Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 132

that was on the dough given these two transparencies representing the
initial and target configurations.
Transformation grids are thought to recreate the true form change
grid as our bread dough did in the above experiment, but there are sev-
eral reasons why they fail to do so. These reasons, some of which were
introduced at the beginning of this chapter, are discussed with specif-
ic reference to deformation approaches below.
Arbitrary choice of deformation
There are many ways in which one object can be deformed into any
other given object. Knowledge of material properties may help us to
expect certain classes of deformations, but in most morphometric
applications, material properties are neither available nor are explicit-
ly considered, and the choice of deformation is arbitrary.
Sir D'Arcy Thompson recognized that the combined action of many
different appropriate forces on any material form could transform one
form into another. Thompson suggested that only simple transforma-
tions should be considered in biology, citing the laws of physics to
support the use of parsimony in choosing a deformation. Parsimony is
a guiding principle in many branches of biology, and in the case of
deformation approaches, minimum energy deformations correspond to
a specific type of parsimony. But without a valid scientific reason for
the adoption of this principle in any particular study, the use of parsi-
mony is difficult to defend. Parsimony is often defended on the basis of
statistical reasons or mathematical convenience, but these reasons are
a poor substitute for biological relevance.
Relationship between the number of landmarks and the choice
of deformations under parsimony
Recall that in our example we have only three landmarks on each of
the objects. With only three landmarks on each object, the simplest
deformation that deforms one object into another is the affine defor-
mation. Goodall and Green (1986) provide an excellent description of
affine deformation. Simply stated, an affine transformation is a trans-
formation such that lines that are parallel in the original grid remain
parallel in the deformed grid. Equivalently, a circle drawn on the orig-
inal grid is transformed into an ellipse in thetransformed grid. Thus,
shear (change on an axis not parallel with the original grid) does not
occur in the affine transformation.