Biomedical Engineering Reference
In-Depth Information
algorithm that allows the constraint to be satisfied to a user-defined tolerance.
Ill conditioning of the stiffness matrix is entirely avoided.
The Euler-Lagrange equations defined in Eq. (12.13) are modified by the
addition of a term that represents the additional image-based force
γ
due to the
augmentation:
d
J
=
0
G
∗
=
G
(
ϕ,
η
)
+
(12.24)
β
γ
· η
The solution procedure involves incrementally increasing
γ
at each computa-
tional timestep and then iterating using a quasi-Newton method [27] until the
energy is minimized. In the context of the FE method described above, the
augmented Lagrangian update procedure for timestep
n
+
1 takes the form:
0
γ
n
+
1
=
γ
n
k
=
0
DO for each augmentation
k
WHILE
(
γ
k
+
1
n
+
1
−
γ
k
k
n
+
1
)
/γ
n
+
1
> TOL
(12.25)
Minimize
G
∗
with
γ
n
+
1
fixed using the BFGS method
Update multipliers using
γ
k
+
1
n
+
1
=
γ
k
ϕ
)
n
+
1
n
+
1
+
(
∂
U
/∂
END DO
This nested iteration procedure, referred to as the Uzawa algorithm [36, 37),
converges quickly in general because the multipliers
γ
are fixed during the
minimization of
G
∗
. In practice, the augmentations are not performed until the
penalty parameter
λ
has been incremented to the maximum value that can be
obtained without solution difficulties due to ill conditioning. At this last timestep,
the augmented Lagrangian method is then used to satisfy the constraint to a user
defined tolerance (usually TOL
=
0.05).
12.2.7
Sequential Spatial Filtering to Overcome
Local Minima
The solution approach described above follows the local gradient to search for a
minimum in the total energy (Eq. 12.4) and therefore it is susceptible to converg-
ing to local minima. This means that the registration process may get “stuck”
by alignment of local image features that produce forces locking the deforma-
tion into a particular configuration. It is often possible to avoid local minima
and converge to a global minimum by first registering larger image features,
