Biology Reference
In-Depth Information
translation and scaling to remove the effects of (differences in) location and scale. We have
not yet rotated the shapes to remove the effects of rotation (that comes later, as we move
from pre-shape space to shape space). Thus, configurations of landmarks that differ only
by a rotation are located at different points in pre-shape space, as are configurations that
differ only in shape. This underscores an important point (which some may find counter-
intuitive): as we said earlier, configurations that differ only by a rotation (such as those
shown in
Figure 4.1B
) do not differ in shape. Because we have not yet removed all three
effects mentioned in Kendall's definition of shape (location, scale and rotation), we have
not yet reached shapes. At present we are concerned with pre-shapes, i.e. configurations
that may differ by a rotation, by a shape change or by some combination of the two. In
pre-shape space, configurations that differ only by rotation are different points, as are con-
figurations that differ only in shape.
Fibers in Pre-Shape Space
To visualize the locations in pre-shape space of configurations that differ only in rota-
tion, we introduce the term
fiber
. A fiber (in the context of our particular discussion of pre-
shape space) consists of the set of all the points in pre-shape space that can be obtained by
rotating a particular centered pre-shape. The fiber is a circular arc that comprises the set of
all points in pre-shape space that can be “reached” by rotating the pre-shape matrix.
Figure 4.5
depicts the concept of fibers as an arc on the surface of a sphere (ignoring the
higher dimensionality of a pre-shape hypersphere). Two fibers are shown: arcs 1 and 2.
Arc 1 is the set of all possible rotations of the pre-shape
Z
1
, and arc 2 is the set of all possi-
Z
2
. For a less abstract visualization of the concept of fibers,
we have drawn a cartoon (
Figure 4.6
) representing four fibers (in columns); the triangles
within a column differ solely by a rotation, whereas those in different columns also differ
in shape. (This visualization is somewhat limited, because a row does not accurately
ble rotations of the pre-shape
FIGURE 4.5
Fibers in pre-shape space. The points Z
1
and Z
2
are the locations of pre-shapes on the hypersphere (centered and
scaled matrices computed from two original matrices X
1
and X
2
,
which are not shown). Curve 1 passing through Z
1
is a fiber, the
set of all centered and scaled pre-shapes differing from Z
1
only
by rotation. Curve 2 is a fiber of pre-shapes differing from Z
2
only by rotation. (The dotted curve is the “equator” of the hyper-
sphere, and does not represent a fiber.)
2
1
Z
2
Z
1
