Whence and what art thou, execrable shape?
John Milton (1667) Paradise Lost II, 681

In a wide variety of disciplines it is of great practical importance to measure, describe and compare the shapes of objects. The field of shape analysis involves methods for the study of the shape of objects where location, rotation and scale information can be removed. In particular, we focus on the situation where the objects are summarized by key points called landmarks. A geometrical approach is favoured throughout and so rather than selecting a few angles or lengths we work with the full geometry of the objects (up to similarity transformations of each object). Statistical shape analysis is concerned with methodology for analysing shapes in the presence of randomness. The objects under study could be sampled at random from a population and the main aims of statistical shape analysis are to estimate population average shapes, to estimate the structure of population shape variability and to carry out inference on population quantities.

Interest in shape analysis at Leeds began with an application in Central Place Theory in Geography. Mardia (1977) investigated the distribution of the shapes of triangles generated by certain point processes, and in particular considered whether towns in a plain are spread regularly with equal distances between neighbouring towns. Our joint interest in statistical shape analysis began in 1986, with an approach from Paul O’Higgins and David Johnson in the Department of Anatomy at Leeds, asking for advice about the analysis of the shape of some mouse vertebrae. Some of their data are used in examples in the book.

In 1986 the journal Statistical Science began, and the thought-provoking article by Fred Bookstein was published in Volume 1. David Kendall was a discussant of the paper and it had become clear that the elegant and deep mathematical work in shape theory from his landmark paper (Kendall 1984) was of great relevance to the practical applications in Bookstein’s paper. The pioneering work of these two authors provides the foundation for the work that we present here. The penultimate chapter on shape in image analysis is rather different and is inspired by Grenander and his co-workers.

This text aims to introduce statisticians and applied researchers to the field of statistical shape analysis. Some maturity in Statistics and Mathematics is assumed, in order to fully appreciate the work, especially in Chapters 4, 6, 8 and 9. However, we believe that interested researchers in various disciplines including biology, computer science and image analysis will also benefit, with Chapters 1–3, 5, 7, 8, 10 and 11 being of most interest.

As shape analysis is a new area we have given many definitions to help the reader. Also, important points that are not covered by definitions or results have been highlighted in various places. Throughout the text we have attempted to assist the applied researcher with practical advice, especially in Chapter 2 on size measures and simple shape coordinates, in Chapter 3 on two-dimensional Procrustes analysis, in Section 5.5.3 on principal component analysis, Section 6.9 on choice of models, Section 7.3.3 on analysis with Bookstein coordinates, Section 8.8 on size-and-shape versus shape and Section 9.1 on higher dimensional work. We are aware of current discussions about the advantages and disadvantages of superimposition type approaches versus distance-based methods, and the reader is referred to Section 12.2.5 for some discussion.

Chapter 1 provides an introduction to the topic of shape analysis and introduces the practical applications that are used throughout the text to illustrate the work. Chapter 2 provides some preliminary material on simple measures of size and shape, in order to familiarize the reader with the topic. In Chapter 3 we outline the key concepts of shape distance, mean shape and shape variability for two-dimensional data using Procrustes analysis. Complex arithmetic leads to neat solutions. Procrustes methods are covered in Chapters 3–5. We have brought forward some of the essential elements of two-dimensional Procrustes methods into Chapter 3 in order to introduce the more in-depth coverage of Chapters 4 and 5, again with a view to helping the reader. In Chapter 4 we introduce the shape space. Various distances in the shape space are described, together with some further choices of shape coordinates. Chapter 5 provides further details on the Procrustes analysis of shape suitable for two and higher dimensions. We also include further discussion of principal component analysis for shape.

Chapter 6 introduces some suitable distributions for shape analysis in two dimensions, notably the complex Bingham distribution, the complex Watson distribution and the various offset normal shape distributions. The offset normal distributions are referred to as ‘Mardia–Dryden’ distributions in the literature. Chapter 7 develops some inference procedures for shape analysis, where variations are considered to be small. Three approaches are considered: tangent space methods, approximate distributions of Procrustes statistics and edge superimposition procedures. The two sample tests for mean shape difference are particularly useful.

Chapter 8 discusses size-and-shape analysis – the situation where invariance is with respect to location and rotation, but not scale. We discuss allometry which involves studying the relationship of shape and size. The geometry of the size-and-shape space is described and some size-and-shape distributions are discussed. Chapter 9 involves the extension of the distributional results into higher than two dimensions, which is a more difficult situation to deal with than the planar case.

Chapter 10 considers methods for describing the shape change between objects. A particularly useful tool is the thin-plate spline deformation used by Bookstein (1989) in shape analysis. Pictures can be easily drawn for describing shape differences in the spirit of D’Arcy Thompson (1917). We describe some of the historical developments and some recent work using derivative information and kriging. The method of relative warps is also described, which provides an alternative to principal component analysis emphasizing large or small scale shape variability.

Chapter 11 is fundamentally different from the rest of the book. Shape plays an important part in high-level image analysis. We discuss various prior modelling procedures in Bayesian image analysis, where it is often convenient to model the similarity transformations and the shape parameters separately. Some recent work on image warping using deformations is also described.

Finally, Chapter 12 involves a brief description of alternative methods and issues in shape analysis, including consistency, distance-based methods, more general shape spaces, affine shape, robust methods, smoothing, unlabelled shape, probabilistic issues and landmark-free methods.

We have attempted to present the essential ingredients of the statistical shape analysis of landmark data. Other books involving shape analysis of landmarks are Bookstein (1991), Stoyan and Stoyan (1994, Part II) (a broad view including non-landmark methods) and Small (1996) (a more mathematical treatment which appeared while our text was at the manuscript stage).

In the last few years the Leeds Annual Statistics Research workshop has discussed various ideas and issues in statistical shape analysis, with participants from a wide variety of fields. The edited volumes of the last three workshops (Mardia and Gill 1995; Mardia et al. 1996c, 1997a) contain many topical papers which have made an impact on the subject.

If there are any errors or obscurities in the text, then we would be grateful to receive comments about them.

Real examples are used throughout the text, taken from biology, medicine, image analysis and other fields. Some of the datasets are available from the authors on the internet and we reprint three of the datasets in Appendix B.

Ian Dryden and Kanti Mardia
Leeds, December 1997