Biomedical Engineering Reference
In-Depth Information
With these notations, at grid level
, the cost function can be modified as:
N
I
A
,
T
n
◦
T
0
B
|
n
U
(
;
A
,
B
,
T
0
,
w
)
=−
T
w
◦
n
=
1
⎡
⎣
m
∈
V
(
n
)
⎤
N
<
s
,
r
>
∈
C
nm
||
w
s
+
P
s
n
−
w
r
+
P
r
m
||
2
⎦
+
α
n
=
1
⎡
⎣
⎤
N
<
s
,
r
>
∈
C
n
||
w
s
+
P
s
n
−
w
r
+
P
r
n
||
2
⎦
,
(8.5)
+
α
n
=
1
where
B
|
n
denotes the restriction of volume
B
to the cube
n
. The minimization
is performed with Gauss-Seidel iterative solver (each cube is iteratively updated
while its neighbors are “frozen”). On each cube, Powell's algorithm [110, 111] is
used to estimate the parametric affine increment.
8.3.2.9
Implementation
The algorithm has been implemented in
C
++
using a template class for volu-
metric images.
2
A synopsis of the algorithm is presented in Fig. 8.3.
8.3.3
Results
8.3.3.1
Intensity Correction
We have evaluated the approach on various MR acquisitions. We present results
on real data of the intensity correction, comparing the EM and SEM approaches
and comparing the number of Gaussian laws used to model the histogram.
We have tested the approach on various T1-MR images and the algorithm has
proved to be robust and reliable. Furthermore, it does not require any spatial
alignment between the images to be corrected and can therefore be applied in
various contexts: MR time series or MR of different subjects. Figure 8.4 presents
cut-planes images of volumetric MR.
Figure 8.5 presents the effect of the correction using a EM algorithm and
Fig. 8.6 the correction using a SEM algorithm. For each estimation scheme, we
test a mixture of five (left) and seven Gaussian distributions to model the his-
togram. In each case, a fourth order parametric correction has been estimated.
2
http://www.irisa.fr/vista/Themes/Logiciel/VIsTAL/VIsTAL.html
