Biology Reference
In-Depth Information
CHAPTER
5
The Thin-plate Spline: Visualizing
Shape Change as a Defo rmation
In geometric morphometrics, the thin-plate spline serves three functions. First, it is used
as a tool for visualizing changes in shape. Second, it provides a coordinate system for the
tangent space, a particularly convenient one for landmark data because the number of
variables is 2K
7 (for three-dimensional data).
Third, the metric underlying the thin-plate spline is also used for superimposing semi-
landmarks. In this chapter, we briefly discuss each of these three purposes, then present a
basic overview of the mathematical idea of a deformation and the mathematical metaphor
underlying the thin-plate spline, which is a particular model of a deformation. We then
discuss how it can be decomposed to yield a useful set of variables. We then summarize a
method for sliding semilandmarks based on the thin-plate spline. In general, we present
a largely intuitive overview before delving more deeply into the mathematics, reserving
more technical details to the appendix of this chapter.
The graphical depiction of shape coordinates is fundamentally limited because they
cannot tell us what happens between landmarks because we have no measurements at
those locations. Sometimes it is obvious what happens between landmarks, as in
Figure 5.1 , where we can see that the snout elongates relative to the eye. That is obvious
because the posterior eye landmark is displaced towards the anterior eye landmark, and
that anterior eye landmark is not displaced towards the snout so the snout must be
lengthening relative to the eye. However, it is not so obvious whether the postorbital
region is elongating (relative either to the head or body). Similarly, it is difficult to judge
whether the head (as a whole) elongates relative to the post-cranial body. The problem is
not that we lack landmarks in the relevant regions; rather, it is that so many landmarks
are displaced relative to the others that it is mentally exhausting to track what happens
between them all. That tracking requires examining the lengths of all the vectors to deter-
mine whether several landmarks are displaced to a similar degree in concert or, instead,
if some are displaced relatively more than others (thereby increasing or decreasing the
2
4 (for two-dimensional data) or 3K
2
 
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