Biomedical Engineering Reference
In-Depth Information
then the deformation
g
is locally invertible. Evaluating this constraint at pixel
coordinates and converting the strict constraints into soft ones using a barrier
function yields the following penalty term
e
−
α
det(
∇
x
g
(
i
))
E
p
=
(9.22)
i
∈
I
Experience shows that, for typical data, this term is never important at the
solution point (to which the optimization converges). It mostly becomes useful
at the beginning of the optimization process when the trial points vary a lot,
especially with some optimizers. In such cases, the penalty term forces the
algorithm to stay in the region of invertible deformations.
Depending on the particular task and the expected properties of the solu-
tion, various regularization terms can be used. We investigated, for example,
a stabilizer penalizing non-linear deformations
∂
2
N
2
g
k
∂
x
l
∂
x
m
E
t
=
d
x
(9.23)
k
,
l
,
m
=
1
and a very simple norm measuring the distance of
g
from identity through the
coefficients
c
j
c
j
2
E
d
=
(9.24)
When the corresponding weight
γ
is small, the regularization mainly
smoothes the deformation function in places where little information is present
in the images. As it gets bigger, the regularization gradually overrides the data
term and the deformation tends towards a smooth function in the sense of the
particular regularization. An alternative to regularization is the virtual spring
mechanism described in section 9.4.7.
9.4.10
Experiments
We now illustrate the application of the presented algorithm to various problems
involving medical images of several modalities. We refer the interested reader
to [88,95,96], where we study in detail the accuracy, speed and robustness of the
algorithm by means of a comprehensive series of experiments in a controlled
environment.
