Image Processing Reference
In-Depth Information
denotes the variance around that mean for tissue
k
, and
1
x
2
G
()
x
=
exp
−
σ
2
σ
2
2
πσ
2
is a zero-mean Gaussian distribution with variance
. It is further assumed that
the prior (before the image data is seen) probability for a certain tissue type
σ
2
k
in
the
i
th voxel is identical for all voxels
pl
(
==π
k
)
(5.2)
i
k
where is assumed to be known a priori. Finally, knowledge about the spatial
smoothness of MR inhomogeneities is incorporated by modeling the entire inho-
mogeneity field, denoted as
π
k
βββ
=,
[
12
…
,
,
β
] ,
T
where
N
is the number of voxels
N
in the image, by a
N
-dimensional zero-mean Gaussian prior probability density
p
()
β
=
G
()
β
(5.3)
Σ
β
where
.
1
1
2
Gx
()
=
exp
−
x
T
Σ
−
1
x
Σ
(
2
π
)
det(
Σ
)
N
Using Bayes' rule, the posterior probability of the inhomogeneity field in a
given log-transformed intensity image
yyy…
N
=,
[
,
,
]
T
is given by the distri-
12
bution
py
(| )()
(| )( )
ββ
ββ
p
py
(|
β
)
=
(5.4)
∫
py
′
p
′
β
′
where
p
(
β
) depends on the parameter
Σ
through Equation 5.3, and where
β
∏
py
(| )
β
=
py
( | )
β
i
i
i
with
∑
py
(|
β
)
=
py l
(|
=
k
,
β
)(
pl
=
k
)
(5.5)
i
i
i
i
i
i
k
through Equation 5.1 and Equation 5.2.
Before Equation 5.4 can be used to assess the inhomogeneity field, appropriate
depends on the parameters
µ
,
σ
, and
π
k
k
k
Search WWH ::
Custom Search