Biology Reference
In-Depth Information
rotate the second, around its centroid, through angle
to minimize that sum of squared
differences (see
Figure 3.9D
). After the rotation, the
X
-and
Y
-coordinates of each landmark
will have the coordinates (
X
cos
θ
θ
2
Y
sin
θ
), (
X
sin
θ
1
Y
cos
θ
). Thus, the rotated form the
second triangle will be:
2
3
ð
2
0
362 cos
θÞ
2
ð
2
0
488 sin
θÞ
2
0
362 sin
θÞ
1
ð
2
0
488 cos
θÞ
:
:
:
:
4
5
ð
0
516 cos
θÞ
2
ð
2
0
089 sin
θÞ
ð
0
516 sin
θÞ
1
ð
2
0
089 cos
θÞ
(3.9)
:
:
:
:
ð
2
0
154 cos
θÞ
2
ð
0
577 sin
θÞ
ð
2
0
154 sin
θÞ
1
ð
0
577 cos
θÞ
:
:
:
:
19
.
2
The value of
θ
that gives us the minimum sum of squared deviations is
so,
2
inserting that into
Equation 3.9
, gives us the coordinates for the second triangle:
2
4
3
5
2
0
:
502
2
0
:
341
0
458
0
254
(3.10)
:
2
:
0
044
0
596
:
:
We now have the superimposed triangles (see
Figure 3.9E
).
PROC
RUSTES SUPERIMPOSITION IN THREE-DIMEN
SIONS
Differences in location, scale and orientation of three-dimensional configurations are
removed by exactly the same operations; the only substantive difference is that we work with
larger matrices, making the computations more tedious (especially for the programmer).
We will not go through them all again because the only difference is the number of columns
to be averaged to compute the centroid, the number of columns from which the centroid coor-
dinates are subtracted, and the number of coordinates that are divided by centroid size.
The one step that is complicated by three-dimensional data is rotation, just as this was the one
complication encountered when we extended the formula for two-dimensional Bookstein
shape coordinates to three-dimensions. We now have to solve for the particular combination
of angles that minimizes that distance. Still, the solution remains conceptually simple and
it is obtained by a singular value decomposition (SVD) of the matrix
X
t
R
X
T
in which
X
R
and
X
T
are the centered and scaled configuration matrices of the reference and target, respectively
(Rohlf, 1990). As Rohlf points out, this is just one example of the general utility of SVD for
finding the angular relationship between two matrices.
SEMILANDMARK SLIDING
As discussed earlier, there may be studies in which we want to incorporate information
about outlines or curves as well as landmarks into the analysis. This is done using
semilandmark techniques, in which points are placed along the curve or outline. A curve
is simply an infinite set of points so, in using semilandmarks, we are approximating this
infinite set with a finite number of points placed along the curve using some algorithmic
approach to optimize this approximation.
