Biology Reference
In-Depth Information
and the total bending energy of the entire plate is:
ð
ð
BE
ð
X
;
Y
Þ
dx dy
Þ
(5.5)
which is the bending energy at each point integrated over the entire surface. The choice of
U(R) minimizes this total bending energy.
For biological purposes, we do not really care about the bending energy of a steel plate.
Rather, we care about the connection between bending energy and the curvature of the
plate (and their connection to spatial scale). Minimizing bending energy minimizes the
curvature of the plate, so when we fit a linear combination of the U (R) function to our
data, we are fitting a function that minimizes the amount of curvature needed to model
the observed deformations.
Suppose we want a linear combination of U (R) values, centered on each of the K land-
marks of our reference form (because we are describing a deformation, we are talking
about changes relative to a reference). We need to describe deformations in the X and Y
directions, so we form the following linear combinations:
X
K
fx
ð
X
;
Y
Þ 5
A
X1
1
A
XX
X
1
A
XY
Y
1
W
Xi
U
ð
X
2
X
i
;
Y
2
Y
i
Þ
(5.6)
i
5
1
X
K
f
Y
ð
X
;
Y
Þ 5
A
Y1
1
A
YX
X
1
A
YY
Y
1
W
Yi
U
ð
X
2
X
i
;
Y
2
Y
i
Þ
(5.7)
i
5
1
where f
X
(X, Y) and f
Y
(X, Y) are the spline functions that describe the deformations along
the X- and Y-directions relative to the reference form, and W
Xi
and W
Yi
are weights of the
functions
U
2
2
Y
i
), centered on the landmark locations of the reference (X
i
, Y
i
).
The A terms describe uniform (or affine) deformations of the target, using what is known
as the six-component uniform model. We need to include those A terms at this stage, but
will discard them later in favor of
(X
X
i
, Y
the two uniform components discussed in the
Appendix (
Equations 5A.23 and 5A.38
).
Fitting the functions to the observed deformations is a standard problem in systems of
linear equations; we can thus cast the problem into matrix form. We form a (K
1
3)
3
2
matrix
of the observed deformations at each of the K landmarks, where the deformation
at the ith landmark is denoted (X
0
i
,Y
0
i
):
V
2
4
3
5
2
4
3
5
X
0
1
Y
0
1
X
0
2
Y
0
2
^^
X
0
K
Y
0
K
00
00
00
f
X
ð
X
1
;
Y
1
Þ
f
Y
ð
X
1
;
Y
1
Þ
f
X
ð
X
2
;
Y
2
Þ
f
Y
ð
X
2
;
Y
2
Þ
^
^
V
5
5
f
X
ð
X
K
;
Y
K
Þ
f
Y
ð
X
K
;
Y
K
Þ
5
LW
(5.8)
0
0
0
0
0
0
