Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 130

True form difference

Least Squares superimposition

3

3

2

1

1 2

a

c

Robust superimposition Edge matching landmarks 2 and 3

3

3

1

2

1

2

b d
Figure 4.2. Superimposition of two triangles (three landmarks) according to various
schemes. In 4.2a the “true form difference” that we created is given. Fig 4.2b shows the
form difference depicted by the robust superimposition method. Fig. 4.2c shows the
form difference depicted by the least squares method. Figure 4.2d provides the form
difference depicted by edge superimposition. None of the superimposition methods
depicted recreate the “true form difference.” The various superimposition schemes pro-
vide varying versions of the overall comparison with dissimilar descriptions of the local
differences between the two original triangles.

cies (or any forms) can never be known. The original relationship
between the transparencies cannot be known because there is no sin-
gle common coordinate system in which to put them. We cannot,
therefore, reproduce the original relationship. Of course, they can be
superimposed using any system of choice, but adoption of a particular
superimposition scheme determines the relationship that will be “dis-
covered.” Comparison of the two forms using one superimposition
scheme will produce one relationship, while comparison of the same
two forms using another superimposition scheme will produce another
relationship. No information is available that enables identification of
which relationship is the true one. For the scientist, this means that