Biology Reference
In-Depth Information
analysis assume a Euclidean space. As mentioned earlier in this chapter, the mathematics
of statistical inference in Kendall's shape space has been developed by Kendall and others.
However, in this section we discuss the replacement of Kendall's shape space with a
Euclidean approximation.
The problem of replacing a curved space with a Euclidean approximation is illustrated
for the special case of triangles in
Figure 4.18
. As before (see
Figure 4.11
), the outer hemi-
sphere is the space constructed by aligning pre-shapes (with centroid size fixed at one) to
minimize the partial Procrustes distance (the square root of the summed squared distances
between corresponding landmarks). The inner sphere is Kendall's shape space, con-
structed by scaling the aligned target shapes to centroid size
). These two spaces
share a common point, the reference shape, because the distance of the reference from
itself is zero, so cos(
cos(
ρ
5
) is one. Tangent to both of these spaces, at the reference shape, is a
Euclidean plane. We also need to decide how we will construct the projection of shapes
onto the tangent plane, which includes deciding (1) which space will be the source of the
configurations projected onto the tangent plane, and (2) what rule we will use to deter-
mine the direction of the projection. (We also need to decide how to choose an appropriate
reference configuration to serve as the tangent point, which is discussed in the next
section.)
Figure 4.18
illustrates two common approaches to projecting from one space onto
another. One approach is to project to the new space from the centroid of some reference
space. In this case, the reference space is the hemisphere of aligned pre-shapes, so the pro-
jections are along the radii of this hemisphere to the tangent space. In this stereographic
projection, the shape represented by points
ρ
B
and
A
(at centroid sizes cos(
ρ
) and one,
C
respectively) map to the same location (
) in the tangent space. The distance in the plane
C
is
greater
than the arc length from the reference to
B
from the reference to
(the
E
D
C
Tangent plane
A
B
p
(
1, 0)
(0, 0)
(1, 0)
FIGURE 4.18
Tangent space to shape spaces of triangles and projections onto the tangent space (
Rohlf, 1999
).
As in
Figure 4.11
, the outer hemisphere is a section through the space of centered and aligned shapes scaled to
unit centroid size, and the inner circle is a section through Kendall's shape space of centered and aligned shapes
scaled to cos(
). The plane is tangent to the sphere and the hemisphere at the point of the reference shape. The
configuration at point
ρ
B
A
represents a triangle in Kendall's shape space;
is the same shape scaled to unit centroid
size.
C
is a stereographic projection of
B
onto the tangent plane.
D
is the orthogonal projection of
A
onto the tan-
gent plane, and
E
is the orthogonal projection of
B
onto the tangent plane.
