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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

α

M

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.

Consider

α

the

model

describing

the

form

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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