Image Processing Reference
In-Depth Information
these
N
complex data points:
N
N
∑∑
1
(
)
y
=
m
2
=
ωω
,
2
+
2
.
(4.34)
n
rn
in
,
n
=
n
=
1
Then, it can be shown that the PDF of
y
is given by
N
−
1
2
y
µ
1
y
2
y
+
µ
σ
−
py
()
=
e
I
,
(4.35)
2
2
y
N
−
1
2
σµ
σ
2
2
2
∑
where [22]. Note that Equation 4.35 is the PDF of the sum
of 2
N
Gaussian-distributed variables. Also, note the following:
µ
2
=
(
µ
2
+
µ
2
)
rn
,
in
,
n
•
If the variance of the Gaussian-distributed components equals one, the
PDF given in Equation 4.35 turns into a noncentral chi-squared distri-
bution. The mean is given by
N
+
µ
2
. The variance is given by
2(2
µ
2
+
N
).
•
If, in addition, the mean of the components equals zero, it turns into
the chi-squared distribution with 2
N
degrees freedom and with mean
and variance equal to
N
and 2
N
, respectively.
4.2.3
PDF
OF
P
HASE
D
ATA
Phase data, which are commonly obtained during flow imaging, are constructed
from the real and imaginary observations
w
{(
w
,
)}
by calculating the arctan-
rn
,
in
,
gent of their ratio for each complex data point:
ω
ω
in
,
φ
=
arctan
.
(4.36)
n
rn
,
The PDF of the phase deviation
∆φ
from the true phase value is given by [24]
A
cos
∆
φ
1
2
A
∫
σ
p
()
∆=
φ
e
−/
A
2
2
σ
2
1
+
s
∆
φ
e
A
2
cos
2
∆/
φ σ
2
2
e x
−/
x
2
2
.
(4.37)
∆
ϕ
π
σ
−∞
Note that the distribution can be expressed solely in terms of the SNR, defined
as
A
/
σ
. A graphical representation of the phase difference PDF as a function of
the SNR is shown in
Figure 4.3
. Although the general expression for the distri-
bution of
∆φ
is complicated, the two limits of the SNR turn out to yield simple
distributions:
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