Biomedical Engineering Reference
In-Depth Information
uses a Gaussian sensor model to describe the image energy density functional:
ϕ
)
=
2
(
T
(
X
)
−
S
(
ϕ
))
2
U
(
X
,
.
(12.8)
λ
is a penalty parameter [28] that enforces the alignment of the template model
with the target image data. As
λ
→∞
,
(
T
(
X
)
−
S
(
ϕ
))
2
→
0, and the image en-
ergy converges to a finite value.
Hyperelastic Warping assumes that
W
is the standard strain energy density
function from continuum mechanics that defines the material constitutive be-
havior. It depends on the right deformation tensor
C
. The right deformation
tensor is independent of rotation and thus hyperelasticity provides an objec-
tive (invariant under rotation) constitutive framework, in contrast to linearized
elasticity (see below, [29]). With these specific assumptions, Eq. (12.4) takes the
form:
W
(
X
,
C
)
d
J
−
ϕ
))
d
J
E
=
U
(
T
(
X
)
,
S
(
(12.9)
β
β
The first variation of the first term in Eq. (12.9) yields the standard weak from
of the momentum equations for nonlinear solid mechanics (see, e.g., [25]). The
first variation of the functional
U
in Eq. (12.8) with respect to the deformation
ϕ
(
X
) in direction
η
gives rise to the image-based force term:
ϕ
)
·
η
=
D
2
(
T
(
X
)
−
S
(
ϕ
))
2
DU
(
·
η
.
(12.10)
(
T
(
X
)
−
S
(
ϕ
+
ε
η
))
ϕ
+
ε
η
))
∂
∂ε
=
λ
(
T
(
X
)
−
S
(
ε
→
0
Noting that
∂
∂ε
ϕ
+
ε
η
))
−
∂
S
(
ϕ
+
ε
η
)
·
∂
(
ϕ
+
ε
η
)
ϕ
+
ε
η
)
∂ε
(
T
(
X
)
−
S
(
ε
→
0
=
∂
(
ε
→
0
(12.11)
=−
∂
S
(
ϕ
)
·
η
,
∂
ϕ
Eqs. (12.10) and (12.11) can be combined to yield:
(
T
(
X
)
−
S
(
ϕ
))
∂
S
(
ϕ
)
DU
(
ϕ
)
·
η
=−
λ
·
η
.
(12.12)
∂
ϕ
This term drives the deformation of the template based on the pointwise differ-
ence in the image intensities and the gradient of the target intensity evaluated
at material points associated with the template.
