Biomedical Engineering Reference
In-Depth Information
7.2 Source-Space Coherence Imaging
In the source-space analysis, the first step estimates voxel time courses using an
inverse algorithm. Since the source is a three-dimensional vector with three (
x
,
y
,
and
z
) components,
1
most source reconstruction algorithms produce component-
wise, multiple time courses at each voxel. Accordingly, we should compute a “rep-
resentative” single time course by projecting the multiple time courses onto the
direction of the source orientation.
However, in practical applications, the source orientation is generally unknown,
and it must be estimated from the data. One quick and easy way to estimate the source
orientation is to use the direction that maximizes the reconstructed voxel power. The
reconstructed source-time courses at the
j
th voxel is expressed as
⊡
⊣
⊤
⊦
,
s
x
(
r
j
,
t
1
)
s
x
(
r
j
,
t
2
)
···
s
x
(
r
j
,
t
K
)
S
j
=
s
y
(
r
j
,
t
1
)
s
y
(
r
j
,
t
2
)
···
s
y
(
r
j
,
t
K
)
(7.1)
s
z
(
r
j
,
t
1
)
s
z
(
r
j
,
t
2
)
···
s
z
(
r
j
,
t
K
)
where we assume that the data is collected at time points,
t
1
,
t
2
,...,
t
K
, and
r
j
indicates the location of the
j
th voxel. Denoting the orientation of the source at the
j
th voxel as
ʷ
j
, the estimate of
ʷ
j
,
ʷ
j
, is obtained using the following maximization:
j
j
S
j
S
T
T
ʷ
j
=
argmax
ʷ
j
ʷ
ʷ
j
.
The optimum estimate is obtained as the eigenvector corresponding to the maximum
eigenvalue of a matrix
S
j
S
T
, i.e.,
j
=
ˑ
max
{
S
j
S
T
ʷ
j
j
}
,
(7.2)
where the notation
, defined in Sect. C.9, indicates the eigenvector corre-
sponding to the maximum eigenvalue of a matrix between the parentheses. The
representative time course at the
j
th voxel,
u
j
(
ˑ
max
{·}
)
t
, is obtained using
u
j
(
t
K
)
=
ʷ
j
S
j
.
T
t
1
),
u
j
(
t
2
), . . . ,
u
j
(
(7.3)
Once a representative voxel time course is obtained at each voxel, the next step
is to compute a voxel-pairwise coherence. This step involves first setting a reference
point, called the seed point, and computing the coherence between the time course
from the seed point and that from another voxel's location, referred to as the target
location. By scanning through all target locations in a brain, a three-dimensional
1
Although the source has two components when the homogeneous spherical conductor model
is used, for arguments in this chapter, we assume that the source vector has three
x
,
y
,and
z
components.
