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a succinct description of this approach as “an attempt to discover the
shape differences between sets will typically involve a matching of the
sets to determine how differences in the coordinates of corresponding
points can be explained through similarity transformations. Any
residual differences that cannot be explained through similarity trans-
formations can be understood to be due to differences in shape.”
The model used in the superimposition approach may be described
in mathematical terms as follows. Suppose we are interested in
describing the difference in two forms M 1 and M 2 . The superimposition
approach postulates that the two forms are related to each other by the
model: M 2
1 t T
D F where t is a translation parameter, R is an
orthogonal matrix and the matrix D F describes the form difference. If
one is interested in shape differences, the model is given by
M 2
M 1 R
> 0 is a real number, taking into considera-
tion the scaling of an object and the residual matrix D S describes the
shape difference. The model component 1 t T
1 t T
D S where
1 R
M 1 R is known as the sim-
ilarity transformation because the resultant objects after this
transformation are geometrically similar to the original object M 1 .
According to the superimposition approach, what remains after the
similarity transformation, namely D S , describes the shape difference.
D F . Given M 1 and M 2 , we want to determine t , R and D F
such that this equation is satisfied. Suppose the objects under consid-
eration are two-dimensional. Then, there are 2 parameters related to
translation — one parameter related to the angle of rotation and 2 K
parameters related to the form difference. It can be noted immediate-
ly that there are 2 K
1 t T
M 2
M 1 R
3 unknowns and 2 K equations, hence, there is no
unique solution. To make the form difference identifiable, superimpo-
sition methods put additional constraints on these parameters. For
example, one Procrustes superimposition approach (Siegel and
Benson, 1982), also known as the least squares fitting criterion, applies
the constraint that the rotation and translation parameters are such
that tr {( M 2
1 t ) T } is minimum. Once the trans-
lation and rotation parameters are fixed in this fashion, the form
difference is well defined. There are other sets of constraints that are
also used in practice. For example, consider the edge superimposition
scheme commonly used in roentgenographic cephalometry. This partic-
ular method of studying form difference translates and rotates M 1 in
such a manner that an edge joining a chosen pair of landmarks match-
es with the corresponding edge in M 2 . This, in effect, imposes the
constraint that one particular row and one other element of D F is zero.
M 1 R
1 t )( M 2
M 1 R
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