Biology Reference
In-Depth Information
disadvantages of aligning landmark configurations along one edge, as in the two-point
registration. Configurations are scaled to unit centroid size and, because of that, baselines
will usually vary in length. Consequently, the two endpoints cannot be superimposed
simultaneously. Instead, their
Y
-coordinates are fixed at zero and their
X
-coordinates are
allowed to vary as necessary to align the
X
-coordinates of the centroids at zero, in effect
sliding the baseline along the
X
-axis (the
Y
-coordinate of the centroid is the average
perpendicular distance of the landmarks from the baseline after scaling to unit centroid
size). Because SBR prevents rotation of the baseline, it yields a more realistic representa-
tion of the data
in this case, of the ontogenetic change in skull shape. Actually, a very
similar picture can be obtained by a GPA on the unreflected data, or on the back-reflected
data obtained by duplicating the averaged and reflected data back across the midline
(
Figure 3.17C
). In general, reconstructing the whole skull makes a more interpretable picture
(one that looks more like the organism), so it might be useful to present results in these
terms even if the statistical analyses used the GPA coordinates computed for the reflected
and averaged half skull.
Of the various superimposition methods discussed in this chapter, the one that is
most widely used is GPA for reasons that will become clearer in the next chapter
this
method is grounded in the mathematical theory of shape. Configurations of landmarks are
manipulated using the three operations that do not alter shape as defined by
Kendall
.
These operations are used in a manner that removes all differences that are not shape
differences. The configurations produced by this procedure are those that map to points
in the shape spaces implied by Kendall's definition of shape. The computed distances
between these configurations are the distances between points in those spaces, or certain
linear approximations of those spaces. The characteristics of these metrics are well known,
providing a secure and stable foundation for biological shape analysis, and the pictures
embody the results.
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