Biology Reference
In-Depth Information
the least squares criterion causes the variances of the influential landmarks to be allocated
to other points, inducing covariances (
Walker, 2000
).
Resistant-fit methods reduce the influence of the Pinocchio effect by taking a “robust”
approach to superimposition. In statistics, “robust” means that the method is relatively
insensitive to outliers in the data. Similarly, a robust superimposition method is relatively
insensitive to a few landmarks with large relative displacements. A wide variety of error
functions has been used as criteria for robust fitting procedures, the interested reader is
referred to
Press et al. (1988)
for a discussion of several alternatives. None of them allow
analytic solutions for the rotation and scaling parameters needed to carry out a superim-
position; instead they use numerical methods (simplex searches) to find the rotation and
scaling necessary to minimize the error function.
The robust approach implemented by RFTRA uses the method of “repeated medians”
to determine the scaling and rotation necessary to superimpose one shape on another
(
Chapman, 1990
). We describe the steps used to find the scaling factor in some depth; then
more briefly describe the steps to find the rotation. For the scaling factor:
1.
Compute the pairwise interlandmark distances in both shapes and then compute the
ratio of each pair of corresponding distances.
2.
For each landmark, find the median of ratios for all segments radiating from that
landmark. This will yield one ratio for each landmark.
3.
Find the median of the medians generated by step 2. This median of medians is the
scaling factor used in the superimposition
in other words, all coordinates of the
second shape are scaled by this factor.
After scaling the second form, the rotation angle used by RFTRA can be determined in
a similar fashion from the same set of line segments. The first step is to compute the
angles between the corresponding segments; the remaining steps find the median angle
associated with each landmark and then the median of the medians.
Rohlf and Slice (1990)
present a generalized resistant-fit method that centers and scales coordinates to a common
size (computed as the median squared interlandmark distances) yielding a matrix
X
j
:
The
X
j
to the coordinates of the first specimen
initial step uses least squares to fit
Y
, which
is the initial reference. A new reference
Y
is then calculated as the median of the rotated
X
j
is then rotated to fit that
specimens, and
Y
. This procedure is iterated until the change
in
,
determines when the iterations cease because there is no explicit fitting criterion being
minimized.
Resistant fit methods are robust because medians are relatively insensitive to outliers.
Consequently, large changes at one or a few landmarks will not appreciably alter the
median scaling factor or the median rotation angle. This makes the resistant-fit methods
resistant to the Pinocchio effect, which helps to highlight the region where the effect
occurs, as in
Figure 3.12
. However, in the absence of the Pinocchio effect, superimpositions
produced by resistant-fit methods usually do not differ greatly from those produced by
GPA.
Figure 3.13
shows both GPA and resistant-fit superimpositions of the real scapulae
that were the basis of the hypothetical example. The real scapulae differ in the relative
length of the ventral process, as in the hypothetical case, but they also differ in the shape
of the anterior edge of the scapula (producing large relative displacements of the two
Y
is smaller than the chosen stopping criterion. The lack of change in the reference,
Y
