Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 148

Defining the difference between two forms

Let us begin with two forms, A and B. We will set the landmark coor-
dinates for forms A and B to differ only in the position of landmark 2,
so that the coordinate locations for these two forms are:

FM (A) and FM (B) denote the form matrices corresponding to the
two objects. Recall that FM ij (A) is the distance between landmarks i
and j in object A and that FM ij (B) is the corresponding distance in
object B . The form matrices for the above objects are:

01 1
10

02 1
20 FM A () . 1 FM B () . 3

1 1 41 . 0 1 2 23 . 0

We present three possible ways to express form difference using the
form matrices, FM (A) and FM (B).

Arithmetic form difference

The difference between forms can be expressed as the arithmetic dif-
ference between like linear distances in the two forms. The collection
of these differences can be expressed as a matrix consisting of elements
that correspond to the arithmetic difference between like linear dis-
tances in the two objects. For example, the absolute form difference
matrix, AFDM (B, A)

FM ij (B) FM ij (A) , for the above two objects, is

given as:

01 0
10 AFDM B A (, ) . 2

0 0 82 . 0