Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 129

in an attempt to minimize the sum of the squared distances between
corresponding landmarks on the forms. In our transparency example,
this process corresponds to fixing the red transparency and rotating
and translating the green transparency so that the sum of the squared
distances between corresponding landmarks on the red and green
transparencies is minimized (see Goodall, 1991; Dryden and Mardia,
1998 for details on the “Generalized Procrustes Analysis” (GPA)
approach to superimposition). The application of GPA to the compari-
son of our triangles results in the difference between the red and green
transparencies being roughly equally distributed among all landmarks.
Methodologically, the measure of similarity (the minimization cri-
terion) differs between approaches. This follows intuitively from the
fact that there is more than one way to superimpose. In practice, min-
imization of an alternate measure will produce an alternate
superimposition of forms. For example, another superimposition
scheme translates and rotates the green transparency so that the sum
of the absolute distances (instead of the sum of the squared distances)
is minimized. This results in some landmarks being matched closely,
while other corresponding landmarks fall farther from one another.
Siegel and Benson (1982) describe this “Robust Procrustes” superim-
position and other related approaches. These are only two examples of
minimization criteria that can be used to superimpose forms. In reali-
ty, there are infinite potential measures of similarity and their
corresponding superimposition schemes, although some may be more
sensible than others. Figure 4.2 illustrates the results of superimpos-
ing our example triangles according to various schemes. Figure 4.2a
shows the “true form difference” that we created (the form difference
that you created in the experiment will be different), Figure 4.2c shows
the form difference depicted by the robust superimposition method,
and Figure 4.2b shows the form difference depicted by the least
squares superimposition method. Figure 4.2d provides the form differ-
ence depicted by edge superimposition. Notice that none of the
schemes recreate the “true form difference,” and that the various
superimposition schemes provide varying versions of the overall com-
parison with dissimilar descriptions of the local differences between
the red and green transparency.
Once the original relative positions of the red and green trans-
parencies are disturbed, we lose the information necessary to put them
back into their original, relative positions. The failure of superimposi-
tion schemes in localizing the true form change stems from the fact
that once disturbed, the original relationship between the transparen-