Biomedical Engineering Reference
In-Depth Information
The problemof estimating the sensor lead field is referred to as the bioelectromagnetic
forward problem. Arguments on how to compute the sensor lead field are presented
in Appendix A.
2.4 Voxel Source Model and Tomographic Source
Reconstruction
Using the lead-field matrix in Eq. (
2.4
), the relationship between the sensor data,
y
(
t
)
, and the source vector,
s
(
r
,
t
)
, is expressed as
y
(
t
)
=
L
(
r
)
s
(
r
,
t
)
d
r
.
(2.6)
ʩ
Here,
d
r
indicates the volume element, and the integral is performed over a volume
where sources are assumed to exist. This volume is called the source space, which
is denoted
ʩ
. Equation (
2.6
) expresses the relationship between the sensor outputs
.
The bioelectromagnetic inverse problem is the problem of estimating the source-
vector spatial distribution,
s
y
(
t
)
and the source distribution
s
(
r
,
t
)
(
r
,
t
)
, from the measurements,
y
(
t
)
. Here, we assume
that we know the sensor lead field
L
, although our knowledge of the sensor lead
field is to some degree imperfect because it must be estimated using an analytical
model or numerical computations.
When estimating
s
(
r
)
is
discrete in space. A common strategy here is to introduce voxel discretization over the
source space. Let us define the number of voxels as
N
, and the locations of the voxels
are denoted as
r
1
,
(
r
,
t
)
from
y
(
t
)
,
s
(
r
,
t
)
is continuous in space, while
y
(
t
)
r
2
,...,
r
N
. Then, the discrete form of Eq. (
2.6
) is expressed as:
N
N
y
(
t
)
=
L
(
r
j
)
s
(
r
j
,
t
)
=
L
(
r
j
)
s
j
(
t
).
(2.7)
j
=
1
j
=
1
where the source vector at the
j
th voxel,
s
for simplicity. We
introduce the augmented lead-field matrix over all voxel locations as
(
r
j
,
t
)
, is denoted
s
j
(
t
)
F
=[
L
(
r
1
),
L
(
r
2
),...,
L
(
r
N
)
]
,
(2.8)
which is an
M
×
3
N
matrix. We define a 3
N
×
1 column vector containing the source
vectors at all voxel locations,
x
(
t
)
, such that
⊡
⊤
s
1
(
t
)
⊣
⊦
.
s
2
(
t
)
x
(
t
)
=
(2.9)
.
s
N
(
t
)
