Biology Reference
In-Depth Information
modeling as deformed, which we will call the reference form (the other is the target). This
terminology should be familiar
the reference form is the same one that we discussed in
Chapter 3. If you do not wish to read further, you do not need to. You can return directly
to the section on decomposing the non-uniform (non-affine) component.
Although we now have a general idea of the procedure, there are still a few ideas that
need to be added. The first is the idea of complex number notation for landmark locations,
which is often used in mathematical derivations (see
Dryden and Mardia, 1998
, for exam-
ple). Consider a landmark configuration consisting of K landmarks in two dimensions,
which we will call
Z
, the reference form. Mathematically, we will say:
K
j
Z
5 f
Z
j
;
Z
j
5ð
X
j
;
Y
j
Þg
(5A.1)
5
1
which means that
is a set of K pairs of landmark positions Z
j
, or (X
j
,Y
j
). It is a useful
mathematical shortcut to think of Z
j
as being a complex number Z
j
5
Z
iY
j
, where i is
the square root of minus one. Complex number notation is often used in texts on the statis-
tics of shape, so understanding this approach is useful.
The next idea is to require that the reference form be rotated to a principal axis align-
ment, so that
X
j
1
0, which will later simplify the mathematics (but may pose pro-
blems for aligning specimens in some software, discussed below). The summation
Σ
j
X
j
Y
j
5
Σ
j
is
from j
K, and all the summations in the derivation are likewise over all K land-
marks. We are also going to assume that the reference has a centroid size of one, so that
5
1toj
5
Σ
j
ð
X
j
1
Y
j
Þ 5
1
;
and a centroid position of (0, 0), so that
Σ
j
X
j
5
0 and
Σ
j
Y
j
5
0.
Mathematical Derivations
Let us consider the two functions of interest: shear, which we will call
S
1
(
λ
), and com-
pression/dilation, which we will call
S
2
(
λ
)(
λ
describes the magnitude of the mapping).
2
will
We will be taking the limit as
λ
-
0 at the end of this derivation, so terms including
λ
Z
Z
0
under these
be discarded. The mappings from a reference form
to a target form
operations are as follows:
K
j
Z
0
Z
0
5 f
Z
j
5ð
X
j
S
1
ðλÞ
: Z
;
1λ
Y
j
;
Y
j
Þg
(5A.2)
-
5
1
K
j
Z
0
;
Z
0
5 f
Z
j
5ð
S
2
ðλÞ
: Z
X
j
;
Y
j
1λ
Y
j
Þg
(5A.3)
-
5
1
You can probably convince yourself that
S
1
describes a shear; the X-coordinates of each
point are displaced a distance proportional to their Y-axis position relative to the centroid.
Similarly, you should be able to recognize that
describes an expansion of the landmarks
along the Y-axis. We do not need to worry about modeling the contraction along the X-
axis, even though it must also be occurring, because the Procrustes GLS superimposition
will take care of that by requiring that the centroid size be fixed at one.
If
S
2
Z
0
are both centered (i.e. have a centroid position of zero), then the Procrustes
superimposition may be approximated as the multiplication of
Z
and
Z
0
by the complex factor
P
z
0
, where:
ZZ
0
Z
0
Z
0
P
Z
0
5
(5A.4)
