Biology Reference
In-Depth Information
forensics community, size is viewed as an important part of the evidence. If the size of
two impressions are not consistent with one another, that serves as grounds to conclude
that the impressions arose from different sources (leaving aside the possibility that the
recording medium has shrunk or expanded). Thus, in forensics settings, there can be an
advantage to working with size and shape simultaneously.
Dryden and Mardia (1992,
1998)
developed distribution models using Procrustes shape variables and centroid size,
treated jointly in distribution functions, but as separate quantities. This approach (some-
times referred to as Procrustes Form Space) has been used in a PCA (
Mitteroecker et al.,
2004
), in which shape data and the log of centroid size were placed in a single data matrix
and analyzed.
An alternative approach to size and shape analysis is to superimpose landmark coordi-
nates of two or more configurations using only translation and rotation, but not using size
changes (scaling). This produces landmark coordinates with 2
k
6 degrees of
freedom, in superimpositions that could be called Procrustes Size Preserving (Procrustes-
SP,
Bush et al., 2011a; Sheets et al., 2013
). Suppose we have
k
landmarks in two dimensions
and want to superimpose the landmarks (
x
t
1
,
y
t
1
,
x
t
2
,
y
t
2
...
3or3
k
2
2
x
tk
,
y
tk
) of a target (
t
) specimen
on a reference (
r
) specimen (
x
r
1
,
y
r
1
,
x
r
2
,
y
r
2
...
x
rk
,
y
rk
) by translations along the
x
and
y
directions (
a
x
and
a
y
) with a rotation through some angle
. We can assume, without loss
of generality, that the reference specimen is centered on the origin (0,0) so that:
θ
X
k
x
ri
5
0
(14.1)
i
5
1
X
k
y
ri
5
0
(14.2)
i
1
5
but note that the reference is not scaled to a centroid size of one. Rather, it is left in the
original measurement units. It is assumed that the target and reference are both measured
in the same units. If we translate and rotate the target specimen, the landmark points are
mapped to:
x
ti
5
ð
x
ti
a
x
Þ
cos
ð
θ
Þ
2
ð
y
ti
2
a
y
Þ
sin
ð
θ
Þ
(14.3)
2
y
ti
5
ð
x
ti
a
x
Þ
sin
ð
θ
Þ
1
ð
y
ti
a
y
Þ
cos
ð
θ
Þ
(14.4)
2
2
We can then calculate the squared distance between the reference and the rotated and
translated target specimen:
X
k
D
2
2
2
1
ð
x
ri
2
ð
x
ti
2
a
x
Þ
cos
ð
θ
Þ
1
ð
y
ti
2
a
y
Þ
sin
ð
θ
ÞÞ
1
ð
y
ri
2
ð
x
ti
2
a
x
Þ
sin
ð
θ
Þ
2
ð
y
ti
2
a
y
Þ
cos
ð
θ
ÞÞ
(14.5)
5
i
5
Next we minimize this squared distance with respect to
a
x
,
a
y
and
. This yields:
θ
P
i
5
1
x
ti
k
a
x
5
(14.6)
