Biomedical Engineering Reference
In-Depth Information
The positive definite, symmetric right and left Cauchy-Green deformation
tensors are, respectively,
C
=
F
T
F
=
U
2
and
B
=
F
F
T
=
U
2
(12.2)
.
The Jacobian
J
is defined as:
J
:
=
det
F
=
ρ
0
(12.3)
ρ
.
12.2.2
Variational Framework
Most deformable image registration methods can be posed as the minimization
of an energy functional
E
that consists of two terms. This can be defined with
respect to the reference or current (deformed) configuration as:
E
(
X
,
ϕ
)
=
W
(
X
,
ϕ
)
dV
−
U
(
T
(
X
,
ϕ
)
,
S
(
X
,
ϕ
))
dV
β
0
β
0
(12.4)
.
ϕ
)
d
J
−
ϕ
))
d
J
=
W
(
X
,
U
(
T
(
X
,
ϕ
)
,
S
(
X
,
β
β
Here,
W
is an energy term that provides regularization and/or some type of
constraint on the deformation map (e.g., one-to-one mapping or no negative
volumes admitted), while
U
represents an energy that depends on the image data
in the template and target images.
β
0
and
β
represent the volumes of integration
in the reference and current configurations, respectively.
The Euler-Lagrange equations are obtained by taking the first variation of
E
(
X
,
ϕ
. This can be thought of as a “virtual
displacement” - a small variation in the current coordinates
x
, denoted
ε
η
.
Here
ε
is an infinitesimal scalar. The first variation of the first energy term in
Eq. (2.4) defines the forces per unit volume that arise from the regularization.
The second energy term in Eq. (2.4) gives rise to an image-based force term. The
first variation of Eq. (2.4) with respect to the deformation
ϕ
) with respect to the deformation
ϕ
(
X
) in direction
η
is
denoted:
d
J
G
(
ϕ,
η
):
=
DE
(
ϕ
)
·
η
=
DW
(
X
,
ϕ
)
·
η
β
(12.5)
d
J
=
0
.
DU
(
T
(
X
,
ϕ
)
,
S
(
X
,
ϕ
))
·
η
+
β
