Biomedical Engineering Reference
In-Depth Information
linear-elastic constraint has the form
2
dx
+
2
dx
C
REG
(
u
)
+
C
REG
(
w
)
=
||
Lu
(
x
)
||
||
L
w
(
x
)
||
(6.3)
and can be used to regularize the transformations. The linear elasticity op-
erator
L
has the form
Lu
(
x
)
=−
α
∇
2
u
(
x
)
−
β
∇
(
∇·
u
(
x
))
+
γ
u
(
x
) where
∇=
∂
. In general,
L
can be any non-
singular linear differential operator [30]. The limitation of using linear differen-
tial operators is that they can't prevent the transformation from folding onto
itself, i.e., destroying the topology of the images under transformation [31]. This
includes the linear elasticity and thin-plate spline models. The linear elasticity
operator is used in this work to help prevent the Jacobian of the transforma-
tion from going negative. At each iteration the Jacobian of the transformation
is checked to make sure that it is positive for all points in
d
which implies that
the transformation preserves topology when transforming images.
The purpose of the regularization constraint is to ensure that the transforma-
tions maintain the topology of the images
T
and
S
. Thus, the elasticity constraint
can be replaced by or combined with other regularization constraints that main-
tain desirable properties of the template (source) and target when deformed.
An example would be a constraint that prevented the Jacobian of both the for-
ward and reverse transformations from going to zero or infinity. A constraint
that penalizes small and large Jacobian values is given by
C
Jac
(
h
)
+
C
Jac
(
g
)
=
(
J
(
h
(
x
)))
2
and
∇
∂
2
∂
x
1
+
∂
2
∂
x
2
+
∂
2
∂
x
1
,
∂
∂
x
2
,
∂
2
=∇·∇=
∂
x
3
∂
x
3
1
J
(
h
(
x
))
2
1
J
(
g
(
x
))
2
+
(
J
(
g
(
x
)))
2
dx
where
J
denotes
the
+
+
Jacobian
operator.
Further
examples
of
regularization
constraints
that
penalize large and small Jacobians can be found in Ashburner
et al
. [21].
6.2.6
Transformation Parameterization
Until now, the forward and reverse transformations have been described as
general functions. In order to estimate the transformations, they must be pa-
rameterized. Examples of transformation parameterizations that are used in
practice include the 3D Fourier series [26], polinomials [32], b-splines [19, 24],
wavelets [17], and vector displacements [14, 15]. We will concentrate on a 3D
Fourier series parameterization in this chapter. In a 3D Fourier series param-
terization, each basis coefficient is interpreted as the weight of a harmonic
component in a single coordinate direction. The discretized displacement fields
