material has resulted in the new title ‘Statistical Shape Analysis, with Applications in R’ for this second edition. There is more emphasis on the joint analysis of size and shape (form) in this edition, treatment of unlabelled size-and-shape and shape analysis, more three-dimensional applications and more discussion of general Riemannian manifolds, providing more context in our discussion of geometry of size and shape spaces. All chapters contain a good deal of new material and we have rearranged some of the ordering of topics for a more coherent treatment. Chapters 6, 13, 14, 16 and 18 are almost entirely new. We have updated the references and give brief descriptions of many of the new and ongoing developments, and we have included some exercises at the end of the book which should be useful when using the book as a class text.

In Chapter 1 we provide an introduction and describe some example datasets that are used in later chapters. Chapter 2 introduces some basic size and shape coordinates, which we feel is an accessible way to understand some of the more elementary ideas. Chapter 3 provides a general informal introduction to Riemannian manifolds to help illustrate some of the geometrical concepts. In Chapter 4 we concentrate on Kendall’s shape space and shape distances, and in Chapter 5 the size-and-shape (form) space and distance.

After having provided the geometrical framework in Chapters 2--5, statistical inference is then considered with a focus on the estimation of mean shape or size-and-shape in Chapter 6. Chapter 7 provides a detailed discussion of Procrustes analysis, which is the main technique for registering landmark data. Chapter 8 contains specific two-dimensional methods which exploit the algebraic structure of complex numbers, where rotation and scaling are carried out via multiplication and translation by addition. Chapter 9 contains the main practical inferential methods, based on tangent space approximations. Chapter 10 introduces some shape distributions, primarily for two-dimensional data. Chapter 11 contains shortened material on offset normal shape distributions compared with the first edition, retaining the main results and referring to our original papers for specific details. Chapter 12 discusses size and shape deformations, with a particular focus on thin-plate splines as in the first edition.

In Chapter 13 we have introduced many recent developments in non-parametric shape analysis, with discussion of limit theorems and the bootstrap. Chapter 14 introduces unlabelled shape, where the correspondence between landmarks is unknown and must be estimated, and the topic is of particularly strong interest in bioinformatics. Chapter 15 lays out some distance-based measures, and some techniques based on multidimensional scaling. Chapter 16 provides a brief summary of some recent work on analysing curves, surfaces and volumes. Although this area is extensive in terms of applications and methods, many of the basic concepts are extensions of the simpler methods for landmark data analysis. Chapter 17 is a more minor update of shapes in images, which is a long-standing application area, particularly Bayesian image analysis using deformable templates. Chapter 18 completes the material with discussion of a wide variety of recent methods, including statistics on other manifolds and the broad field of Object Data Analysis.

There are many other books on the topic of shape analysis which complement our own including Bookstein (1991); Stoyan and Stoyan (1994); Stoyan et al. (1995); Small (1996); Kendall et al. (1999); Lele and Richtsmeier (2001); Grenander and Miller (2007); Bhattacharya and Bhattacharya (2008); Claude (2008); Davies et al. (2008b); da Fontoura Costa and Marcondes Cesar J. (2009); Younes (2010); Zelditch et al. (2012); Brombin and Salmaso (2013); Bookstein (2014) and Patrangenaru and Ellingson (2015). A brief discussion of other books and reviews is given in Section 18.7.

Our own work has been influenced by the long-running series of Leeds Annual Statistical Research (LASR) Workshops, which have now been taking place for 40 years (Mardia et al. 2015). A strong theme since the 1990s has been statistical shape analysis, and a particularly influential meeting in 1995 had talks by both Kendall and Bookstein among many others (Mardia and Gill 1995), and the proceedings volume was dedicated to both David Kendall and Fred Bookstein.

We are very grateful for the help of numerous colleagues in our work, notably at the University of Leeds and The University of Nottingham. We give our special thanks to Fred Bookstein and John Kent who provided many very insightful comments on the first edition and we are grateful for Fred Bookstein's comments on the current edition. Their challenging comments have always been very helpful indeed. Also, support of a Royal Society Wolfson Research Merit Award WM110140 and EPSRC grant EP/K022547/1 is gratefully acknowledged.

We would be pleased to hear about any typographical or other errors in the text.

Ian Dryden and Kanti Mardia
Nottingham, Leeds and Oxford,
January 2016

with Applications in R

Second Edition