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independent components. The uniform part can be further decomposed into twelve inde-
pendent components; but only five of these change shape.
The numbers of uniform and non-uniform components can be explained if we consider
the possible deformations of the simplest three-dimensional shape, a tetrahedron of four
landmarks. All deformations of a tetrahedron, like all deformations of a triangle, must be
uniform; only when a fifth point is added can we detect non-uniform transformations (i.e.
transformations that differ between regions of the tetrahedron). With just four landmarks
a deformation can have twelve components, all of them uniform. Seven of the uniform
components do not change shape
which leaves five uniform components that do change shape. With each additional land-
mark beyond the fourth, there are three possible non-uniform components of deformation
(because there are three directions in which that point might move relative to the others),
hence 3(K
they are the ones removed by superimposition
4).
The components of the non-uniform part of a three-dimensional deformation are
defined in nearly the same terms as the components of the non-uniform part of a two-
dimensional deformation. Again, we use the thin-plate spline model to describe the defor-
mation at any point in space as f
X
, f
Y
and f
Z
, which describe the X-, Y- and Z-components
of the deformation:
2
X
K
f
X
ð
X
;
Y
;
Z
Þ 5
A
X1
1
A
XX
X
1
A
XY
Y
1
A
XZ
Z
1
W
Xi
U
ð
X
2
X
i
;
Y
2
Y
i
;
Z
2
Z
i
Þ
i
5
1
X
K
f
Y
ð
X
;
Y
;
Z
Þ 5
A
Y1
1
A
YX
X
1
A
YY
Y
1
A
YZ
Z
1
W
Yi
U
ð
X
2
X
i
;
Y
2
Y
i
;
Z
2
Z
i
Þ
(5.14)
i
5
1
X
K
f
Z
ð
X
;
Y
;
Z
Þ 5
A
Z1
1
A
ZX
X
1
A
ZY
Y
1
A
ZZ
Z
1
W
Zi
U
ð
X
2
X
i
;
Y
2
Y
i
;
Z
2
Z
i
Þ
i
5
1
where U(X
2
X
i
, Y
2
Y
i
, Z
2
Z
i
) is a function of the interlandmark distances given by:
q
ð
X
2
2
2
5
2
X
i
Þ
1 ð
Y
2
Y
i
Þ
1 ð
Z
2
Z
i
Þ
R
i
(5.15)
Again, we have more columns to accommodate the third dimension. The more substan-
tive difference is that U
R
2
ln
R
2
. As in the two-dimensional case, the next steps are to solve for the spline coefficients
(the values of A and W) and the eigenvectors of the bending energy matrix (the partial
warps).
In both the two-dimensional and three-dimensional cases, the thin-plate spline is
only used to solve for the non-uniform components of the deformation; a different
approach is taken to solve for the uniform components.
Bookstein (1996)
shows that
the approach he developed to construct a pair of basis vectors for the uniform part of
a two-dimensional deformation can be extended to the three-dimensional case. This
5 j
R
j
in contrast to the two-dimensional case in which U
5
