Biomedical Engineering Reference
In-Depth Information
Equation (
2.7
) is then rewritten as
⊡
⊤
s
1
(
t
)
⊣
⊦
=
s
2
(
t
)
y
(
t
)
=[
L
(
r
1
),
L
(
r
2
),...,
L
(
r
N
)
]
Fx
(
t
).
(2.10)
.
s
N
(
t
)
Here, since the augmented lead-field matrix
F
is a known quantity, the only unknown
quantity is the 3
N
×
1 column vector,
x
(
t
)
. This vector
x
(
t
)
is called the voxel source
vector.
The spatial distribution of the source orientation,
ʷ
(
)
, may be a known quantity
if accurate subject anatomical information (such as high-precision subject MRI) can
be obtained with accurate co-registration between the MRI coordinate and the sensor
coordinate. In this case, the inverse problem is the problem of estimating the source
magnitude,
s
r
. Let us consider a situation
in which the source orientations at all voxel locations are predetermined. Defining
the orientation of a source at the
j
th voxel as
(
r
,
t
)
, instead of the source vector,
s
(
r
,
t
)
ʷ
j
, the lead field at the
j
th voxel is
expressed as the column vector
l
j
, which is obtained as
l
j
=
L
(
r
j
)
ʷ
j
,
according to
Eq. (
2.5
). Thus, the augmented lead field is expressed as an
M
×
N
matrix
H
defined
such that
H
=[
L
(
r
1
)
ʷ
1
,
L
(
r
2
)
ʷ
2
,...,
L
(
r
N
)
ʷ
N
]=[
l
1
,
l
2
,...,
l
N
]
,
(2.11)
whereby Eq. (
2.10
) can be reduced as follows:
⊡
⊣
⊤
⊦
s
1
(
t
)
s
2
(
t
)
y
(
t
)
=[
L
(
r
1
),
L
(
r
2
),...,
L
(
r
N
)
]
.
s
N
(
t
)
⊡
⊤
⊡
⊤
ʷ
1
0
···
0
s
1
(
t
)
⊣
.
⊦
⊣
⊦
s
2
(
t
)
0
ʷ
2
·
=[
L
(
r
1
),
L
(
r
2
),...,
L
(
r
N
)
]
.
s
N
(
.
.
.
.
·
0
)
t
0
···
0
ʷ
N
⊡
⊤
s
1
(
t
)
⊣
⊦
=
s
2
(
t
)
=[
L
(
r
1
)
ʷ
1
,
L
(
r
2
)
ʷ
2
,...,
L
(
r
N
)
ʷ
N
]
Hx
(
t
).
(2.12)
.
s
N
(
t
)
