Manifolds of greatest interest in this book are spaces of shapes of k-ads in ℝm, with a k-ad being a set of k labeled points, or landmarks, on an object in ℝm. This chapter introduces these shape spaces.

Introduction

The statistical analysis of shape distributions based on random samples is important in many areas such as morphometrics, medical diagnostics, machine vision, and robotics. In this chapter and the chapters that follow, we will be interested mainly in the analysis of shapes of landmark-based data, in which each observation consists of k > m points in m dimensions, representing k landmarks on an object, called a k-ad. The choice of landmarks is generally made with expert help in the particular field of application. Depending on the way the data are collected and recorded, the appropriate shape of a k-ad is the maximal invariant specified by its orbit under a group of transformations.

For example, one may look at k-ads modulo size and Euclidean rigid body motions of translation and rotation. The analysis of this invariance class of shapes was pioneered by Kendall (1977, 1984) and Bookstein (1978). Bookstein's approach is primarily registration-based, requiring two or three landmarks to be brought into a standard position by translating, rotating and scaling the k-ad. We would prefer Kendall's more invariant view of a shape identified with the orbit under rotation (in m dimensions) of the k-ad centered at the origin and scaled to have a unit size.