Biology Reference
In-Depth Information
consequence of applying these constraints to the data. However, unlike the notebook
example, we still have
2K
variable coordinates in our data matrix; none of them have been
removed or constrained. We have not lost degrees of freedom by removing coordinates,
because the loss of degrees of freedom is shared by
all
coordinates
each coordinate has
lost some fraction of a degree of freedom because each is partially constrained by the
operations of centering, scaling and rotation. Consequently, we have too many variable
coordinates for the degrees of freedom. The primary advantage of the thin-plate spline
methods (discussed in Chapter 5) is that we can work with
2K
4 variables, so that the
number of variables and the number of degrees of freedom are the same. The situation is
even worse in the case of three-dimensional data, when we have 3
K
variable coordinates
but only 3
K
2
7 degrees of freedom.
2
SUMMARY
Because there are several different morphometric spaces and distances, some with only
slightly different names, we summarize them below.
The
configuration space
is the set of all matrices representing landmark configurations
that have the same number of landmarks and coordinates. This space has
K
M
dimen-
3
sions, where
K
is the number of landmarks and
M
is the number of coordinates.
The
pre-shape space
is the set of all
K
M
configurations with a centroid size of one, cen-
tered at the origin. This space is the surface of a hypersphere of radius one. Because of the
centering, configurations that differ only in position are represented as the same point in
pre-shape space. Similarly, because of the scaling, configurations that differ only in cen-
troid size are represented by the same point in pre-shape space. Consequently, this space
has
KM
3
1) dimensions;
M
dimensions are lost due to centering, and one dimension
is lost due to scaling. In pre-shape space, the set of all configurations that may be con-
verted into one another by rotation lies along a circular arc called a
fiber,
which lies on the
surface of the pre-shape hypersphere. The distance between shapes in pre-shape space is
the length of the shortest arc across the surface connecting the fibers representing those
shapes, and is called the
Procrustes distance
. Because the radius of the pre-shape hyper-
sphere is one, the length of the arc is also the value (in radians) of the angle subtended (
2
(M
1
).
To construct a
shape space
, we select one point on each fiber, removing differences in
rotation. The number of axes on which a configuration can be rotated is a function of the
number of landmark coordinates:
M
(
M
ρ
1)/2. This also specifies the number of dimen-
sions that are lost in the transition from pre-shape space to shape space (1 if
M
2
2, 3 if
5
M
3). The construction of a shape space begins with the selection of one shape in a con-
venient orientation to serve as the
reference
configuration. Every other shape (called a
target
configuration) is placed in the orientation that corresponds to the location on its fiber that
is closest to the reference. This orientation is the position that minimizes the square root of
the sum of the squared differences between the coordinates of corresponding landmarks.
When minimized simply by rotation, this quantity is called the
partial Procrustes distance
.
Configurations that satisfy this condition are said to be in
partial Procrustes superimposition
on the reference. The partial Procrustes distance is the length of the chord of the arc
between the fibers in pre-shape space.
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