Biomedical Engineering Reference
In-Depth Information
T
=
ˉ
−
ˉ
˃
ˉ
,
(7.109)
2
ˉ
ˉ
(
ˉ
˃
=
,...,
where
and
are the average and the variance of
1
B
). Since these
2
ˉ
ˉ
and
˃
are obtained at each voxel, the values at the
j
th voxel are denoted
ˉ(
j
)
2
. The maximum value of
T
obtained at the
j
th voxel is denoted
T
max
and
˃
ˉ
(
j
)
(
j
)
.
Defining a total number of voxels as
N
V
,wehave
T
max
T
max
(
1
), . . . ,
(
N
V
)
to form
a null distribution. We then sort these values in an increasing order:
T
max
(
1
T
max
(
2
T
max
(
N
V
),
)
≤
)
≤ ··· ≤
(
k
where
T
max
is the
k
th minimum value.
We set the level of the statistical significance to
)
, and choose
T
max
ʱ
(
˜
p
)
where
p
˜
=
ʱ
N
V
and
ʱ
N
V
indicates the maximum integer not greater than
ʱ
N
V
.The
th
threshold value for the
j
th voxel,
ˉ
(
j
)
, is finally derived as
th
T
max
ˉ
(
j
)
=
(
˜
p
)˃
ˉ
(
j
)
+
ˉ(
j
)
.
(7.110)
At the
j
th voxel, we evaluate the statistical significance of the imaginary coherence
value by comparing it with
th
th
,it
is considered to be statistically significant; if not, it is considered to be statistically
insignificant.
ˉ
(
j
)
. When the metric value is greater than
ˉ
(
j
)
7.9 Mean Imaginary Coherence (MIC) Mapping
Guggisberg et al. [
3
] have proposed to compute a metric called the mean imaginary
coherence. Defining the coherence computed between the
j
th and
k
th voxels as
ˆ
j
,
k
(
f
)
, the mean imaginary coherence for the
j
th voxel,
M
j
(
f
)
, is obtained using
⊡
⊤
N
V
tanh
−
1
|
(ˆ
j
,
k
(
)
|
1
N
V
⊣
⊦
.
M
j
(
f
)
=
tanh
f
(7.111)
k
=
1
|
(ˆ
j
,
k
(
On the right-hand side, the absolute value of the imaginary coherence
f
))
|
is averaged across all voxel connections. In Eq. (
7.111
),
2
log
1
e
2
z
1
+
r
−
1
tanh
−
1
z
=
(
r
)
=
and
r
=
tanh
(
z
)
=
e
2
z
1
−
r
+
1
are the inverse hyperbolic and hyperbolic functions, respectively. The idea of using
these functions is to average the voxel coherence values in the Fisher's Z-transform
domain. We may use the corrected imaginary coherence instead of using the imagi-
nary coherence in Eq. (
7.111
).
