Biomedical Engineering Reference
In-Depth Information
ʣ
y
is given by
However,
N
ʣ
y
=
ʲ
−
1
I
L
j
ʥ
j
L
j
,
+
(4.74)
j
=
1
where the
M
×
3 lead fieldmatrix at the
j
th voxel,
L
(
r
j
)
, is denoted
L
j
for simplicity.
In agreement with Eq. (
4.49
), since log
|
ʣ
y
|
is a concave function, we can find
3
×
3 auxiliary parameter matrices,
Z
j
,(
j
=
1
,...,
N
) that satisfy
N
tr
Z
j
ʥ
j
−
Z
o
≥
log
|
ʣ
y
|
,
(4.75)
j
=
1
where
Z
o
is a scalar term that depends on
Z
j
. Regarding the second term in the
right-hand side of Eq. (
4.73
), we have the relationship
x
k
ʥ
−
1
x
k
y
k
ʣ
−
1
2
y
k
=
min
x
k
ʲ
y
k
−
Fx
k
+
y
⊡
⊣
ʲ
⊤
⊦
.
N
t
k
)
ʥ
−
1
2
s
j
(
=
min
x
k
y
k
−
Fx
k
+
s
j
(
t
k
)
(4.76)
j
j
=
1
Let us use
Z
to collectively express
Z
1
,...,
Z
N
, and use
s
to collectively express
s
j
(
t
k
)
, where
j
=
1
,...,
N
and
k
=
1
,...,
K
. The alternative cost function,
F(
ʥ
,
s
,
Z
)
, is obtained as
⊡
⊤
K
N
1
K
F(
ʥ
,
⊣
ʲ
2
t
k
)
ʥ
−
1
s
j
(
⊦
s
,
Z
)
=
y
k
−
Fx
k
+
s
j
(
t
k
)
j
k
=
1
j
=
1
tr
Z
j
ʥ
j
N
+
−
Z
o
.
(4.77)
j
=
1
This
F(
ʥ
,
forms an upper bound of the true cost function in Eq. (
4.73
). Accord-
ingly, when we minimize
F(
ʥ
,
s
,
Z
)
s
,
Z
)
with respect to
ʥ
,
s
, and
Z
, we can minimize
the true cost function
F(
ʥ
)
.
4.8.1 Update Equation for Z
j
Using the same arguments in Sect.
4.6.2
, the update equation for
Z
j
is derived as
∂
∂
ʥ
j
Z
j
=
log
|
ʣ
y
|
.
(4.78)
