Biomedical Engineering Reference
In-Depth Information
distribution. Because of this, we use the voxel variance instead of the voxel precision
in this section. We define prior voxel variance of the
j
th voxel as
ʽ
j
, which is equal
T
. We rewrite
to 1
/ʱ
j
, and define the column vector
ʽ
such that
ʽ
=[
ʽ
1
,...,ʽ
N
]
the cost function in Eq. (
4.31
)using
ʽ
,
K
1
K
y
k
ʣ
−
1
F(
ʽ
)
=
log
|
ʣ
y
|+
y
k
,
(4.47)
y
k
=
1
where the model data covariance
ʣ
y
is expressed as
ʣ
y
=
ʲ
−
1
I
H
T
+
H
ʥ
,
(4.48)
ʥ
ʥ
=
and
is the covariance matrix of the voxel prior distribution, defined as
(
[
ʽ
1
,...,ʽ
N
]
)
diag
.
The first term in Eq. (
4.47
), log
|
ʣ
y
|
, is a concave function of
ʽ
1
,...,ʽ
N
. Thus,
for an arbitrary
ʽ
, we can find
z
that satisfies the relationship
z
T
ʽ
−
z
o
≥
log
|
ʣ
y
|
,
(4.49)
where
z
is the column vector
z
=[
z
1
,...,
z
N
]
, which contains auxiliary variables
z
j
(
j
N
), and
z
o
is a scalar term that depends on
z
.
The second term in Eq. (
4.47
) can be rewritten using
=
1
,...,
x
k
ʥ
−
1
x
k
y
k
ʣ
−
1
2
y
k
=
min
x
k
ʲ
y
k
−
Hx
k
+
.
(4.50)
y
The proof for the equation above is presented in Sect.
4.10.3
. Therefore, we have the
relationship
x
k
ʥ
−
1
x
k
K
K
1
K
1
K
y
k
ʣ
−
1
2
y
k
=
min
x
ʲ
y
k
−
Hx
k
+
.
(4.51)
y
k
=
1
k
=
1
Using auxiliary variables
z
and
x
(where
x
collectively expresses
x
1
,...,
x
K
),
we define a new cost function
F(
ʽ
,
,
)
x
z
, such that
ʲ
x
k
ʥ
−
1
x
k
K
1
K
F(
ʽ
,
2
z
T
x
,
z
)
=
y
k
−
Hx
k
+
+
ʽ
−
z
o
.
(4.52)
k
=
1
Equations (
4.49
) and (
4.51
) guarantee that the relationship
F(
ʽ
,
x
,
z
)
≥
F(
ʽ
),
always holds. That is, the alternative cost function
F(
ʽ
,
x
,
z
)
forms an upper bound
. When we minimize
F(
ʽ
,
of the true cost function
F(
ʽ
)
x
,
z
)
with respect to
ʽ
,
x
and
z
, such
ʽ
also minimizes the true cost function
F(
ʽ
)
.
