Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 126

think of this as the average triangle for the first group. Put another
transparency on top of the first and, using a green pen, place three
points on the second transparency in close proximity to the red points
on the first transparency, but change the location of each of the points
any way you like. Label these points from 1 to 3 corresponding to the
numbering system for the first triangle. Think of the triangle that you
just drew in green as the average of a second group.
Now, take a third transparency and lay it over the first two without
disturbing the relative positions of the points drawn in red and in green
on the two original transparencies. On the third transparency, draw
vectors connecting the landmark drawn in red and labeled “1” to the
landmark drawn in green and labeled “1,” then connect the red land-
mark “2” to the green landmark “2,” and finally connect the red
landmark “3” to the green landmark “3,” with a vector. These vectors,
originating at the red point and ending at the green point, justifiably
represent the form difference or form change required to go from trian-
gle 1 to triangle 2, providing precise information pertaining to the
direction and magnitude of change local to each landmark. The trans-
parency that contains these vectors represents the “true form change.”
The vectors enable “localization” of the changes from one form to the
other.
We refer to the transparency with red markings as the “red trans-
parency,” the transparency with the green markings as the “green
transparency,” and the transparency with the vectors depicting the
change from red to green as the “true form change” transparency. Take
away the true form change transparency and set it aside for future use.
Move the red and green transparencies away from one another in an
arbitrary fashion (e.g., throw them on the floor and pick them up). This
exercise simulates the effect of the loss of a common coordinate system.
Once the common coordinate system is lost, our job is to determine
whether the vectors that depict the true form change can be recreated
using only the landmark data recorded on the red and green trans-
parencies.
The conditions we have described in this experiment reflect the sit-
uation of the scientist who wants to compare two objects. All that the
scientist has are the red and green transparencies. Nothing inherent
to those transparencies provides information regarding the common
coordinate system that enabled the recording of the true form change.
Superimposition approaches posit that the true form change depicted
as vectors that represent the magnitude and direction of change can be
found using these red and green transparencies. Next, we present