Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 73

marks. This transformation eliminates the nuisance parameters of
translation, rotation, and reflection. One important consequence of any
such transformation adopted for the purpose of eliminating nuisance
parameters is that the original parameters of interest, namely, the
mean form and the perturbation structure, are transformed as well. As
a result, only the transformed parameters are estimable and not the
original parameters. But are these transformed parameters useful for
scientific interpretation? Our experience is that in the case of land-
mark coordinate data used in biological analysis, the transformed
parameters described are meaningful and interpretable.
In the next section, we explain the necessary data transformation and
corresponding parameter transformation that eliminate the nuisance
parameters of translation, rotation, and reflection. We also demonstrate
that the transformed parameters are biologically interpretable.
3.5 A definition of form
To demonstrate that the suggested transformation does not affect the
study of form, we must begin with a precise definition of the concept of
form of an object.
Definition: The form of an object is the characteristic that remains
invariant under any translation, rotation or reflection of the object.
To clarify this definition, consider the simple situation of a triangle,
defined by the location of three landmarks. Suppose we rotate or trans-
late or reflect the triangle by an arbitrary amount. Any such movement
of the triangle results in changes in the coordinate locations of the
three vertices. Although no changes have been made regarding the rel-
ative location of the landmarks, a new set of coordinates is required to
define the new location of the three landmarks once the triangle has
been translated, rotated, or reflected. This means that the landmark
coordinate matrix changes upon reflection, translation, or rotation and
that the landmark coordinate matrix is not invariant with respect to
translation, rotation, and/or reflection.
Now, consider characterizing the form of a triangle as a vector of
distances between all possible pairs of landmarks (a vector of three dis-
tances in the case of a triangle). This vector of inter-landmark
distances is equivalent to the definition of the triangle by landmark
locations with one subtle but important difference: a coordinate system
is not required to record the inter-landmark distances. While rotation,