1
Introduction

1.1 Definition and motivation

Objects are everywhere, natural and man-made. Geometrical data from objects are routinely collected all around us, from sophisticated medical scans in hospitals to ubiquitous smart-phone camera images. Decisions about objects are often made using their sizes and shapes in geometrical data, for example disease diagnosis, face recognition and protein identification. Hence, developing methods for the analysis of size and shape is of wide, growing importance. Locating points on objects is often straightforward and we initially consider analysing such data, before extending to curved outlines, smooth surfaces and full volumes.

Size and shape analysis is of great interest in a wide variety of disciplines. Some specific applications follow in Section 1.4 from biology, chemistry, medicine, image analysis, archaeology, bioinformatics, geology, particle science, genetics, geography, law, pharmacy and physiotherapy. As many of the earliest applications of shape analysis were in biology we concentrate initially on biological examples and terminology, but the domain of applications is in fact very broad indeed.

The word ‘shape’ is very commonly used in everyday language, usually referring to the appearance of an object. Following Kendall (1977) the definition of shape that we consider is intuitive.

Definition 1.1 Shape is all the geometrical information that remains when location, scale and rotational effects are removed from an object.

An object’s shape is invariant under the Euclidean similarity transformations of translation, scaling and rotation. For example, the shape of a human skull consists of all the geometrical properties of the skull that are unchanged when it is translated, rescaled or rotated in an arbitrary coordinate system. Two objects have the same shape if they can be translated, rescaled and rotated to each other so that they match exactly, that is if the objects are similar. In Figure 1.1 the two mouse vertebrae outlines have the same shape. In practice we are interested in comparing objects with different shapes and so we require a way of measuring shape, some notion of distance between two shapes and methods for the statistical analysis of shape.

Image described by caption.

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.

Sometimes we are also interested in retaining scale information (size) as well as the shape of the object, and so the joint analysis of size and shape (or form) is also very important.

Definition 1.2 Size-and-shape is all the geometrical information that remains when location and rotational effects are removed from an object.

Two objects have the same size-and-shape if they can be translated and rotated to each other so that they match exactly, that is if the objects are rigid-body transformations of each other. ‘Size-and-shape’ is also frequently denoted as form and we use the terms equivalently throughout the text.

A common theme throughout the text is the geometrical transformation of objects. The terms superimposition, superposition, registration, transformation, pose and matching are often used equivalently for operations which involve transforming objects, either with respect to each other or into a specified reference frame.

An early writing on shape was by Galileo (1638), who observed that bones in larger animals are not purely scaled up versions of those in smaller animals; there is a shape difference too. A bone has to become proportionally thicker so that it does not break under the increased weight of the heavier animal, see Figure 1.2. The field of geometrical shape analysis was initially developed from a biological point of view by Thompson (1917), who also discussed this application.

Image described by caption.

Figure 1.2 From Galileo (1638) illustrating the differences in shapes of the bones of small and large animals.

How should a scientist wishing to investigate a shape change proceed? Even describing an object’s shape is difficult. In everyday conversation an object’s shape is usually described by naming a second more familiar shape which it looks like, for example a map of Italy is ‘boot shaped’. This leads to very subjective descriptions that are unsuitable for most applications. A practical way forward is to locate a finite set of points on each object, which summarize the key geometrical information.

1.2 Landmarks

Initially we will describe a shape by locating a finite number of points on each specimen which are called landmarks.

Definition 1.3 A landmark is a point of correspondence on each object that matches between and within populations.

There are three basic types of landmarks in our applications: scientific, mathematical and pseudo-landmarks. In the literature there have been various synonyms for landmarks, including vertices, anchor points, control points, sites, profile points, ‘sampling’ points, design points, key points, facets, nodes, model points, markers, fiducial markers, markers, and so on.

A scientific landmark is a point assigned by an expert that corresponds between objects in some scientifically meaningful way, for example the corner of an eye or the meeting of two sutures on a skull. In biological applications such landmarks are also known as anatomical landmarks and they designate parts of an organism that correspond in terms of biological derivation, and these parts are called homologous (e.g. see Jardine 1969). In Figure 1.3 we see some anatomical landmarks located on the skull of a macaque monkey, viewed from the side. This application is described further in Section 1.4.3. Another example of a scientific landmark is a carbon Cα atom of an amino acid on a protein backbone, as seen in Section 1.4.9.

Image described by caption.

Figure 1.3 Anatomical landmarks located on the side view of a macaque monkey skull.

Mathematical landmarks are points located on an object according to some mathematical or geometrical property of the figure, for example at a point of high curvature or at an extreme point. The use of mathematical landmarks is particularly useful in automated recognition and analysis.

Pseudo-landmarks are constructed points on an object, located either around the outline or in between scientific or mathematical landmarks. For example, Lohmann (1983) took equally spaced points on the outlines of micro-fossils. In Figure 1.4 we see six mathematical landmarks at points of high curvature and seven pseudo-landmarks marked on the outline inbetween each pair of landmarks on a second thoracic (T2) mouse vertebra. Continuous curves can be approximated by a large number of pseudo-landmarks along the curve. Hence, continuous data can also be studied by landmark methods, although one needs to work with discrete approximations and the choice of spacing of the pseudo-landmarks is crucial. Examples of such approaches include the analysis of hand shapes (Grenander et al. 1991; Mardia et al. 1991; Cootes et al. 1992), resistors (Cootes et al. 1992, 1994), mitochondrial outlines (Grenander and Miller 1994), carotid arteries (Cheng et al. 2014; Sangalli et al. 2014) and mouse vertebrae (Cheng et al. 2016). Also, pseudo-landmarks are useful in matching surfaces, when points can be located on a grid over each surface, for example the cortical surface of the brain (Brignell et al. 2010) or the surface of the hippocampus (Kurtek et al. 2011).

Image described by caption.

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.

Bookstein (1991) also demarks landmarks into three further types, which are of particular use in biology. Type I landmarks occur at the joins of tissues/bones; type II landmarks are defined by local properties such as maximal curvatures, and type III landmarks occur at extremal points or constructed landmarks, such as maximal diameters and centroids.

A further type of landmark is the semi-landmark which is a point located on a curve and allowed to slip a small distance in a direction tangent to another corresponding curve (Bookstein 1996a,c; Green 1996; Gunz et al. 2005). The term ‘semi-’ is used because the landmark lies in a lower number of dimensions than other types of landmarks, for example along a one-dimensional (1D) curve in a two-dimensional (2D) image (see Section 16.3).

A further situation that may arise is the combination of landmarks and geometrical curves. For example, the pupil of the eye may be represented by a landmark at the centre surrounded by a circle, with the radius as an additional parameter. Yuille (1991) and Phillips and Smith (1993, 1994) considered such representations for analysing images of the human face.

Definition 1.4 A label is a name or number associated with a landmark, and identifies which pairs of landmarks correspond when comparing two objects. Such landmarks are called labelled landmarks.

The landmark with, say, label 1 on one specimen corresponds in some meaningful way with landmark 1 on another specimen. A labelling is usually known and given as part of the dataset. For example, in labelling the anatomical landmarks on a skull the labelling follows from the definition of the points. When we refer to just ‘shape’ of landmarks we implicitly mean the shape of labelled landmarks, that is labelled shape.

Unlabelled landmarks are those where no labelling correspondence is given between points on different specimens. It may make sense to try to estimate a correspondence between landmarks, although there is usually some uncertainty involved. This approach is common in bioinformatics for example, as seen in Chapter 14.

In some applications there is no natural labelling, and one must treat all permutations of labels as equivalent. The unlabelled shape of an object is the geometrical information that is invariant under permutations of the labels, and translation, rotation and scale.

Example 1.1 Consider the simple example in Figure 1.5. The six triangles (A, B, C, D, E and F) are constructed from triples of labelled points (1,2,3). Triangles A and B have the same size and the same labelled shape because they can be translated and rotated to be coincident. Triangle C has the same labelled shape as A and B (but has a larger size) because it can be translated, rotated and rescaled to be coincident with A and B. Triangle D has a different labelled shape but, if ignoring the labelling, it has the same unlabelled shape as A, B and C. Triangle E has a different shape to D but it can be reflected and translated to be coincident, and so D and E have the same reflection shape. Triangle F has a different shape from all the rest.   □

Image described by surrounding text and caption.

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.

In the majority of this book the methodology is appropriate for landmark data or other point set data. Following Kendall (1984) our notation will be that there are k landmarks in m dimensions, where we usually have k ≥ 3 and m = 2 or m = 3. Extensions to size and shape analysis methods for outline data, surface data and volume data are then considered in the latter chapters of the book, and many of the basic ideas from landmark shape analysis are very helpful for studying these more complex applications.

1.3 The shapes package in R

The statistical package and programming language R is an extremely powerful and wide ranging environment for carrying out statistical analysis (Ihaka and Gentleman 1996; R Development Core Team 2015). The progam is available for free download from http://cran.r-project.org and is a very widely used and popular platform for carrying out modern statistical analysis. R is continually updated and enhanced by a dedicated and enthusiastic team of developers. R has thousands of contributed packages available, including the shapes package (Dryden 2015), which includes many of the methods and datasets from this book. There are numerous introductory texts on using R, including Crawley (2007). For an excellent, comprehensive summary of a wide range of statistical analysis in R, see Venables and Ripley (2002).

We shall make use of the shapes package in R (Dryden 2015) throughout the text. Although it is not necessary to follow the R commands, we believe it may be helpful for many readers. To join in interactively the reader should ensure that they have downloaded and installed the base version of R onto their machine. Installation instructions for specific operating systems are given at http://cran.r-project.org. After successful installation of the base system the reader should install the shapes package. The reader will then be able to repeat many of the examples in the book by typing in the displayed commands.

The first command to issue is:

library(shapes)

which makes the package available for use. In order to obtain a quick listing of the commands type:

library(help=shapes)

Also, at any stage it is extremely useful to use the ‘help’ system, by typing:

help(commandname)

where commandname is the name of a command from the shapes package. For example,

help(plotshapes)

gives information about a basic plotting function for landmark data.

1.4 Practical applications

We now describe several specific applications that will be used throughout the text to illustrate the methodology. Some typical tasks are to study how shape changes during growth; how shape changes during evolution; how shape is related to size; how shape is affected by disease; how shape is related to other covariates such as sex, age or environmental conditions; how to discriminate and classify using shape; and how to describe shape variability. Various methodologies of multivariate analysis have been used to answer such questions over the last 75 years or so. Many of the questions in traditional areas such as biology are the same as they have always been and many of the techniques of shape analysis are closely related to those in multivariate analysis. One of the practical problems is that small sample sizes are often available with a large number of variables, and so high dimension, low sample size issues (large p, small n) are prevalent (e.g. Hall et al. 2005). We shall describe many new techniques that are not part of the general multivariate toolkit. As well as traditional biological applications many new problems can be tackled with statistical size and shape analysis.

1.4.1 Biology: Mouse vertebrae

In an experiment to assess the effects of selection for body weight on the shape of mouse vertebrae, three groups of mice were obtained: Large, Small and Control. The Large group contains mice selected at each generation according to large body weight, the Small group was selected for small body weight and the Control group contains unselected mice. The bones form part of a much larger study and these bones are from replicate E of the study (Falconer 1973; Truslove 1976; Johnson et al. 1985, 1988; Mardia and Dryden 1989b).

We consider the second thoracic vertebra T2. There are 30 Control, 23 Large and 23 Small bones. The aims are to assess whether there is a difference in size and shape between the three groups and to provide descriptions of any differences. Each vertebra was placed under a microscope and digitized using a video camera to give a grey level image, see Figure 1.4. The outline of the bone is then extracted using standard image processing techniques (for further details see Johnson et al. 1985) to give a stream of about 300 coordinates around the outline. Six landmarks were taken from the outline using a semi-automatic procedure described by Mardia (1989a) and Dryden (1989, Chapter 5), where an approximate curvature function of the smoothed outline is derived and the mathematical landmarks are placed at points of extreme curvature as measured by this function. In Figure 1.6 we see the six landmarks and also in between each pair of landmarks, nine equally spaced pseudo-landmarks are placed.

Image described by caption.

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.

The dataset is available in the R package shapes and the three groups can be accessed by typing:

library(shapes)
data(mice)

The dataset is stored as a list with three components: mice$x is an array of the coordinates in two dimensions of the six landmarks for each bone; mice$group is a vector of group labels; and mice$outlines is an array of 60 points on each outline in two dimensions, containing the landmarks and pseudo-landmarks. To print the k × m × n array of landmarks in R, type mice$x and to print the group labels type mice$group (‘c’for Control, ‘l’ for Large and ‘s’ for Small). In order to plot the landmark data we can use:

par(mfrow=c(1,3))
joins<-c(1,6,2:5,1)
plotshapes(mice$x[,,mice$group=="c"],joinline=joins)
title("Control")
plotshapes(mice$x[,,mice$group=="l"],joinline=joins)
title("Large")
plotshapes(mice$x[,,mice$group=="s"],joinline=joins)
title("Small")

Here the plotshapes function plots 2D (x, y) coordinates of each object, and lines are drawn between the landmarks given in the joinline option. Here lines are drawn from landmark 1 to 6 to 2 to 3 to 4 to 5 and finally back to 1 on each object. The plot is given in Figure 1.7.

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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.

In order to plot the outline data we can use:

par(mfrow=c(1,3))
joins<-c(1:60,1)
plotshapes(mice$outlines[,,mice$group=="c"],joinline=joins,col=2)
title("Control")
plotshapes(mice$outlines[,,mice$group=="l"],joinline=joins,col=2)
title("Large")
plotshapes(mice$outlines[,,mice$group=="s"],joinline=joins,col=2)
title("Small")

and the result is given in Figure 1.8. Here the points are drawn in red (col=2) and the lines are drawn to connect points 1 through 60 and back to 1.

Image described by surrounding text and caption.

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.

Note that the coordinates in mice$x are also available in the shapes package individually by group: qcet2.dat, qlet2.dat, and qset2.dat, which can be useful for short-cuts in coding.

It is of interest to examine size and shape differences in the three groups, and how shape is related to size.

1.4.2 Image analysis: Postcode recognition

A random sample of handwritten British postcodes was collected and digitized (Anderson 1997), and an example digit ‘3’ is shown in Figure 1.9. It is of interest to classify each of the handwritten characters so that mail can be automatically sorted. The problem is a classic one in image analysis and many methods have been suggested, with varying degrees of success (e.g. see Hull 1990). The location and size of the characters are not so important for recognition but orientation information may be crucial, for example an ‘M’ must not be confused with a ‘W’. Some successful attempts at reading handwritten numbers include Simard et al. (1993); Hastie and Tibshirani (1994); and Hastie and Simard (1998). A survey of relevant work is given by Plamondon and Srihari (2000), and a related topic is hand-drawn gesture recognition (e.g. see Mardia et al. 1993).

Image described by caption.

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.

Anderson (1997) obtained mathematical landmarks and pseudo-landmarks on the digital images by hand, and in particular for the digit 3 there were 13 landmarks, as shown in Figure 1.9. It is of interest to examine the average shape and variability in shape in the data, which can then be used as a prior model for digit recognition from images of handwritten postcodes.

The landmark data are given the dataset digit3.dat in the shapes package. The data are displayed in Figure 1.10 using the R command:

Image described by surrounding text and caption.

Figure 1.10 The thirty digit 3 configurations, each with 13 landmarks.

plotshapes(digit3.dat,joinline=1:13)

1.4.3 Biology: Macaque skulls

In an investigation into sex differences in the crania of a species of macaque Macaca fascicularis (a type of monkey), random samples of 9 male and 9 female skulls were obtained by Paul O’Higgins (Hull-York Medical School) (Dryden and Mardia 1993). A subset of seven anatomical landmarks was located on each cranium and the three-dimensional (3D) coordinates of each point were recorded.

It is of interest to assess whether there are any size and shape differences between the sexes. If there are any differences, then a description of the differences is required. An artist’s impression of the 3D skull with the anatomical landmarks is given in Figure 1.11.

Image described by caption.

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.

The data are obtained by typing:

library(shapes)
data(macaques)

The 3D landmarks are available in the 7 × 3 × 18 dimensional array macaques$x, and the genders are in macaques$group.

We plot the data in Figure 1.12 using the command shapes3d as follows:

Image described by surrounding text and caption.

Figure 1.12 The macaque skull data with seven landmarks from 18 individuals, with each landmark displayed by a different colour.

joins<-c(1,2,5,2,3,4,1,6,5,3,7,6,4,7)
colpts<-rep(1:7,times=18)
shapes3d(macaques$x,col=colpts,joinline=joins)

The command shapes3d uses the rgl library in R, which in turn uses OpenGL graphics. The 3D plots can be easily rotated and moved in the graphics window by clicking and moving the mouse, thus giving a good idea of the 3D geometry of the configuration.

Note that the coordinates in macaques$x are also available in the shapes package individually by group: macf.dat, and macm.dat.

1.4.4 Chemistry: Steroid molecules

Dryden et al. (2007) and Czogiel et al. (2011) analyse a dataset of steroids, which are small molecules with a wide variety of uses. The dataset consists of between 42 and 61 atoms for each of 31 steroid molecules. The three-dimensional coordinates, atom type, van der Waals radius and partial charges of each atom are given. The collection of steroids has been considered by a number of authors, including Wagener et al. (1995). This particular version of the data was constructed by Jonathan Hirst and James Melville (School of Chemistry, University of Nottingham). The steroids have different binding affinities to the corticosteroid binding globulin (CBG) receptor, and so each molecule has an activity class of either ‘1’ high, ‘2’ intermediate or ‘3’ low binding affinity. It is of interest to examine how the shape (‘steric’) properties of the molecules are related to activity class. This dataset is quite challenging in that the molecules have different numbers of atoms, and the correspondence between atoms (labelling) is not given. However, the 17 carbon atoms in the three cyclohexane rings and one cyclopentane ring are common to all the steroids, and these do correspond in a sensible way. The carbon rings are plotted in Figure 1.13 using the following commands.

Image described by surrounding text and caption.

Figure 1.13 The first 17 carbon atoms in the 31 steroid molecules.

data(steroids)
joins<-c(1:6,1,6,5,4,7:10,5,4,7,11:14,8,14:17,13)
shapes3d(steroids$x[,,],col=rep(1:17,times=31),joinline=joins)

1.4.5 Medicine: Schizophrenia magnetic resonance images

Bookstein (1996b) considers 13 landmarks taken on near midsagittal 2D slices from magnetic resonance (MR) brain scans of 14 control volunteers and 14 schizophrenia patients. It is of interest to study any shape differences in the brain between the two groups, either in average shape or in shape variability. If shape differences between the two groups can be established, then this should enable researchers to gain an increased understanding about the condition. In Figure 1.14 we see the 13 landmarks on a 2D slice from a scan of a schizophrenia patient. The landmarks are: 1, splenium, posteriormost point on corpus callosum; 2, genu, anteriormost point on corpus callosum; 3, top of corpus callosum, uppermost point on arch of callosum (all three landmarks registered to the diameter of the callosum); 4, top of head, a point relaxed from a standard landmark along the apparent margin of the dura; 5, tentorium of cerebellum at dura; 6, top of cerebellum; 7, tip of fourth ventricle; 8, bottom of cerebellum; 9, top of pons, anterior margin; 10, bottom of pons, anterior margin; 11, optic chiasm; 12, frontal pole, extension of a line from 1 through 2 until it intersects the dura; and 13, superior colliculus.

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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.

The data are plotted in Figure 1.15 using the following commands.

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Figure 1.15 The dataset of 13 landmarks per individual from the schizophrenia study, with circles for controls and triangles for patients.

data(schizophrenia)
plotshapes(schizophrenia$x,symbol=as.integer(schizophrenia$group))

1.4.6 Medicine and law: Fetal alcohol spectrum disorder

Another important application of shape analysis is in the assessment of fetal alcohol spectrum disorders (FASDs). An MR image from the corpus callosum of a prisoner with a landmark (Rostrum) and 39 semi-landmarks is displayed in Figure 1.16. The shape of the corpus callosum has been used in court cases in expert witness testimony to help assess whether or not a defendant had been affected by FASD. Statistical shape analysis has been used successfully to help waive the death penalty for many defendants. For further details see Mardia et al. (2013a) and Section 7.10.

Image described by caption.

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.

1.4.7 Pharmacy: DNA molecules

Molecular dynamics simulations are a widely used and powerful method of gaining an understanding of the properties of molecules, particularly biological molecules such as DNA. The simulations are undertaken with a computer package, for example AMBER (Salomon-Ferrer et al. 2013), and involve a deterministic model being specified for the molecule. The model consists of point masses (atoms) connected by springs (bonds) moving in an environment of water molecules, also treated as point masses and springs. At each time step the equations of motion are solved to provide the next position of the configuration in space. The simulations are very time-consuming to run – for example several weeks of computer time may be needed to generate a few nanoseconds of data.

We consider the statistical modelling of a specific DNA molecule configuration in water. In particular, we concentrate on the simple case of 22 phosphorous atoms, where the k = 22 atom locations are recorded in angstroms (in m = 3 dimensions) and are observed over a period of time. The temporal data are highly correlated. There are many questions of interest, for example, can we describe the main features of geometric variability, can we estimate the full configuration space of the molecule, and can we simulate the molecule over much longer time scales using fast statistical techniques. In the shape package there is a very small set of DNA data with n = 30 observations, which can be obtained using:

data(dna.dat)

A plot of the data is given in Figure 1.17.

Image described by caption.

Figure 1.17 A small dataset of 22 phosphorous atoms from a DNA molecule at n = 30 time points.

1.4.8 Biology: Great ape skulls

In an investigation to assess the cranial differences between the sexes of ape, 29 male and 30 female adult gorillas (Gorilla gorilla), 28 male and 26 female adult chimpanzees (Pan), and 30 male and 24 female adult orangutans (Pongo) were studied. The data are described in detail by O’Higgins (1989) and O’Higgins and Dryden (1993). Eight landmarks are chosen in the midline plane of each skull as shown in Figure 1.18. The landmarks are anatomical landmarks and are located by an expert biologist.

Image described by caption.

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).

The dataset is available in the R package shapes and can be accessed by typing:

data(apes)

The dataset is stored as a list with two components: apes$x is an array of coordinates in eight landmarks in two dimensions for each skull; and apes$group is a vector of group labels. To print the k × m × n array of landmarks in R, type apes$x and to see the group labels type apes$group (‘gorf’for female gorillas, ‘gorm’ for male gorillas, ‘panf’ for female chimpanzees, ‘panm’ for male chimpanzees, ‘pongof’ for female orangutans, and ‘pongom’ for male orangutans). In order to plot the landmark data we can use:

par(mfcol=c(2,3))
plotshapes(apes$x[,,apes$group=="gorf"],col=1)
title("Female Gorillas")
plotshapes(apes$x[,,apes$group=="gorm"],col=2)
title("Male Gorillas")
plotshapes(apes$x[,,apes$group=="panf"],col=3)
title("Female Chimpanzees")
plotshapes(apes$x[,,apes$group=="panm"],col=4)
title("Male Chimpanzees")
plotshapes(apes$x[,,apes$group=="pongof"],col=5)
title("Female Orang Utans")
plotshapes(apes$x[,,apes$group=="pongom"],col=6)
title("Male Orang Utans")

Again the plotshapes function plots 2D (x, y) coordinates of each object, and here we have drawn each group using a different colour. The plot is given in Figure 1.19.

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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.

It is of interest to assess whether there is a size difference between the sexes and whether there are any shape differences between the sexes in the face and braincase regions. A biologist would also be interested in geometrical descriptions of the shape difference, and how shape relates to size and other covariates.

Note that the coordinates in apes$x also available in the shapes package individually by group: gorf.dat, gorm.dat, panf.dat, panm.dat, pongof.dat and pongom.dat.

1.4.9 Bioinformatics: Protein matching

A protein is a sequence of amino acids, of which there are 20 types, and each amino acid has a one-letter code: A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y. A protein structure is the 3D configuration of atoms determined by the sequence and the 3D size and shape of the protein structure determines its function. The structure is summarized using key atoms, such as α-carbon atoms in each amino acid, and other secondary structure features such as α-helices and β-sheets (e.g. Mardia 2013b). Predicting the 3D structure of the protein from the amino acid sequence (protein folding) is one of the grand challenges in science.

Green and Mardia (2004, 2006) considered matching two proteins from the PDB data bank http://www.rcsb.org/pdb (Berman et al. 2000). The particular pair of proteins that were compared had PDB codes 1a27 (Human Type I 17Beta-Hydroxysteroid Dehydrogenase) (Mazza 1997) with 63 points; and 1cyd (Mouse lung carbonyl reductase) (Tanaka et al. 1996) with 40 points. The points are the active sites of the proteins and consist of the coordinates of the centres of gravity of the amino acids that make up the nicotinamide adenine dinucleotide phosphate (NADP) binding sites of two proteins. The 20 amino acid types of each active site are labelled in one of four groups as follows:

  • Group 1: Hydrophobic. A, F, I, L, M, P, V.
  • Group 2: Charged. D, E, K, R.
  • Group 3: Polar. C, H, N, Q, S, T, W, Y.
  • Group 4: Glycine. G.

Representations of the full proteins are given in Figure 1.20.

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Figure 1.20 The proteins 1a27 (a) and 1cyd (b) from the PDB databank (Tanaka et al. 1996; Mazza 1997; Berman et al. 2000).

It is of interest to align the two molecules in order to find common similar geometrical parts. The size and shape of the proteins are of interest, and scale invariance is not required here. In this application the landmarks are unlabelled, and estimation of the correspondence between subsets of the active sites of each protein is required.

1.4.10 Particle science: Sand grains

A dataset of the curved outlines of sand grain profiles is available. There are 24 sea sand and 25 river sand grain profiles in two dimensions. The original data were provided by Dietrich Stoyan (Stoyan and Stoyan 1994; Stoyan 1997). On each particle outline there are 50 points, which were were extracted at approximately equal arc lengths by the method described in Kent et al. (2000, section 8.1). The sea particles are from the Baltic Sea and the river particles from the Caucasian River Selenchuk. It is of interest to describe the differences in shape variability of the sand grains between the two groups. Here we are interested in the shapes of the continuous closed curves. The points on each outline are unlabelled – there is no natural correspondence between points. The data are displayed in Figure 1.21.

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Figure 1.21 The sand particle outlines: (a) sea sand; and (b) river sand.

data(sand)
plotshapes(sand$x[,,sand$group=="sea"],
   sand$x[,,sand$group=="river"],joinline=c(1:50,1))

1.4.11 Biology: Rat skull growth

We consider a well-known dataset of landmarks located on X-rays of rat skulls as they grow. The data are described in Bookstein (1991) and studied by several other authors including Goodall and Lange (1989); Monteiro (1999); Le and Kume (2000a); Kent et al. (2001); and Kenobi et al. (2010). The rats were carefully X-rayed at ages 7, 14, 21, 30, 40, 60, 90 and 150 days, and there are 18 rats with complete sets of eight landmarks in two dimensions at each age. The X-rays were taken of the skulls of the same rats recorded throughout their lifetimes, and it is of interest to describe the size and shape changes as the rats undergo growth.

The data for the 18 rats with complete data are displayed in Figure 1.22 and are available in the shapes package using:

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Figure 1.22 The eight landmarks on the 18 rat skulls, observed at eight time points.

data(rats)

with the landmark coordinates in rats$x, identification number in rats$no and the time in rats$time.

1.4.12 Biology: Sooty mangabeys

Twelve landmarks are taken from the midline of the skulls of a type of monkey, sooty mangabey (Cercocebus atys), in a study described by O’Higgins and Dryden (1992), see Figure 1.23. The specimens ranged from young juveniles to an adult female and an adult male. The objective is to describe the size and shape differences in the individuals in the series from the young juveniles to the older juveniles, and then to the adults. A further problem is to examine whether the individuals can be modelled by a regression line in shape space.

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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).

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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.

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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.

The data are available in the shapes package using:

data(sooty)

The data is an array of size 10 × 2 × 7, with the first two configurations being centred, rotated coordinates for the smallest juvenile and the male adult, and then the last five observations contain the original data for the three juveniles, then the female adult and the male adult.

1.4.13 Physiotherapy: Human movement data

Kume et al. (2007) consider an application in the study of human movement data, consisting of k = 4 landmarks (lower back, shoulder, wrist and index finger) moving in time. The landmark coordinates are obtained by recording the 3D locations of small reflective markers using a system of seven video cameras. Each individual in the study sat at a table and was asked to move his or her index finger towards a target point which was positioned straight, or to the right or or to the left at different angles. The data were collected by Dr James Richardson, Université Paris Sud, France. We are interested in the shapes of the landmark configurations, and we concentrate on the shapes of the configurations in the plane of the table and this subset was considered by Kume et al. (2007). We shall concentrate on a subset of the data where five curves are available, which are labelled a, b, c, d, e. The dataset consists of the projected view of the movements in the plane of the table at which the subject is sitting. For each of the individuals we have 10 equally spaced time points (after initially linearly transforming the times to [0, 1] for each curve). The data are available as a four-dimensional array in the shapes package using:

data(humanmove)

The array is of dimension 4 × 2 × 10 × 5 and represents the coordinates of the k = 4 landmarks in m = 2 dimensions with 10 time points for each of the 5 movements.

1.4.14 Genetics: Electrophoretic gels

A technique for the identification of proteins involves the comparison of electrophoretic gel images (Horgan et al. 1992). Two examples of such images are gel A and gel B shown in Figure 1.24. The images were obtained from particular strains of parasites which carry malaria. The objective is to use the gel image to identify the strain of parasite.

In each gel there are a number of black spots, where each spot can be one of two types – invariant or variant. The invariant spots are present for all parasites and the arrangement of variant spots enables identification of the parasite. A problem with the technique when used in the field is that the gels are prone to deformations and so the gel images first need to be ‘registered’ (transformed) so that direct comparisons can be made.

In this application, 10 invariant spots have been picked out by an expert, as shown in Figure 1.25. The spots are available in the shapes package using:

data(gels)

The invariant spots are used to match gel A to gel B, either by a similarity transformation or by a more complicated transformation. A question of interest is: can a matching procedure be made resistant to some outlier points, for example mislabelled points?

1.4.15 Medicine: Cortical surface shape

In a study of the shape of cortical surfaces of schizophrenia patients and normal volunteers, some structural MR images were taken of the brain (Brignell et al. 2010). The dataset consists of n = 68 3D MR images of the brain from 29 male healthy controls, 25 male schizophrenia patients, 9 female healthy controls, and 5 female schizophrenia patients. The MR images are proton density weighted images and were collected by Sean Flynn at the University of British Columbia, Canada. Each volunteer’s image consists of 256 × 256 × 256 voxels (3D pixels of size 1 mm3). The cortical surface was extracted from each dataset using image processing techniques [specifically the brain extraction tool of Smith (2002)]. Finally 62 501 points were located on the surface of the cortex above a transverse plane passing through the anterior and posterior commisures, with the brain midline plane being located in a sagittal plane. We see an example of the cortical surface points of one of the subjects in Figure 1.26. It is of interest to describe the cortical surface shape, and whether there are any differences in shape between the schizophrenia group and the control group.

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Figure 1.26 A set of 62 501 cortical surface points in three dimensions. The colouring indicates the ordering of the points.

A subset of the data are available in the shapes package:

data(cortical)

which contains the 250 points on the outline intersecting the axial slice containing the commisures (the bottom-most axial cross-section in Figure 1.26).

1.4.16 Geology: Microfossils

The microfossil Globorotalia truncatulinoides is a microscopic planktonic found in the ooze on the ocean bed. Lohmann (1983) published 21 mean outlines of the microfossil which were based on random samples of organisms taken at different latitudes in the South Indian Ocean. Figure 1.27 shows the three mathematical landmarks selected on each of the outlines, where one landmark has been placed at the origin. The coordinates of the landmarks are extracted from Figure 7 of Bookstein (1986). It is of interest to examine whether the size of the organisms is related to the shape, and whether size or shape are related to the covariate of latitude. A more basic problem would be to obtain an estimate of the average shape of the fossils and to describe the structure of the shape variability.

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Figure 1.27 Landmarks from 21 mean outlines of microfossils.

The data are available in the shapes package using:

data(shells)

1.4.17 Geography: Central Place Theory

Central Place Theory was postulated by Christaller (1933) and is the situation where towns are distributed on a regular hexagonal lattice over a homogeneous area (with towns at hexagon centres, see Figure 1.28). Mardia et al. (1977) consider this hypothesis for a map of 44 places in 6 counties in Iowa, namely Union, Ringgold, Clarke, Decatur, Lucas and Wayne.

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Figure 1.28 The Voronoi polygons (unbroken lines) and Delaunay triangulation (broken lines) for a completely regular configuration, that is ideal central places.

In order to examine whether Central Place Theory holds, one could examine the shapes of the triangles formed by a town and its neighbours to see if they are more equilateral than expected under a randomness hypothesis. A convenient triangulation of the towns is a Delaunay triangulation (Mardia et al. 1977; Green and Sibson 1978; Okabe et al. 2000). In Figure 1.28 we see that Voronoi polygons for ideal central places would be hexagons and Delaunay triangles would be equilateral triangles. In Figure 1.29 we see a Delaunay triangulation and the Voronoi polygons for the Iowa data. An important question to ask is: are the Delaunay triangles more equilateral than expected by chance?

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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.

The points here are unlabelled (there is no correspondence in the vertices of the triangles). Also the triangles are correlated due to neighbouring triangles sharing points. Kendall (1983, 1989) also studied shape in Delaunay triangulations, in order to investigate the Central Place Theory hypothesis, following Mardia et al. (1977).

1.4.18 Archaeology: Alignments of standing stones

Consider a map of the 52 megalithic sites that form the ‘Old Stones of Land’s End’ in Cornwall, UK, given in Figure 1.30. It was proposed by Alfred Watkins, in the early 1920s, that these and other megalithic sites were placed in deliberate straight lines, called ley lines. One approach is to consider the shapes of all possible triangles and to see if there are more ‘flat’ triangles (triangles with the largest angle close to 180°) than expected under a randomness hypothesis. The points are unlabelled, and in this dataset there are triangles in two dimensions.

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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.

This dataset is particularly important in the history of shape analysis because it motivated D.G. Kendall’s pioneering work. Analysis of these data is considered by Broadbent (1980), Kendall and Kendall (1980), Small (1988) and Stoyan et al. (1995) among others.

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