Biology Reference
In-Depth Information
FIGURE 4.13
Centered triangles computed from
X
and
W
.
Computation of the centroids of
3
3
is given by
Equation
4.15
; computation of the landmark coordinates after centering is
given by
Equations 4.16
X
and
W
4.18
. Vertices are numbered to indicate
their homology.
X
2
1
2
W
1
The centered triangles are not in pre-shape space. To put them there, we need to rescale
each so that its centroid size is one. The formula for centroid size is:
t
X
X
K
M
2
CS
ðXÞ
5
1
ð
X
ij
C
j
Þ
(4.19)
2
i
j
5
1
5
which is the square root of the sum of the squared distances of the landmarks from the
centroid. Given that the centroids of
X
centered
and
W
centered
are both at (0, 0), we can simply
sum the squared coordinates:
q
ð
2
2
2
2
2
2
2
CS
ðX
centered
Þ
5
Þ
1
ð
2
Þ
1
ð
Þ
1
ð
2
Þ
1
ð
Þ
1
ð
Þ
1
:
0
0
:
667
1
:
0
0
:
667
0
1
:
333
(4.20)
5
2
:
160
q
ð
2
2
2
2
2
2
2
CS
ðW
centered
Þ
5
0
837
Þ
1
ð
1
127
Þ
1
ð
1
193
Þ
1
ð
2
0
207
Þ
1
ð
0
357
Þ
1
ð
1
333
Þ
:
:
:
:
:
:
(4.21)
2
311
5
:
Dividing each coordinate of the centered triangle by its centroid size produces the pre-
shape matrices:
2
3
2
3
1
0
667
0
463
0
309
2
2
:
2
:
2
:
1
4
5
5
4
5
X
pre
-
shape
5
1
667
01
0
0
463
0
309
(4.22)
2
:
:
2
:
2
160
:
333
0
000
0
617
:
:
:
2
3
2
3
0
837
1
127
0
362
0
488
2
:
2
:
2
:
2
:
1
4
5
5
4
5
W
pre
-
shape
5
1
193
0
207
0
516
0
089
(4.23)
:
2
:
:
2
:
2
311
:
0
357
1
333
0
154
0
577
2
:
:
2
:
: