Biomedical Engineering Reference
In-Depth Information
forces arising from the differences in the image intensities and gradients in
Eq. (12.12), and
F
int
is the vector of internal forces resulting from the stress di-
vergence. The material and geometric stiffnesses combine to give the mechanics
regularization stiffness:
K
R
(
B
NL
)
T
σ
B
NL
d
ν
+
(
B
L
)
T
c
B
L
d
ν.
(12.22)
=
β
β
The contribution of the image-based energy to the tangent stiffness is:
N
T
k
N
d
J
.
K
I
(12.23)
=−
β
Together, the terms in Eq. (12.22) and Eq. (12.23) form the entire tangent stiffness
matrix. In our FE implementation, an initial estimate of the unknown incremen-
tal nodal displacements is obtained by solving Eq. (12.21) for
u
and this solution
is improved iteratively using a quasi-Newton method [27].
12.2.6
Solution Procedure and Augmented Lagrangian
In the combined energy function in Eq. (12.9), the image data may be treated
as either a soft constraint, with the mechanics providing the “truth”, as a hard
constraint, with the mechanics providing a regularization, or as a combination.
For typical problems in deformable image registration, it is desired to treat the
image data as a hard constraint. Indeed, the form for
U
specified in Eq. (12.8) is
essentially a penalty function stating that the template and target image intensity
fields must be equal over the domain of interest as
λ
→∞
. The main problem
with the penalty method is that as the penalty parameter
λ
is increased, some
of the diagonal terms in the stiffness matrix
K
I
become very large with respect
to others, leading to numerical ill-conditioning of the matrix. This results in
inaccurate estimates for
K
−
1
, which leads to slowed convergence or divergence
I
of the nonlinear iterations.
To circumvent this problem, the augmented Lagrangian method is used [33,
35]. With augmented Lagrangian methods, a solution to the governing equations
at a particular computational timestep is first obtained with a relatively small
penalty parameter
λ
. Then the total image-based body forces
∂
U
/∂
ϕ
are incre-
mentally increased in a second iterative loop, resulting in progressively better
satisfaction of the constraint imposed by the image data. This leads to a stable
