Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 139

(landmarks 1, 2, 3, and 4) does not extend as far forward in the Apert
individual. The upper face (landmarks 1 and 2) is shifted posteriorly,
whereas the lower face and palate (landmarks 3, 4, and 6) are shifted
posteriorly and superiorly.
In Figure 4.4c the same two mean forms are compared using anoth-
er superimposition scheme that matches landmark 7 (tuberculum
sella) exactly and orients the forms on a line stretching from landmark
7 to landmark 10 (basion). In this comparison, similar to the previous
superimposition scheme, there is no difference in the forms local to the
sella region (landmarks 7, 8, and 9). However, landmark 11 is displaced
in the Apert individual, in a pattern that is opposite to what we
obtained by the first superimposition. All facial landmarks of the Apert
individual (landmarks 1, 2, 3, 4, and 6) are displaced posteriorly and
superiorly.
Finally, in Figure 4.4d , we show the superimposition obtained
according to the least squares criterion. This provides another quite
distinct description of how the two mean forms differ from each other.
In this comparison, landmarks local to the pituitary fossa (landmarks
7, 8, and 9) on the Apert cranial base are displaced inferiorly. The ante-
rior face (landmarks 1, 2, 3, and 4) of the Apert individual is displaced
superoposteriorly. There is minimal difference local to the posterior
portion of the palate (landmark 6). The posterior neurocranium of the
Apert individual is displaced inferiorly, indicating a shallower neuro-
cranium in the unaffected individual.
Now, let us remind ourselves of the reasons for comparing these
forms. One reason involved providing the surgeon with information on
how the Apert syndrome skull differs from the normal skull. A surgeon
might design varying corrective procedures depending upon which
superimposition scheme is applied to the data. It should be clear from
this example that the application of the superimposition approach pro-
vides very different scientific inferences depending on the choice of the
external constraint.
Now consider the comparison of the same two mean forms using the
thin-plate spline methodology. One of the attractions of the thin-plate
spline methodology is the ability to present the form change graphi-
cally as the way in which a square grid changes under the estimated
deformation. In Figure 4.5 we show the deformation grid that results
from the comparison of the normal and Apert data when the constraint
is based on the minimum bending energy thin-plate spline ( Figure
4.5b ) , and when the constraint is based on W

D 1 where D is the
matrix of squared distances between all pairs of landmarks in the