Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 75

3.6 Coordinate system-free representation of form

We have described a coordinate system-free representation of a trian-
gle as the Euclidean distances among all possible landmark pairs. This
can be easily extended to an object with any number of landmarks,
that number represented by K , and any number of dimensions repre-
sented by D , such that K is strictly larger than D . Because all the
information about the form of an object defined on the basis of land-
mark coordinates is summarized in the collection of all distances
between pairs of landmarks, we call such a collection of distances (put
in matrix form) a form matrix . The number of unique pair-wise linear
distances in a form matrix is L where L K ( K 1 )/ 2 .

Definition of form matrix: The form matrix or Euclidean distance
matrix corresponding to the landmark coordinate matrix A is a
matrix consisting of all possible pair-wise distances between
landmarks. The form matrix of A is defined as:

This is a square, symmetric matrix where d ij is the Euclidean dis-
tance between landmarks i and j . The form matrix for a given object
provides all the relevant information about the form of that object that
can be obtained from landmark coordinates.
As an illustration, we present a simple two-dimensional example.
We strongly recommend that the reader conduct these calculations and
draw the original triangle on a piece of graph paper. The distances
between landmarks can be obtained easily using a ruler.

Let the landmark coordinate matrix be This is a tri-

angle with vertices at (0,0), (1,0), and (0,1) for landmarks 1, 2, and 3,
respectively. The form matrix corresponding to this triangle is given by
the collection of all pair-wise distances in a matrix, namely,