Statistical Shape Analysis, 2nd Edition

11 Offset normal shape distributions

11.1 Introduction

Rather than conditioning we could consider the marginal distribution of shape after integrating out the similarity transformations. These distributions will be called offset shape distributions. We consider multivariate normal configurations of k-points in , that is the model for the landmarks is:

(11.1)

where Ω is symmetric positive definite. We shall concentrate initially on the important m = 2 dimensional case in this chapter. Kendall (1984) introduced the case where μ consists of coincident points and Ω was proportional to the identity matrix. Mardia and Dryden (1989a,b) and Dryden and Mardia (1991b) further developed the shape distributions for general μ and Ω, in two dimensions. These distributions have been called ‘Mardia–Dryden’ distributions in the literature (Bookstein 1991, 2014; Lele and Richtsmeier 1991; Kendall 1991b; Stuart and Ord 1994).

11.1.1 Equal mean case in two dimensions

11.1.1.1 Isotropic case

Result 11.1 Consider the k ≥ 3 landmarks in to be i.i.d. as bivariate normal with equal means and covariance matrix proportional to the identity, that is

(11.2)

The resulting shape distribution is uniform ...