Biomedical Engineering Reference
In-Depth Information
mapping of source coherence—a source coherence image with respect to the seed
location—can be obtained. We can further scan not only the target location but also
the seed location to obtain a six-dimensional coherence image, called the voxel-to-
voxel coherence matrix.
Let us define the spectra of the seed and the target voxels as
˃
S
(
f
)
and
˃
T
(
f
)
,
respectively. The coherence
ˆ(
f
)
is obtained by computing the correlation of these
spectra,
)˃
S
(
˃
T
(
f
f
)
ˆ(
f
)
=
,
(7.4)
2
2
|
˃
T
(
f
)
|
|
˃
S
(
f
)
|
where the superscript
indicate
the ensemble average. In practical applications, this ensemble average is computed by
averaging across multiple trials. When only a single continuous data set is measured,
the single data set is divided into many trials and coherence is obtained by averaging
across these trials.
It is apparent in Eq. (
7.4
) that if the seed and target spectra contain common
components that do not result from true brain interactions, then the coherence may
contain spurious components. In source coherence imaging, the leakage of the imag-
ing algorithm is a major source of such spurious coherence.
∗
indicates the complex conjugate, and the brackets
·
7.3 Real and Imaginary Parts of Coherence
We here take a look at the nature of real and imaginary parts of coherence before
proceeding with the arguments of the leakage influence on the coherence imaging.
Let us define the time course from the seed voxel as
u
S
(
t
)
and the time course from
the target voxel as
u
T
(
t
)
. The cross correlation of the seed and target time courses,
R
(˄ )
, is then defined such that
∞
R
(˄ )
=
u
T
(
t
+
˄ )
u
S
(
t
)
=
u
T
(
t
+
˄ )
u
S
(
t
)
d
t
.
(7.5)
−∞
The cross correlation is related to the cross spectrum density through
∞
e
−
i
2
ˀ
f
˄
d
f
R
(˄ )
=
ʨ(
f
)
,
(7.6)
−∞
where the cross spectrum is
)˃
S
(
ʨ(
f
)
=
˃
T
(
f
f
)
.
(7.7)
