Biology Reference
In-Depth Information
and the expression Z
0
refers to the complex conjugate of the complex vector
Z
0
representing
the landmark configuration after the compression/dilation. The Procrustes superimposition
of
P
z
0
Z
0
. To get the vectors that describe the uniform deformation, we just
subtract the starting position
Z
0
on
Z
is thus
Z
from
P
z
0
Z
0
and then divide through by the magnitude of the
Z
0
2
deformation
λ
, yielding (
P
Z)/
λ
as the set of vectors describing the deformation.
z
0
Further Derivation of the Uniform Components
To find
P
Z
0
for the
S
2
(
λ
) mapping (compression/dilation), we note that the numerator
of
P
z
0
is:
X
ZZ
0
5
ð
X
j
1
iY
j
Þ 3 ð
X
j
2
i
ð
Y
j
1λ
Y
j
ÞÞ
(5A.5)
j
which expands to:
X
X
j
2
2
2
5
ð
iX
j
Y
j
2
iX
j
λ
Y
j
1
iX
j
Y
j
2 ð
iY
j
Þ
2 ð
iY
j
Þ
λÞ
(5A.6)
j
Because i
2
1 and the products of X
j
Y
j
sum to zero (under the alignment specified
earlier), we can simplify this to:
52
X
ð
X
j
1
Y
j
1
Y
j
λÞ
5
(5A.7)
j
Now add the constraint that
P
j
ð
X
j
1
Y
j
Þ 5
1 because we scaled the reference to unit
centroid size, and we have:
X
Y
j
ZZ
0
5
1
1λ
(5A.8)
j
Now we simplify the denominator of
P
z
0
:
X
Z
0
Z
0
5
ð
X
j
1
iY
j
ð
1
1λÞÞ 3 ð
X
j
2
iY
j
ð
1
1λÞÞ
(5A.9)
j
X
X
2
ð
X
j
1
Y
j
ð
1
X
j
1
Y
j
ð
1
2
5
1λÞ
Þ 5
1
2
λ1λ
Þ
(5A.10)
j
j
X
X
j
1
Y
j
1
2Y
j
λ1
Y
j
λ
2
5
(5A.11)
j
As mentioned before,
P
j
ð
X
j
1
Y
j
Þ 5
2
1
;
and terms including
λ
can be discarded in the
limit of small
λ
, so that:
X
Z
0
Z
0
D
Y
j
1
1
2
λ
(5A.12)
j
This leaves us with:
1λ
P
j
Y
j
Þ
ð
1
ZZ
0
Z
0
Z
0
5
Pz
0
λ
P
j
Y
j
Þ
5
(5A.13)
ð
1
1
2
