Biology Reference
In-Depth Information
tests to these shape coordinates is that the variances of landmark locations are not inde-
pendent of the mean location of that landmark relative to the baseline. Variance thus will
not be homogeneous at all landmarks. That is because the variance of the baseline end-
points is transferred to the other coordinates. Additionally, the use of the baseline induces
correlations between landmarks so methods like principal components analysis should not
be applied to Bookstein shape coordinates (
Bookstein, 1996; Dryden and Mardia, 1998
).
PROCRUSTES SUPERIMPOSITION
In Greek mythology, Procrustes was a bandit who fit his visitors (victims) to a bed by
stretching or truncating them, minimizing the difference between his visitors and the bed.
The method that we will now use to obtain shape coordinates is Procrustean in the sense
that it minimizes the differences between landmark configurations. Unlike the mythologi-
cal Procrustes, who altered the shape of his victims, the mathematical Procrustes superim-
position method does not alter shape because it uses the three operations that do not
alter shape: translation, scaling and rotation. Presumably Procrustes' guests would have
preferred that he had done likewise!
A step-wise description of the method was presented by
Rohlf and Slice (1990)
:
1.
Center each configuration of landmarks at the origin by subtracting the coordinates of
its centroid from the corresponding (
X
or
Y
) coordinates of each landmark. This
translates each centroid to the origin (and the coordinates of the landmarks now reflect
their deviation from the centroid).
2.
Scale the landmark configurations to unit centroid size by dividing each coordinate of
each landmark by the centroid size of that configuration.
3.
Choose one configuration to be the reference, then rotate the second configuration to
minimize the summed squared distances between homologous landmarks (over all
landmarks) between the forms.
When there are more than two forms, all are rotated to optimal alignment on the first; the
average shape is then calculated and all are rotated to optimal alignment on the average
shape, which is the new reference. At this point, the average shape is recalculated. If it differs
from the previous reference, the rotations are recalculated using this newest reference. When
the newest reference is the same as the previous, the iterations stop (usually only a few itera-
tions are necessary). The final reference is the one that minimizes the average distances of
shapes from the reference. Note that this result does not depend on the shape of the first spec-
imen used in the alignment; instead, it depends on the distribution of shapes in the sample.
While the terminology and different variations on Procrustes superimposition will be
discussed in greater detail later, it is worth mentioning that the procedure just discussed is
called partial Procrustes Superimposition, which simply indicates that the centroid size
was fixed at one. This is currently the standard Procrustes technique, and a lot of workers
will use the term Procrustes to refer to what is really a partial Procrustes superimposition.
The key is to look for a centroid size of one in the description of the methods used. When
an iterative method is used to estimate the mean form, as discussed above, the entire
process is called a Generalized Procrustes Analysis (GPA).
