Biomedical Engineering Reference
In-Depth Information
2
, the problem reduces to the thin-plate spline image registration
problem given by
When
L
=∇
∂
2
2
2
u
i
(
x
)
∂
2
u
(
x
)
||
2
dx
=
C
=
||∇
2
x
1
i
=
1
+
2
∂
∂
2
2
u
i
(
x
)
∂
x
1
∂
x
2
2
u
i
(
x
)
∂
dx
1
dx
2
(6.12)
+
2
x
2
subject to the constraints that
u
(
p
i
)
=
q
i
−
p
i
for
i
=
1
,...,
M
.
It is well known [1, 2, 39] that the thin-plate spline displacement field
u
(
x
)
that minimizes the bending energy defined by Eq. (6.12) has the form
i
=
1
ξ
i
φ
(
x
−
p
i
)
+
Ax
+
b
M
u
(
x
)
=
(6.13)
where
φ
(
r
)
=
r
2
log
r
and
ξ
i
are 2
×
1 weighting vectors. The 2
×
2 matrix
A
=
[
a
1
,
a
2
] and the 2
×
1 vector
b
define the affine transformation where
a
1
and
a
2
are 2
×
1 vectors.
The thin-plate spline interpolant
φ
(
r
)
=
r
2
log
r
is derived assuming infinite
boundary conditions, i.e.,
is assumed to be the whole plane
R
2
. The thin-plate
spline transformation is truncated at the image boundary when it is applied to
an image. This presents a mismatch in boundary conditions at the image edges
when comparing forward and reverse transformations between two images. It
also implies that a thin-plate spline transformation is not a one-to-one and onto
mapping between two image spaces. To overcome this problem and to match the
periodic boundary conditions assumed by the intensity-based consistent image
registration algorithm, approximate periodic boundary conditions are imposed
on the registration problem (see [34] for details).
The inverse consistent landmark-based, thin-plate spline (CL-TPS) image
registration algorithm is solved by minimizing the cost function given by
2
2
dx
C
=
ρ
||
L
u
(
x
)
||
+||
Lw
(
x
)
||
2
dx
subject to
p
i
+
u
(
p
i
)
=
q
i
and
q
i
+
w
(
q
i
)
=
p
i
for
i
=
1
,...,
M
(6.14)
2
+
χ
||
u
(
x
)
−
w
(
x
)
||
+||
w
(
x
)
−
u
(
x
)
||
The first integral of the cost function defines the bending energy of the thin-
plate spline for the displacement fields
u
and
w
associated with the forward
