Biomedical Engineering Reference
In-Depth Information
4.8 Extension to Include Source Vector Estimation
In Sect.
4.6
, we derive the convexity-based algorithm when the source orientation
at each voxel is known, i.e., when the relationship between the sensor data and the
voxel source distribution is expressed as in Eq. (
4.3
).
In this section, we derive the convexity-based algorithm when the source orien-
tation at each voxel is unknown and the relationship,
y
k
=
Fx
k
+
ʵ
,
(4.69)
holds, where
x
k
is given by
s
1
(
T
s
N
(
x
k
=
t
k
), . . . ,
t
k
)
,
(4.70)
and
s
j
(
1 source vector at the
j
th voxel. In other words, we describe an
extension that enables estimating the source vector at each voxel. The algorithms in
the preceding sections use a diagonal prior covariance (or precision) matrix, and the
use of the diagonal matrix is possible because the source orientation is known. There-
fore, a naïve application of those algorithms to cases where the source orientation is
unknown generally results in the shrinkage of source vector components. Namely,
the naïve application possibly leads to a solution where only a single component of
a source vector has nonzero value and other components have values close to zero,
leading to incorrect estimation of source orientations.
Since the unknown parameter at each voxel is not a scalar quantity but the 3
t
k
)
is the 3
×
1
vector quantity in this section, the algorithm being developed here uses the nondi-
agonal covariance matrix of the prior distribution. That is, the prior distribution is
assumedtobe[
1
];
×
N
p
(
x
k
|
ʥ
)
=
1
N(
s
j
(
t
k
)
|
0
,
ʥ
j
),
(4.71)
j
=
where
ʥ
j
is a 3
×
3 covariancematrix of the source vector
s
j
(
t
k
)
. Thus, the covariance
ʥ
×
matrix of the voxel source distribution,
,isa3
N
3
N
block diagonal matrix
expressed as
⊡
⊤
ʥ
1
0
···
0
⊣
⊦
.
0
ʥ
2
···
0
ʥ
=
(4.72)
.
.
.
.
.
0
00
···
ʥ
N
The cost function in this case has the same form:
K
1
K
y
k
ʣ
−
1
F(
ʥ
)
=
log
|
ʣ
y
|+
y
k
.
(4.73)
y
k
=
1
