Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 147

one can plot a single point in the L -dimensional Euclidean space corre-
sponding to each distinct K -landmark object. The collection of all such
points is called the form space corresponding to K -landmark objects.

Definition of the form space of K-landmark objects: The form
space of K -landmark objects is a collection of points in
L 2 dimensional Euclidean space such that each point in

this collection corresponds to a form matrix of some K -landmark
object.
Every possible form with K-landmarks corresponds to one and only one
point in the identified form space. Similar to the three-landmark case,
this form space occupies only a subset of L-dimensional Euclidean
space (e.g., only that part of the space with positive numbers). When
the forms being studied include more than three landmarks (K > 3),
the additional constraints that specify the form space subset are com-
plicated. The exact mathematical description of these constraints is
provided in Part 2 of this chapter.

K(K 1)

4.7.3 Studying the difference between two forms using the form space

Suppose we have two objects. Any two objects will do, but let's think
back to the red and green transparencies described previously. Given
our definition of form space, there is a single point in the form space
corresponding to the red transparency and all other three-landmark
objects with the same form. Similarly, there is another single point in
the form space corresponding to the green transparency and all other
objects with the same form. Since we are dealing with three landmark
objects, we can plot them in the form space as points R and G in Figure
4.7 . How can the difference between points R and G be determined?
There are many different ways to describe the difference between
points R and G. We will present a few possibilities that have proved
useful in our own research. As long as this description depends solely
on the location of points R and G in the defined form space, and not on
any other extraneous information, the description will satisfy the
invariance requirement. Definitions of form difference should satisfy
the invariance requirement and be biologically interpretable.