Biology Reference
In-Depth Information
THE S
PACES OF THREE-DIMENSIONAL CONFIGURAT
IONS
As discussed above, the set of all possible configurations of
K
landmarks with
M
coordi-
nates is called a configuration space, and this space has
K
M
dimensions. Centering, scal-
ing and rotating to a specific alignment all select subspaces with fewer dimensions.
Because the same operations were used to select these subspaces, the same formulae can
be used to determine their dimensions. Centering removes
M
dimensions because the cen-
troid has
M
coordinates, so the space of centered coordinates has
KM
3
2
M
dimensions,
which is
3K
3. Scaling removes one dimension because we are still using
centroid size, which is a one-dimensional scalar. Consequently, the space of centered and
scaled configurations (pre-shapes) has
KM
2
3when
M
5
M
2
2
1 dimensions (
Equation 4.9
), which is
3K
1)/2 dimensions
(
Equation 4.10
), which are the number of orthogonal axes on which an
M
-dimensional
configuration can be rotated. When
M
4 when
M
3. Rotation to a standard orientation removes
M(M
2
5
2
3, there are three axes, and the space of aligned
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configurations (a shape space) has
3K
7
dimensions.
When we impose on two-dimensional configurations of landmarks
(K
2
2 matrices) the
requirements of centering at the origin and scaling to unit centroid size, we generate a
pre-shape space that has the form of the surface of a hypersphere with a radius of one, cen-
tered on the origin. When we impose the same requirements on three-dimensional configura-
tions, we again get a pre-shape space that is the surface of a hypersphere with a radius of one,
centered on the origin. Pre-shape spaces generated by these operations have the same general
shape (differing only in the number of dimensions), regardless of the values of
K
and
M.
The pre-shape spaces described above contain every possible rotation of every possible
M
-dimensional shape that can be formed of
K
landmarks. Each shape is represented by
the set of all possible rotations of that shape, and the distance between shapes is the mini-
mum distance between these sets. As mentioned above, the set of all possible rotations of
a shape is called a
fiber.
This name seems apt when
M
3
2; there is only one axis of rota-
tion, so we can visualize a one-dimensional string lying in the pre-shape space. When
M
5
3, calling the set of rotations a fiber may seem less appropriate because there are
now three orthogonal axes of rotation, which does not fit our mental image of a one-
dimensional string. However, the actual concept is still the same (the set of all possible
rotations), and it is just as useful. Because different fibers represent different shapes, they
do not intersect; and if they do not intersect, we can find the shortest distance between
them. That distance is the difference between centered and rescaled configurations that is
not due to the rotation of one relative to the other. Therefore, regardless of the values of
K
and
M
, the distance between two shapes in the same pre-shape space is the distance
between two points on the surface of a hypersphere. Now that we are again on (relatively)
familiar ground, we can see that we must solve for the rotation of the target that mini-
mizes the partial Procrustes distance (the chord length), which can then be converted to
the Procrustes distance (arc length) or the full Procrustes distance (the cosine of the angle
subtended by the arc). Having a third set of coordinates makes the computation more
tedious, but the procedure is the same.
The shape spaces we generate by the operations described above are hyperspheres tan-
gent to their respective pre-shape spaces at the location of the reference shape. If centroid
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