A HEAD-AND-FACE ANTHROPOMETRIC SURVEY - The National ...

FIGURE 10
Facial Landmarks Recorded for Each Subject
Cartesian Coordinates are collected in raw space (scanner axis system) and then
normalized for translation, rotation, and scale. This is referred to as Generalized
Procrustes Analysis, or GPA (Rohlf, 2000).
GPA is a method for the statistical analysis of coordinate data. It is used specifically to
partition the variation of a given sample of forms into components of size and shape. In
order to achieve this partition and filter out any variation due to differences in position,
orientation, or scale, the landmark configurations have to be normalized using
Procrustes superimposition (Gower 1975, Rohlf & Slice 1990). Using a least squarestype
of algorithm, this technique yields coordinate data that are optimally superimposed
and rescaled to the same centroid size. Centroid size is the square root of the sum of
the squared distances between pairs of homologous landmarks of each specimen and
the mean of the total sample.
In the simplest case, Procrustes superimposition is performed with only two
configurations, and it involves the following steps:
1. Each configuration's centroid (average of all xyz coordinates) is
translated to the origin of the coordinate system.
2. Each configuration is rescaled by setting centroid size to 1.
3. Each configuration is rotated until the Procrustes distance for all homologous
landmarks is minimized. The orthogonal rotation matrix, H, to rotate X
2
to a
least-squares orientation with respect to X
1
is calculated as:
t
H=
VΣU
t
t
where U and V are obtained from the singular value decomposition of XX
1 2
= UDV
and Σ is a diagonal matrix of 1s with the same sign as the corresponding elements of
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