Biology Reference
In-Depth Information
MORPHOMETRIC SPACES
Given the definition of shape, we can now develop the mathematical idea of morpho-
metric spaces. We begin by defining some additional terms.
The Configuration Matrix
A configuration matrix represents an entire configuration of landmarks. It is a
K
M
matrix of Cartesian coordinates that describes a particular set of
K
landmarks in
M
dimen-
sions (
Dryden and Mardia, 1998
). When we talk about a
K
3
M
matrix, we mean that the
matrix has
K
rows and
M
columns; each of the
K
rows represents a specific landmark on a
specimen, with
M
Cartesian coordinates. For example, the simplest shape we might want
to study is a triangle with landmarks located at the three vertices of the triangle. Calling
the coordinates of the first vertex
X
1
and
Y
1
, and those of the second vertex
X
2
and
Y
2
,
and those of the third vertex
X
3
and
Y
3
, the configuration matrix of triangle
3
X
is:
2
3
X
1
Y
1
X
2
Y
2
X
3
Y
3
4
5
X
5
(4.1)
It is often useful to represent this same landmark configuration as a
row vector
, in which
the landmark coordinates are listed along a single row in
K
M
columns:
3
X
5
½
X
1
Y
1
X
2
Y
2
X
3
Y
3
(4.2)
This contains exactly the same information, represented slightly differently. Given a set
of landmark coordinates in row vector form, you can easily convert it to a configuration
matrix (the representation you might prefer at any given time depends on the particular
task or software at hand).
For example, the configuration matrix of the triangle shown in
Figure 4.2
is:
2
4
3
5
2
1
2
1
X
5
1
1
01
(4.3)
2
The row vector representing the same triangle would be:
X
5
½
2
1
11
101
(4.4)
2
2
Configuration Space
The configuration space is a set of all possible
K
M
matrices describing all possible
sets of landmark configurations for that given
K
and
M
. For example, a 16
3
2 dimensional
configuration space is the space of all configurations having 16 two-dimensional land-
marks. That space encompasses
all
possible configurations for those 16 landmarks with
two coordinates. Should we record the locations of 16 landmarks on a two-dimensional
3
