Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 200

It is easy to show that the form difference matrix D F obtained by the
least squares constraint will be different than the one obtained by the
edge superimposition constraint.
Similar to the discussion of the deformation approach, the principle
of invariance suggests that we should restrict ourselves to inferences
that are invariant to the choice of the external constraint. In Part 1,
Section 5 of this chapter, we have illustrated the practical conse-
quences of the lack of invariance of the superimposition approach. In
the following, we discuss the relationship between invariance and
identifiability in a mathematically rigorous fashion and show that the
transformations considered by the deformation and superimposition
approaches are non-identifiable.

4.16 Matrix transformations, invariance,
and identifiability issues

The problem of studying form differences can be put in terms of study-
ing transformation of one K

D matrix. The
invariance ideas described in Chapter 3 can then be easily extended to
the two-sample problem. We then discuss the non-identifiability of the
transformations considered by the superimposition and deformation
approaches. Our argument is that non-identifiable transformations
should not be used for statistical or scientific inferences. We then show
that identifiability is equivalent to invariance. In the next section we
discuss invariant procedures to study form difference.
Consider the class of all transformations that map the space of

D matrix into another K

K D matrices to the space of K D matrices. Let denote the space of

all K

D matrices. Let H: be the collection of all transformations

that map any given K

D matrix. For exam-
ple, the affine transformation discussed earlier belongs to this class, so
do the thin-plate splines and superimposition transformations belong
to this class.
Recall that two K

D matrix to another K

D matrices are equivalent to each other if they
are rotations, reflections, and/or translations of each other. We can par-
tition the space into equivalency classes, each class consisting of

equivalent matrices. Let O( M ) denote the class of matrices equivalent
to a given matrix M .
Let h 1 and h 2 be two members of H such that h 1 ( MR 1 1 t 1 ) h 2

1 t . That is, the resultant matrices from both transforma-
tions acting on equivalent matrices are also equivalent to each other.

( MR 2 1 t 2 ) R