Biomedical Engineering Reference
In-Depth Information
is given by
x
∗
|
2
a
H
2
|
v
|
ʣ
v
x
b
|
2
|
ˆ
R
|
=
=
.
(7.95)
b
H
a
H
vv
∗
x
∗
[
ʣ
vv
a
][
ʣ
xx
b
]
x
2
is defined as the maximum of
2
,
The canonical squared residual coherence
|
ˈ
R
|
|
ˆ
R
|
which is obtained by solving the optimization problem,
ʣ
v
x
b
2
2
a
H
subject to
a
H
1 and
b
H
|
ˈ
R
|
=
max
a
,
ʣ
vv
a
=
ʣ
xx
b
=
1
.
(7.96)
,
b
This maximization problem is exactly the same as that in Eq. (
7.60
), and the solu-
tion of this maximization is known to be the maximum eigenvalue of the matrix
ʣ
−
1
vv
ʣ
v
x
ʣ
−
1
H
xx
ʣ
v
x
. That is, the canonical residual coherence is derived as
2
=
S
max
{
ʣ
−
1
vv
ʣ
v
x
ʣ
−
1
H
|
ˈ
R
|
xx
ʣ
v
x
}
,
(7.97)
ʣ
vv
ʣ
v
x
is derived using Eq. (
7.94
).
where
is derived using Eq. (
7.93
), and
7.6.4 Computing Coherence When Each Voxel
has Multiple Time Courses
When the source vector has
x
,
y
, and
z
components, most source imaging algorithms
generate three time courses corresponding to the
x
,
y
, and
z
components at each
voxel. A usual way is to obtain a single representative time course at each voxel and
compute coherence using such representative time courses, as described in Sect.
7.2
.
Here, we describe an alternative method to compute voxel coherence when each
voxel has multiple time courses [
13
].
The method is a straightforward application of the canonical coherence. Let us
assume that the voxel time courses are expressed as in Eq. (
7.1
). The voxel spectra
at the target and seed voxels are respectively denoted using 3
×
1 vectors
x
and
y
,
which are given by
⊡
⊤
⊡
⊤
x
1
(
f
)
y
1
(
f
)
⊣
⊦
⊣
⊦
,
x
(
f
)
=
x
2
(
f
)
and
y
(
f
)
=
y
2
(
f
)
(7.98)
x
3
(
f
)
y
3
(
f
)
where
x
1
(
are the spectra obtained from the source time courses
in the
x
,
y
and
z
directions at the target voxel, and
y
1
(
f
)
,
x
2
(
f
)
, and
x
3
(
f
)
)
,
y
2
(
)
, and
y
3
(
)
are
the spectra obtained from the source time courses in the
x
,
y
and
z
directions at
the seed voxel. Using these
x
and
y
, we compute canonical magnitude coherence
|
ˈ
|
f
f
f
|
ˈ
|
ˈ
I
in
Eq. (
7.83
)or
ˈ
I
in Eq. (
7.85
). This method is equivalent to determining the source
in Eq. (
7.66
)or
in Eq. (
7.68
), and the canonical imaginary coherence
