Statistical Shape Analysis, 2nd Edition

10 Shape and size-and-shape distributions

Of fundamental interest are probability distributions in shape spaces and pre-shape spaces, which provide models for statistical shape analysis. We shall also consider the joint distribution of size and shape later, but first we concentrate on shape. Since the shape space is non-Euclidean, special care is required. There are several approaches we could consider for stochastic modelling, for example we could specify a probability distribution in:

1. the configuration space
2. the pre-shape space
3. the shape space (directly)
4. a tangent space.

If the probability distribution also includes certain transformation variables (as in 1. and 2.) which are not of interest, then we could either integrate them out (a marginal approach) or condition on them (a conditional approach). Note that 2. is an ambient space model whereas 3. is a quotient space model (see Section 3.2.1).

The work in this chapter is primarily for m = 2 dimensional landmarks, and some extensions to higher dimensions will be considered in Section 11.6. The main inference procedure in this chapter is the method of maximum likelihood.

10.1 The uniform distribution

We gave the volume measure in pre-shape space in Equation (4.24) and the volume measure in shape space in Equation (4.26), using Kent’s polar coordinates of Section 4.3.3. We normalize the volume measure in shape space to give the uniform measure dγ in shape space. Given original landmarks z^o recall that Kent’s polar coordinates ...