Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 127

superimposition approaches in detail to gain an understanding of why
this claim is invalid.
4.4 Superimposition-based approaches
The most intuitive way to compare two objects is to overlay one on top
of the other, or to superimpose them, in order to visualize the differ-
ences between them. Many scientific approaches to the study of form
have developed from this simple idea. Boas (Boas, 1905) proposed a
superimposition approach at the beginning of the twentieth century,
although his role in the establishment of this approach for the field of
anthropology was not broadly recognized until recently (Cole, 1996).
The invention of x-ray techniques and the consequent development of
Roentgen cephalometry (Broadbent, 1975) early in this century
spawned a number of research projects focusing on the collection and
analysis of longitudinal data. Radiology permitted the study of growth
of internal structures and the field of Roentgen cephalometry grew out
of the simple comparison of head x-rays by overlaying two-dimension-
al tracings of the pertinent data. The first comparisons were done by
manually matching x-rays to the center of the pituitary fossa (marked
by a landmark called “sella”) and orienting on a line that stretched
from sella to a point marking the intersection of the nasal bones with
the frontal bone (the landmark nasion; see see 4.1 ) .Methods of
standardization and various quantitative approaches were later devel-
oped so that comparisons could be made between data collected from
various longitudinal series using more automated techniques (Nanda,
1956). Additionally, two- and three-dimensional superimposition-based
approaches have been proposed to determine localized differences
between forms. For example, a recent comparison of cranial profiles of
evolving lineages of Homo used Procrustes superimposition of land-
marks and semi-landmarks located on the internal surface of the
frontal bone (Bookstein et al., 2000).
The fundamental idea behind superimposition-based approaches is
quite intuitive. The following steps relate closely to those presented in
the transparency experiment and explain the basic idea behind all
superimposition approaches. Fix the first object (the red transparen-
cy). Translate and rotate the second object (the green transparency) so
that the landmarks on the green transparency lie as close as possible
to the corresponding landmarks on the red transparency. Then draw
the vectors that connect the red landmark 1 to the green landmark 1,