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.

Digital images today play a vital role in science and technology, and also in many aspects of our daily life. This book seeks to advance the analysis of images, especially digitized ones, through the statistical analysis of shapes. Its focus is on the analysis of landmark-based shapes in which a k-ad, that is, a set of k labeled points or landmarks on an object or a scene, is observed in two or three dimensions, usually with expert help, for purposes of identification, discrimination, and diagnostics.

In general, consider the k-ad to lie in ℝm (usually, m = 2 or 3) and assume that not all the k points are the same. Then the appropriate shape of the object is taken to be the k-ad modulo a group of transformations.

For example, one may first center the k-ad, by subtracting the mean of the k-ad from each of the k landmarks, to remove the effect of location. The centered k-ad then lies in a hyperplane of dimension mk - m, because the sum of each of the m coordinates of the centered k points is zero. Next one may scale the centered k-ad to unit size to remove the effect of scale or size. The scaled, centered k-ad now lies on the unit sphere Sm(k-1)-1 in a Euclidean space (the hyperplane) of dimension m(k - 1) and is now called the preshape of the k-ad.

This chapter develops nonparametric Bayes procedures for classification, hypothesis testing and regression. The classification of a random observation to one of several groups is an important problem in statistics. This is the objective in medical diagnostics, the classification of subspecies, and, more generally, the target of most problems in image analysis. Equally important is the estimation of the regression function of Y given X and the prediction of Y given a random observation X. Here Y and X are, in general, manifold-valued, and we use nonparametric Bayes procedures to estimate the regression function.

Introduction

Consider the general problem of predicting a response Y ∈ Y based on predictors X ∈ X, where Y and X are initially considered to be arbitrary metric spaces. The spaces can be discrete, Euclidean, or even non-Euclidean manifolds. In the context of this book, such data arise in many chapters. For example, for each study subject, we may obtain information on an unordered categorical response variable such as the presence/absence of a particular feature as well as predictors having different supports including categorical, Euclidean, spherical, or on a shape space. In this chapter we extend the methods of Chapter 13 to define a very general nonparametric Bayes modeling framework for the conditional distribution of Y given X = x through joint modeling of Z = (X, Y). The flexibility of our modelling approach will be justified theoretically through Theorems, Propositions, and Corollaries 14.1, 14.2, 14.3, 14.4, and 14.5.