Biomedical Engineering Reference
In-Depth Information
Here, the hyperplane
j
=
1
tr
Z
j
ʥ
j
−
Z
o
forms a tightest upper bound of the
concave function log
|
ʣ
y
|
. Let us express the lead field matrix at the
j
th voxel using
its column vectors,
=
l
x
(
r
j
)
=
l
x
,
l
z
,
L
j
r
j
),
l
y
(
r
j
),
l
z
(
l
y
,
where explicit notation of the voxel location
is omitted. Using exactly the same
derivation from Eqs. (
4.55
)to(
4.58
), we can derive the update equation
(
r
j
)
∂
∂
ʥ
j
Z
j
=
log
|
ʣ
y
|
⊡
⊤
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
1
,
1
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
1
,
2
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
1
,
3
⊣
⊦
=
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
2
,
1
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
2
,
2
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
2
,
3
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
3
,
1
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
3
,
2
∂
log
|
ʣ
y
|
/∂
[
ʥ
j
]
3
,
3
⊡
⊤
l
x
ʣ
−
1
l
x
l
x
ʣ
−
1
l
y
l
x
ʣ
−
1
l
z
y
y
y
⊣
⊦
=
l
y
ʣ
−
1
l
x
l
y
ʣ
−
1
l
y
l
y
ʣ
−
1
L
j
ʣ
−
1
=
l
z
L
j
,
(4.79)
y
y
y
y
l
z
ʣ
−
1
l
x
l
z
ʣ
−
1
l
y
l
z
ʣ
−
1
l
z
y
y
y
[
ʥ
j
]
,
m
indicates the
(,
)
ʥ
j
.
where
m
th element of the matrix
4.8.2 Update Equation for s
j
(
t
k
)
The update equation for
x
k
can be obtained using
x
k
ʥ
−
1
x
k
2
x
k
=
argmin
x
k
ʲ
y
k
−
Fx
k
+
.
(4.80)
The solution is given by Eq. (
4.61
) with replacing
H
with
F
, which is rewritten as
⊡
⊤
⊡
⊤
⊡
⊤
L
1
L
2
L
N
s
1
(
t
k
)
ʥ
1
0
···
0
⊣
⊦
=
⊣
⊦
⊣
⊦
ʣ
−
1
s
2
(
t
k
)
0
ʥ
2
···
0
y
k
.
(4.81)
.
.
.
0
.
.
.
y
s
N
(
t
k
)
00
···
ʥ
N
Therefore, the source vector at the
j
th voxel
s
j
(
t
k
)
is given by
t
k
)
=
ʥ
j
L
j
ʣ
−
1
s
j
(
y
k
,
(4.82)
y
and Eq. (
4.82
) is the update equation for
s
j
(
t
k
)
.
