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coordinates while smoothly interpolating the interlandmark space. The smoothness of the
interpolation is maximized by minimizing the bending energy required to generate the
deformation. Typically the TPS results are viewed as grid-based deformations, where
the differences in interlandmark space are visible as deformations in the grid surface.
Figure 12.10 shows an example of TPS visualization using mean configurations from samples
of Euro-American and African-American males from the Terry Collection generated using
the program PAST (Hammer, 2011). The two-dimensional thin-plate spline was produced
by mapping the Euro-American male consensus configuration onto the African-American
male consensus configuration. This figure shows the morphological differences between
the two groups both in the face and in the vault shape viewed from a lateral perspective.
The areas requiring the greatest degree of warp to map the corresponding landmark loca-
tions are indicated by the darker colors in the grayscale gradient across the grid. The minimal
degree of interlandmark deformation in the spline grid indicates that the differences between
these two mean shape configurations are limited and mainly confined to a few landmarks of
the anterior face (alare and dacryon) and the inferior cranial vault (basion, mastoideale, and
asterion).
Thin-plate spline analysis can also be used to generate relative warp scores by deforming
each individual configuration onto the grand mean configuration. These scores can be used
for subsequent statistical analysis.
Semilandmark Methods
Semilandmarks (defined earlier) can now be analyzed by allowing the semilandmarks to
slide along vectors tangent to the curve or surface until some criterion is minimized. Initially
FIGURE 12.10 Thin-plate spline deformation grid showing that morphological variation between mean forms
for samples of Euro-American and African-American males is largely confined to the anterior face and inferior
cranial vault. The configurations are shown from a lateral view with landmarks for the face at the left of the image
and vault landmarks to the right.
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