Biology Reference
In-Depth Information
FIGURE 4.15
Optimal alignment of
W
to
X
will be
achieved by rotating
d
3
W
around its centroid through an
unknown angle
to minimize the square root of the sum
of the squares of distances
d
1
,
d
2
, and
d
3
.
θ
X
θ
d
2
d
1
W
what is being minimized. The criterion that leads to the shape space discussed earlier is
minimization of the square root of the sum of the squared distances between the corre-
sponding landmarks (the distances
d
1
,
d
2
,and
d
3
shown in
Figure 4.15
). This quantity can
be computed directly from the squared differences between the corresponding coordinates
of the landmarks:
q
ð
2
2
2
2
D
X
11
2
X
21
Þ
1
ð
Y
11
2
Y
21
Þ
1?1
ð
X
13
2
X
23
Þ
1
ð
Y
13
2
Y
23
Þ
(4.25)
5
(There are other criteria that lead to other superimpositions of the two triangles; one is dis-
cussed below, others in Chapter 5.)
With this criterion in hand, we can solve for the unique value of
θ
at which
D
is mini-
19
.
2
. When we insert this value into the matrix
mized. In our example, that value is
θ
52
for
W
pre-shape
, rotated (
Equation 4.24
), we get:
2
3
0
502
0
341
2
:
2
:
4
5
W
pre
-
shape
;
rotated
5
0
458
0
254
(4.26)
:
2
:
0
044
0
596
:
:
Under the conditions set out above, this is the optimal alignment to the reference form:
2
4
3
5
0
463
0
309
2
:
2
:
X
5
0
:
463
2
0
:
309
(4.27)
pre
-
shape
0
:
000
0
:
617
Figure 4.16
shows the two triangles under these conditions.
