Biomedical Engineering Reference
In-Depth Information
where
x
k
u
k
H
c
=[
H
,
B
]
and
z
k
=
.
Here,
H
c
and
z
k
can respectively be called the extended lead field matrix and the
extended source vector. The second step applies the Champagne algorithm to the
target data using the extended lead fieldmatrix
H
c
where
B
is given from the first step.
The prior probability for the extended source vector
z
k
is:
,
ʦ
−
1
p
(
z
k
)
=
p
(
x
k
)
p
(
u
k
)
=
N(
x
k
|
0
)N(
u
k
|
0
,
I
)
x
k
exp
exp
2
u
k
u
k
1
/
2
1
/
2
2
1
2
x
k
ʦ
I
2
1
=
−
−
ˀ
ˀ
exp
z
k
1
/
2
2
1
2
z
k
ʦ
,
ʦ
),
=
−
=
N(
z
k
|
0
(4.45)
ˀ
matrix
ʦ
where the
(
N
+
L
)
×
(
N
+
L
)
is the prior precision matrix of the extended
source vector, expressed as
⊡
⊤
ʱ
1
···
00
···
0
.
.
.
.
.
.
.
.
⊣
⊦
ʦ
0
0
···
ʱ
N
0
···
0
ʦ
=
=
.
(4.46)
0
I
0
···
01
···
0
.
.
.
.
.
.
.
···
0
···
00
···
1
In this modified version of the Champagne algorithm, we use the same update equa-
tion for
replaced with
ʦ
ʱ
with
ʦ
. Thus, when updating the hyperparameter
ʱ
, only
components up to the
N
th diagonal element of
ʦ
are updated, and the rest of the
elements are fixed to 1.
4.6 Convexity-Based Algorithm
In this section, we describe an alternative algorithm that minimizes the cost function
in Eq. (
4.31
). The algorithm is called the convexity-based algorithm [
1
,
2
]. It is faster
than the EM algorithm. Unlike the MacKay update, this algorithm is guaranteed to
converge. The algorithm also provides a theoretical basis for the sparsity analysis
described in Sect.
4.7
.
4.6.1 Deriving an Alternative Cost Function
The convexity-based algorithm makes use of the fact that log
|
ʣ
y
|
is a concave
function of 1
/ʱ
1
,...,
1
/ʱ
N
, which is the voxel variance of the prior probability
