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fact, affine. The affine deformation can be described using the follow-
ing notation. Let
M
1
and
M
2
denote the landmark coordinate matrices
corresponding to the two objects in their original positions. These
matrices consist of the coordinates of the landmarks drawn on the
transparencies before they are disturbed or thrown onto the floor.
These are
3
2
matrices with each row representing the two-dimen-
sional coordinates of the corresponding landmark. Under the
assumption that the transformation from
M
1
to
M
2
is affine, we find
this transformation by solving the equation:
M
2
1
t
where
A
is
M
1
A
a
2
2
matrix corresponding to the affine transformation (degree of
change parallel to the initial grid),
t
is a
1
2
vector corresponding to
the translation required to go from
M
1
to
M
2
, and 1 is a
3
1
matrix of
1's. This is a linear system of six equations and six unknowns with a
unique solution. The matrix
A
fully describes the deformation from
M
1
to
M
2
. However, we have no knowledge of
M
1
and
M
2
in their original
versions or positions. All that we know are the rotated and translated
versions of
M
1
and
M
2
, namely,
M
1
*
and
M
2
*
, where
M
1
*
1
t
1
and
M
1
R
1
1
t
2
,R
1
and R
2
denote the unknown rotation parameters
(
R
1
,R
2
are orthogonal matrices), and
t
1
and
t
2
are the unknown transla-
tion parameters.
The question then becomes, can we obtain the full and correct
description of the true affine deformation by working with
M
1
*
M
2
*
M
2
R
2
and
1
t
*
, can we obtain
the full and correct description of the true affine deformation? The
answer is yes if, and only if,
A
*
M
2
*
? That is, if we solve the equation:
M
2
*
M
1
*
A
*
A
. But a simple mathematical fact is
that
A
*
does not equal
A
. Although the singular values of
A
*
and
A
are
equal, the left and right eigenvectors are different due to the rotation
matrices (
R
1
,R
2
). The lack of a common coordinate system precludes us
from finding full information, which includes not just the eigenvalues
but also the eigenvectors, corresponding to the affine deformation.
4.6 The relationship between mathematical and
scientific invariance
We have shown that the number of landmarks and the lack of a com-
mon coordinate system limit what can be learned about the true form
change from the deformation-based description of form change. The
superimposition approach cannot provide the vectors that depict the
true form change vectors. Similarly, transformation grids cannot recre-
ate the true deformation.
Both approaches fail due to the
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