Biology Reference
In-Depth Information
The Shape of Pre-Shape Space
The two requirements imposed on this space mean that the summed squared land-
mark positions add up to one. The consequences of that property can be understood by
considering the set of points satisfying the restriction in an ordinary two-dimensional
space: the set of points is centered on the origin (0,0), and each point in the set has
coordinates satisfying the equation
X
2
Y
2
1. The set of points is a circle of radius
one, centered on the origin. This circle is a one-dimensional subspace (a curve) inhabit-
ing a two-dimensional space (a plane). Knowing that all points are equidistant from the
center means that we need specify only the direction of a point from the center to
define it uniquely; thus, the location of any point on the circle can be described suffi-
ciently by a single dimension (direction). Extending this to a three-dimensional space,
we now have the set of all points (
X
,
Y
,
Z
) centered on the origin (0,0,0) such that
X
2
1
5
Y
2
Z
2
1. This is the surface of a sphere of radius one, centered on the origin,
and it is a two-dimensional subspace within a three-dimensional space. Again, the con-
straint that all points are on the surface allows us to describe the location of a point by
giving a direction from the center; the only difference from the circle is that we now
need two components to describe that direction (e.g. latitude and longitude). So, in talk-
ing about a pre-shape space, we are talking about the surface of a hypersphere centered
on the origin, which is the generalization of an ordinary sphere in
K
1
1
5
M
dimensions.
3
In that general case, we have:
X
X
K
M
2
1
ðX
Þ
5
1
(4.8)
ij
i
1
j
5
5
which states that the sum of all squared landmark coordinates is one. That hypersphere is
simply the equivalent of a sphere in more than three dimensions.
We can determine the number of dimensions in pre-shape space by considering the
number of dimensions that were lost in the transition from configuration space. One
dimension is lost in fixing centroid size to one, eliminating the size dimension of the con-
figuration space. Another
M
dimensions are lost in centering the configurations; eliminat-
ing the
M
dimensions needed to describe location (the coordinates of the centroid). Thus,
in moving from configuration space to pre-shape space, we moved to a space that has
M
1 fewer dimensions, which is:
1
KM
2
ð
M
1
Þ
5
KM
M
1
(4.9)
1
2
2
3 dimen-
sions; so the pre-shape space for triangles has three dimensions. For three-dimensional
configurations of landmarks, pre-shape spaces have
3K
For two-dimensional configurations of landmarks, pre-shape spaces have
2K
2
4 dimensions.
Returning to the three-dimensional sphere (because most of us have trouble imagining
spaces having more than three dimensions), you should be imagining pre-shape space to
be a hollow ball of radius one, centered at the origin (0, 0, 0). Arrayed on the two-dimen-
sional surface of this ball are points representing individual configurations of landmarks.
The two restrictions we have imposed on our configuration matrices mean that the config-
urations in this set do not differ in scale or location; we have used the operations of
2
