Biology Reference
In-Depth Information
M
ODELING SHAPE CHANGE AS A DEFORMATIO
N
A deformation is a smooth function that maps points in one form to corresponding
points in another form. Intuitively, smoothness means that the function goes on without
interruptions or abrupt changes. More precisely, it means that the function is continuously
differentiable (it can be differentiated, its first derivative can be differentiated, and so can
its second, and so forth). To be differentiable, a function must be continuous. For example,
the function Y
X
3
5
is continuous, but the absolute value function Y
5 j
X
j
is not because it
has a sharp corner at X
5
0 and so is not differentiable at that point. The Dirichelet func-
(1 when X is rational; 0 when X is irrational) is also not continuous
it is not dif-
ferentiable anywhere. To be continuous, it is not enough to have a first derivative, that
first derivative must also be a differentiable function. That deformations are continuously
differentiable is important, because it means that the function must extend between land-
marks
5
tion Y
it cannot be defined only at certain discrete points and disappear in the regions
between them.
If a function blows up (becomes infinite or non-differentiable) between points, we can-
not use it to interpolate values between them. This is important because we are using the
thin-plate spline as an interpolation function, inferring what happens between landmarks
from data at given anatomical points. If it is unreasonable to interpolate, it is unreasonable
to use the thin-plate spline for that purpose. It is also unreasonable to interpolate between
far distant landmarks, just as it is unreasonable to extrapolate a linear regression far
beyond the range of the observed data. If our landmarks are far apart, we have too few
data to draw conclusions about what happens between them. For example, in
Figure 5.2
we are assuming that the changes in regions between post-cranial landmarks can be
inferred from landmarks on the dorsal and ventral periphery. That assumption can be
questioned, because if we actually had more landmarks in that region we might find
abrupt changes
small regions where the grid dramatically compresses or expands. We
are simply assuming that no such localized changes occur.
Another case in which it would be inappropriate to think of shape change as a deforma-
tion is when there is change concentrated at a single landmark. That is equivalent to a
function with an abrupt change, which violates the assumption of continuity. Such discon-
tinuities can be detected as displacement of one shape coordinate against a background of
invariant points. That pattern may be rare, but one close to it has actually been found in
data (
Myers et al., 1996
). In that study, prairie deer mice (Peromyscus maniculatus bairdii)
fed different diets were found to have skulls that differ only in the location of the tips of
the incisors relative to the other skull landmarks. This is an extreme case of a Pinocchio
effect (as discussed in Chapter 3). Such highly local changes should be ruled out before
any deformation-based method is applied; if such highly localized change is found, it is
better to rely on shape coordinates.
There is one other case in which a deformation-based approach might be unwise;
when the interpolation spans a large amount of extra-organismal space
that is, when
it is interpolating the changes over regions of “tissue” outside the organism. This can
happen when landmarks are located at tips of long structures, or on structures that
extend far laterally. Normally this is not a serious problem because we can simply
