Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 205

4.18 Form space based on Euclidean Distance
Matrix representation

We noted in Chapter 3 that every object in D -dimensional space rep-
resented by K landmarks corresponds to some K D matrix, and,

conversely, it is also true that corresponding to every K

D matrix,
there exists an object in D -dimensional space represented by K land-
marks.
Let X be any K

D matrix. Let FM ( X ) be the form matrix corre-
sponding to X . Recall that the form matrix is a square, symmetric
matrix with zeros along the diagonals and consists of all pairwise
inter-landmark distances in the object X . Let us collect all the upper
diagonal entries of the form matrix in a vector of length K ( K 1)/2 .It

is easy to see that this vector can be represented as a point in the
Euclidean space of dimension K ( K

1)/2 . Thus, every object in D-
dimensional space represented by K landmarks can also be
represented by a single point in a Euclidean space of dimension
K ( K 1)/2 .

Now, conversely, suppose that we are given an arbitrary point in the
Euclidean space of dimension K ( K

1)/2 . When does such a point cor-
respond to some D dimensional object with K landmarks?
One condition is obvious because the distances are always positive,
such a point should belong to the positive quadrant of the Euclidean
space. However, this is not a sufficient condition. In Part 1 of this chap-
ter, we noticed that corresponding to a collection of three positive
numbers or equivalently, a point in the positive quadrant of a three
dimensional Euclidean space, there corresponds a triangle or a three
landmark object in two dimensions, if and only if the sum of any two
numbers exceeds the third number. The following discussion general-
izes this result to the case when we have K landmarks and D
dimensional objects. We first introduce some additional notation and
then state the necessary and sufficient conditions under which a point
in the positive quadrant of a Euclidean space of dimension K ( K

1)/2
corresponds to a D -dimensional object represented by K landmarks.
Let d be a point in the positive quadrant of the Euclidean space of
dimension K ( K 1)/2 . Let d T

( d 12 , d 13 ,…, d 1 K , d 23 , d 24 ,…, d 2 K ,… d ( K -1) K )
be the elements of this vector. Construct two new matrices using the
individual elements of this vector: