Biomedical Engineering Reference
In-Depth Information
Gray-White) can also be modeled by additional Gaussian laws to model partial-
volume effects. To do so, a mixture of seven models can be used instead.
In every case (EM or SEM algorithm, five or seven Gaussian models), we
model each class
κ
by a Gaussian distribution of mean
µ
κ
(respectively,
ν
κ
) for
image
I
1
(respectively, image
I
2
).
Parametric Correction.
To align the intensities of the anatomical tissues
and to interpolate smoothly the correction, we choose a polynomial correction
function of order
p
(see [63] for a similar modeling of intensity correction) such
that
g
p
(
x
)
=
i
=
p
i
=
0
i
x
i
i
θ
.
The coefficients
θ
are estimated such as to minimize
the following cost:
l
=
n
g
p
(
µ
j
)
−
ν
j
2
.
l
=
1
The intensity correction aims at aligning the mean values of each classes
while interpolating smoothly between the samples. This least-square problem
amounts to inverting a linear system of order
p
. The resulting correction can
then be applied to the voxel intensities of volume
I
1
.
8.3.2.4
Robust Estimators
Cost function Eq. (8.1) does not make any difference between relevant data
and inconsistent data, nor between neighboring pairs where the field is smooth
and neighboring pairs where the field is discontinuous. Therefore, we introduce
robust functions [77] and more precisely two robust M-estimators [14], the first
one on the data term and the second one on the regularization term. We do not
describe in details the properties of robust M-estimators, referring the reader to
[14, 98] for further explanations. The cost function (8.1) can then be modified as:
s
∈
S
ρ
1
(
∇
f
(
s
,
t
)
·
w
s
+
f
t
(
s
,
t
)
)
+
α
ρ
2
(
||
w
s
−
w
r
||
)
.
(8.2)
U
(
w
;
f
)
=
<
s
,
r
>
∈
C
According to some properties of robust M-estimators [14, 24], it can be shown
that the minimization of
U
(Eq. 8.1) is equivalent to the minimization of an
augmented function, noted
U
:
U
(
w
,δ,β
;
f
)
=
s
∈
S
δ
s
(
∇
f
(
s
,
t
)
·
w
s
+
f
t
(
s
,
t
))
2
+
ψ
1
(
δ
s
)
+
α
2
β
sr
||
w
s
−
w
r
||
+
ψ
2
(
β
sr
)
,
(8.3)
×
<
s
,
r
>
∈
C
