Biology Reference
In-Depth Information
FIGURE 4.17
The relationships among the Procrustes distance,
ρ
D
p
, full Procrustes distance
D
F
5
sin(
ρ
), and partial Procrustes distance
D
p
5
/2). The configuration at point B represents a triangle in
Kendall's shape space.
2 sin(
ρ
sin (
ρ
/2)
W
X
D
F
B
cos (
ρ
)
ρ
/2
that would further reduce the distance between the shapes
B
(see
Figure 4.17
). As indicated in
Figure 4.17
, that distance (
D
F
, the full Procrustes distance)
is measured along a line segment orthogonal to the radius of
X
and
W
; we are taking
W
to
W
pre-shape,rotated
, passing
through
is small (0.0894 radians); its cosine is near one (0.996)
so we need make only a very slight adjustment to convert the coordinates of
X
pre-shape
. In our example,
ρ
W
pre-shape,
rotated
to
W
shape
:
2
4
3
5
5
2
4
3
5
2
0
:
5021
2
0
:
3414
2
0
:
5001
2
0
:
3401
W
ð
Þ
5
cos
0
:
089
0
:
4583
2
0
:
2542
0
:
4564
2
0
:
2532
(4.29)
shape
0
:
439
0
:
5956
0
:
0437
0
:
5932
W
This is the triangle with the same shape as
, but it is now in Kendall's shape space
X
pre-shape
. Because the full Procrustes distance was used to
determine the coordinates of the landmarks in
with the reference at triangle
W
shape
, we can say that
W
shape
is in full
Procrustes superimposition on the reference form
X
pre-shape
.
TANGENT SPACES
The curvature of shape space makes statistical inference more difficult in this space
than it is in Euclidean spaces and most of the familiar methods of multivariate statistical
