Biomedical Engineering Reference
In-Depth Information
function for the multimodality I-SICLE algorithm is given by
i
=
1
σ
i
I
2
C
=
|
T
i
(
h
(
x
))
−
S
i
(
x
)
|
2
dx
+|
S
i
(
g
(
x
))
−
T
i
(
x
)
|
2
2
dx
||
Lu
(
x
)
||
+||
L
w
(
x
)
||
(6.10)
+
ρ
2
2
dx
+
χ
||
u
(
x
)
−
w
(
x
)
||
+||
w
(
x
)
−
u
(
x
)
||
where
σ
i
are relative weighting factors for each of imaging modalities and
ρ
and
χ
define the relative importance of the bending energy minimization and the
inverse consistency terms. The constants
σ
i
, define the relative importance of
each modality with respect to the regularization terms of the cost function. The
I-SICLE algorithm has been applied to registration of brain images [35], skull
images [26] lung images [36, 37], and inner ear images [38].
6.4.2
Inverse Consistent Landmark-Based Thin-Plate
Spline (CL-TPS) Image Registration
Before describing the inverse consistent landmark-based, thin-plate spline
(CL-TPS) image registration algorithm, we discuss the traditional unidirec-
tion thin-plate spline registration algorithm. The unidirectional landmark-based,
thin-plate spline (UL-TPS) image registration algorithm [1, 2, 39] registers a tem-
plate image
T
with a target image
S
by matching corresponding landmarks iden-
tified in both images. Registration at non-landmark points is accomplished by
interpolation such that the overall transformation smoothly maps the template
into the shape of the target image.
The unidirectional landmark image registration problem can be thought of
as a Dirichlet problem [40] and can be stated mathematically as finding the
displacement field
u
that minimizes the cost function
2
dx
C
=
||
Lu
(
x
)
||
(6.11)
subject to the constraints that
u
(
p
i
)
=
q
i
−
p
i
for
i
=
1
,...,
M
where
p
i
and
q
i
are corresponding landmarks in the target and template images, respectively.
The operator
L
denotes a symmetric linear differential operator [41] and is used
to interpolate the displacement field
u
between the corresponding landmarks.
