Biomedical Engineering Reference
In-Depth Information
a recursive manner. Since the auxiliary variable
x
k
is the posterior mean of the voxel
source distribution, the value of the auxiliary variable
x
k
is equal to the Bayesian
estimate of the source distribution,
x
k
, after the iterative procedure is terminated.
¯
4.7 The Origin of the Sparsity
Why does the Champagne algorithm produce sparse solutions? This section tries to
answer this question, and consider the origin of the sparsity by analyzing the cost
function of the Champagne algorithm [
7
]. For simplicity, we set
K
1, and by
omitting the time index,
x
1
and
y
1
are denoted
x
and
y
.UsingEq.(
4.47
), we have
=
log
y
y
T
ʣ
−
1
y
ʽ
=
argmin
ʽ
|
ʣ
y
|+
.
As shown in Eq. (
4.50
), we have
ʥ
−
1
x
ʣ
−
1
y
y
T
2
x
T
y
=
min
x
ʲ
y
−
Hx
+
.
Combining the two equations above, the cost function for estimating
x
is given by
2
x
=
argmin
x
F
:
F
=
ʲ
y
−
Hx
+
ˆ(
x
),
(4.64)
where the constraint
ˆ(
x
)
is expressed as
⊛
⊞
x
T
N
x
j
ʽ
j
+
ʥ
−
1
x
⊝
⊠
.
ˆ(
x
)
=
min
ʽ
+
log
|
ʣ
y
|
=
min
ʽ
log
|
ʣ
y
|
(4.65)
j
=
1
ʽ
as in Eq. (
4.48
). Therefore, we further introduce a simplification by assuming that the
columns in the matrix
H
are orthogonal, i.e., the relationship
l
i
However, computing
ˆ(
x
)
in Eq. (
4.65
) is not so easy because log
|
ʣ
y
|
contains
l
j
=
I
i
,
j
is assumed
to hold. In this case, using Eq. (
4.48
), we get
N
(ʲ
−
1
log
|
ʣ
y
|=
log
+
ʽ
j
),
j
=
1
and, accordingly,
x
j
ʽ
j
+
N
(ʲ
−
1
ˆ(
x
)
=
min
ʽ
log
+
ʽ
j
)
.
(4.66)
j
=
1
