Biomedical Engineering Reference
In-Depth Information
The thin-plate spline method uses the physical model of a thin steel plate [21]
with small vertical displacement given by the scalar field
g
and calculates
J
as
the
strain energy
of the plate:
∂
2
+
2
∂
2
∂
2
∇
2
g
∂
x
2
2
g
∂
x
∂
y
2
g
∂
y
2
2
g
2
d
x
d
y
J
(
g
)
=
+
d
x
d
y
=
(9.1)
where
∇
2
denotes the Laplacian and the right equality is obtained by inte-
gration by parts under some conditions on the solution space. The Lapla-
cian energy (9.1) is a member of a more general family of scale and rota-
tion invariant cost functions which satisfy the requirements of section 9.3.1,
see also [85, 86]. It is also the simplest criterion that does not penalize affine
transforms.
The criterion for the vector form
g
is taken simply as the sum of the strain
energies of the
x
and
y
components,
J
(
g
)
=
J
(
g
x
)
+
J
(
g
y
). As the constraints
g
(
x
i
)
=
z
i
can be broken into two independent sets for
g
x
and
g
y
, it follows
that minimizing
J
for
g
is equivalent to minimizing separately for
g
x
and
g
y
.
Consequently, we can concentrate on the scalar case here.
9.3.2.1
Interpolation Formula
The correspondence function
g
(
x
,
y
) minimizing (9.1) under interpolation con-
straints
g
(
x
i
,
y
i
)
=
z
i
is given by
N
g
(
x
,
y
)
=
1
λ
i
(
x
−
x
i
)
+
a
0
x
+
a
1
y
+
a
2
i
=
(
x
−
x
i
)
2
with
x
−
x
i
=
+
(
y
−
y
i
)
2
=
r
(9.2)
where
(
r
)isa
(
r
)
=
r
2
log
r
. It is called radial because it only depends on the
Euclidean distance
r
to its associated data point [87].
The generating function
(
x
)
=
(
x
,
y
) solves the associated Euler-Lagrange
(or fundamental) equation
(
x
2
x
,
y
+
y
2
)
=
δ
(
x
,
y
)
(9.3)
∇
where
∇
4
is a two times iterated Laplacian and
δ
(
x
,
y
) is the Dirac distribution.
The linear polynomial
a
0
x
+
a
1
y
+
a
2
in (9.2) is called a
kernel term
and it ap-
pears because it does not contribute to the criterion. The unknown parameters
λ
i
and
a
0
,
a
1
,
a
2
are determined from the interpolation constraints
g
(
x
i
,
y
i
)
=
z
i
