Biomedical Engineering Reference
In-Depth Information
7.4 Effects of the Leakage
7.4.1 Leakage Effects on the Magnitude Coherence
We analyze the effects of leakage of imaging algorithms in source coherence imaging.
When the leakage is taken into account, the estimated source time courses at the seed
and the target voxels,
u
S
(
t
)
and
u
S
(
t
)
, can be expressed as
u
S
(
t
)
=
u
S
(
t
)
+
d
1
u
T
(
t
)
+
c
S
(
t
),
(7.13)
u
T
(
t
)
=
u
T
(
t
)
+
d
2
u
S
(
t
)
+
c
T
(
t
).
(7.14)
where
u
S
(
are the true source time courses of the seed and the target
locations. In the above equations,
d
1
u
T
(
t
)
and
u
T
(
t
)
t
)
indicates the leakage of the target signal
in the estimated seed signal, and
d
2
u
S
(
indicates the leakage of the seed signal in
the estimated target signal. The real-valued constants
d
1
and
d
2
express the relative
amount of these leakages.
In the equations above,
c
S
(
t
)
express the interference terms, which may
include the leakage from the third source, contributions from external disturbances,
and that of the sensor noise. The influence of these interference terms can be consid-
ered separately from the influence of the leakage terms [
7
]. In the arguments here,
ignoring
c
S
(
t
)
and
c
T
(
t
)
t
)
and
c
T
(
t
)
, the estimated spectra at the seed and target voxels,
˃
S
(
f
)
˃
T
(
)
and
f
, are expressed as
˃
S
=
˃
S
+
d
1
˃
T
,
(7.15)
˃
T
=
˃
T
+
d
2
˃
S
.
(7.16)
where we omit the explicit notation of
for simplicity.
The magnitude coherence between the seed and the target voxels is expressed as
(
f
)
.
˃
T
˃
S
|
ˆ
|=
(7.17)
2
2
|
˃
T
|
|
˃
S
|
Using Eqs. (
7.15
) and (
7.16
), we have
˃
T
˃
S
=
˃
T
˃
S
+
2
2
d
1
d
2
˃
S
˃
T
,
d
1
|
˃
T
|
+
d
2
|
˃
S
|
+
(7.18)
2
d
1
˃
T
˃
S
,
2
2
d
1
|
˃
T
|
2
|
˃
T
|
=|
˃
S
|
+
+
(7.19)
2
d
2
˃
T
˃
S
.
2
2
d
2
|
˃
S
|
2
|
˃
S
|
=|
˃
T
|
+
+
(7.20)
The equations above show that even when there is no true source interaction, i.e.,
even when
|
ˆ
|
˃
T
˃
S
=
˃
S
˃
T
=
0 and
0,
has a nonzero value, which is equal to
