Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 136

from a normal child at 2 1 / 2 years of age (broken line in Figure 4.3 ) and
one of a 2 1 / 2 -year-old child with Apert syndrome (solid line in Figure
4.3 ) . Following traditional procedures, we locate sella in the center of
the pituitary fossa and nasion at the intersection of the nasal and the
frontal bones. Next, we match the two tracings by registering on the
landmark sella (matching its location exactly in the two configurations)
and rotating the two tracings until the lines that stretch from sella to
nasion overlay one another. This is the superimposition seen in Figure
4.3a . Now take the same two tracings and register on sella, but super-
impose on a line stretching from sella to the landmark basion (the most
anterior point on the rim of foramen magnum). This superimposition is
shown in Figure 4.3b . It is obvious that the two superimposition
schemes produce varying summaries of the differences between the two
forms. The superimposition scheme on the top shows the face of the
Apert individual elevated and projected posteriorly as compared to nor-
mal, while that on the bottom shows the Apert face projected superiorly
and posteriorly but to a different degree and in a different direction. The
posterior portions of the calvaria match fairly well in the comparison at
the bottom, but show marked differences using the superimposition
shown at top. Since decisions regarding pathological processes and clin-
ical treatment are made on the basis of these types of comparisons, it is
critical to understand that the results vary depending upon (are not
invariant to) the superimposition scheme used. This example is not
unlike superimposition methods currently being applied in neuro-
science, where specific features of the brain are mapped by registering
on a “model” brain or atlas.
To underscore the influence of the choice of a superimposition
scheme in the context of the analysis of biomedical data, we further
scrutinize the example given above. Eleven two-dimensional landmarks
were located on the cephalometric radiographs of a sample of children
diagnosed with Apert syndrome from the Center for Craniofacial
Anomalies, University of Illinois, Chicago, and on the radiographs of
unaffected children of similar age from the Bolton-Brush Growth
Series, Cleveland, OH. The landmark data were originally presented in
Richtsmeier (1985). Figure 4.4a shows the placement of the landmarks
on a radiographic tracing (upper left quadrant) and comparison of the
estimated mean forms for the two samples using various superimposi-
tion schemes ( Figures 4.4b , c, d ). Landmarks corresponding to the
unaffected sample mean form are shown as solid diamonds, whereas
the open diamonds correspond to the landmark locations on the mean
form of the Apert sample. The superimposition shown in Figure 4.3b