Biomedical Engineering Reference
In-Depth Information
The variations are calculated by taking the Gateaux derivative [25] of the func-
tional
U
evaluated at
ϕ
+
ε
η
with respect to
ε
and then letting
ε
→
0. For general
forms of
W
and
U
,
∂
W
∂
d
J
+
∂
U
∂
d
J
=
0
.
G
(
ϕ,
η
)
=
(12.6)
ϕ
·
η
ϕ
·
η
β
β
12.2.3
Linearization
Equation (12.6) is highly nonlinear and thus an incremental-interative solution
method is necessary to obtain the configuration
ϕ
that satisfies the equation
[27]. The most common approach is based on linearization of the equations
and an iterative solution using Newton's method or some variant. Assuming
that the solution at a configuration
ϕ
∗
is known, a solution is sought at some
small increment
ϕ
∗
+
u
. Here again,
u
is a variation in the configuration
or a virtual displacement. The linearization of Eq. (12.6) at
ϕ
∗
in the direction
u
is:
∂
W
∂
d
J
ϕ
+
∂
U
L
ϕ
∗
G
=
G
(
ϕ
∗
,
η
)
+
DG
(
ϕ
∗
,
η
)
·
u
=
η
·
∂
ϕ
β
k
)
·
u
d
J
,
η
·
(
D
(12.7)
+
+
β
2
U
∂
∂
2
W
∂
∂
where
k
:
=
∂
is the
image stiffness
and
D
:
=
∂
is the
regularization stiff-
ness
. These 2nd derivative terms (Hessians) describe how small perturbations
of the current configuration affect the contributions of
W
and
U
to the overall
energy of the system.
12.2.4
Particular Forms for
W
and
U
—Hyperelastic
Warping
In Hyperelastic Warping, a physical representation of the template image is
deformed into alignment with the target image, which remains fixed in the refer-
ence configuration. The scalar intensity field of the template,
T
, is not changed
directly by the deformation, and thus it is represented as
T
(
X
). Since the values
of
S
at material points associated with the deforming template change as the
template deforms with respect to the target, it is written as
S
(
ϕ
). The formulation
