Statistical Shape Analysis, 2nd Edition

12 Deformations for size and shape change

12.1 Deformations

12.1.1 Introduction

In biology and other disciplines one is frequently interested in describing the difference in shape or size-and-shape between two or more objects. A measure of distance, such as the Riemannian distance, gives us a numerical measure of size and shape comparison, but this global measure does not indicate locally where the objects differ and the manner of the difference. The biological goal in many studies is to depict or describe the morphological changes in a study.

First of all we consider the case when only two objects are available and we wish to describe the size and shape difference between them. Sometimes there will be a natural ordering of the two objects, for example a juvenile growing into an adult, and sometimes not, for example a male and a female of a species. In order to describe the difference in size and shape we compute the transformation of the space in which the first object lies into the space of the second object. The transformation will give us information about the local and global shape differences.

Definition 12.1 By global differences we mean large scale changes (with small amounts of bending per unit Procrustes distance), such as an overall affine transformation. Local differences are on a smaller scale (with larger amounts of bending per unit Procrustes distance), for example highlighting changes in a few nearby landmarks. Global differences are smooth changes between ...