Biology Reference
In-Depth Information
λ
P
j
Y
j
Þ
λ
P
j
Y
j
;
We can now expand the term 1
=ð
1
1
2
as 1
2
2
keeping only first order
λ
terms in
for this power series expansion. This gives us:
0
1
0
1
X
X
X
ZZ
0
Z
0
Z
0
D
@
A
1
@
A
D
Pz
0
5
Y
j
Y
j
Y
j
1
1λ
2
2
λ
1
2λ
(5A.14)
j
j
j
to first order in
.
Now we can calculate the landmark coordinates after the operation of the compres-
sion/dilation (S
2
(
λ
λ
)) and Procrustes superimposition (which is just a multiplication by P
0
,
Z
Z
0
is already centered):
since
0
@
1
A
3
8
<
0
@
0
@
1
A
1
A
;
0
@
0
@
1
A
1
A
9
=
K
X
X
X
Pz
0
Z
0
5
Z
0
5
Y
j
Y
j
Y
j
1
2λ
Z
j
5
X
j
1
2λ
ð
Y
j
1λ
Y
j
Þ
1
2λ
:
;
j
j
j
1
(5A.15)
j
5
P
z
0
Z
0
is then:
The vector describing the displacement from
Z
to
8
<
9
=
0
@
0
@
0
@
1
A
2
1
A
;
0
@
0
@
1
A
2
1
A
1
A
K
X
X
Pz
0
Z
0
Y
j
Y
j
2
Z
5
X
j
1
2λ
X
j
ð
Y
j
1λ
Y
j
Þ
1
2λ
Y
j
(5A.16)
:
;
j
j
j
5
1
8
<
9
=
0
@
0
@
1
A
;
0
@
1
A
1
A
K
X
Y
j
X
j
2
Y
j
X
j
Y
j
Y
j
2λ
Y
j
52
X
j
λ
λ
Y
j
2λ
(5A.17)
:
;
j
j
5
1
2
Noting that
λ
D
0
;
we can simplify this to:
8
<
:
0
0
1
0
1
1
9
=
;
K
X
Y
j
X
j
@
@
A
; λ
@
A
A
Y
j
Y
j
52
X
j
λ
Y
j
2λ
(5A.18)
j
5
j
1
8
<
:
0
0
1
0
1
1
9
=
;
K
X
X
@
@
A
;
@
A
A
Y
j
Y
j
5λ
X
j
2
Y
j
1
2
(5A.19)
j
j
5
j
1
γ 5Σ
j
Y
j
and
2Σ
j
Y
j
5Σ
j
X
j
, so that
We now define
α5
1
γ 1α5
1. After making these
substitutions and dividing through by
λ
, we have:
Pz
0
2
Z
0
Þ
V
2
5
ð
K
j
5 fð2γ
X
j
; α
Y
j
Þg
(5A.20)
5
1
λ
which is the vector of the displacements at each landmark point (X
j
, Y
j
) produced by the
mapping
S
2
per unit of
λ
. All we need to do now is to normalize this set so that the length
of the vector is one.
