Biomedical Engineering Reference
In-Depth Information
only necessary to compute the transformations and the inverse of the trans-
formations at the discrete voxel locations. Let
d
denote the discrete center
locations of the voxels in the coordinate system
. The discrete inverse trans-
formation is computed using the following procedure only at the discrete voxel
points.
For each
y
∈
d
do
{
Set
δ
=
[1
,
1
,
1]
T
,
x
=
y
, iteration = 0.
While (
||
δ
||
>
threshold) do
{
δ
=
y
−
h
(
x
)
x
=
x
+
2
iteration
=
iteration
+
1
if (iteration
>
max iteration) then
Report algorithm failed to converge and exit.
h
−
1
(
y
)
=
x
}
The threshold is typically set between 10
−
2
and 10
−
4
and the maximum num-
ber of iterations is set to 1000. In practice, the algorithm converges when the
minimum Jacobian of
h
is greater than zero although we have not proved this
mathematically. Reducing the value of the threshold gives a more accurate in-
verse but increases the iteration time. This algorithm normally converges quickly
and is computationally efficient. However, this algorithm has a tendency to get
stuck in osciations and is detected by the if statement. The inverse at these oscil-
latory points can be estimated using gradient descent to solve the minimization
||
y
−
h
(
x
)
||
for
x
keeping
y
fixed. Alternatively, the failure of the algorithm to
converge at a point can be ignored since it will not have a signficant effect on
the registration and will be corrected at the next iteration.
6.2.5
Regularization Constraint
Minimizing the cost function in Eq.(6.2) does not ensure that the transforma-
tions
h
and
g
are diffeomorphic transformations except for when
C
ICC
=
0.
Continuum mechanical models such as linear elasticity [22, 29] and viscous
fluid [15, 22] can be used to regularize the transformations. For example, a
