Image Processing Reference
In-Depth Information
coils [21], and in phase-contrast MR (PCMR) images [22,23]. In general, a
(random) PCMR pixel variable denoted by
m
can be written as
K
∑
1
2
,
m
=
s
(4.26)
k
k
=
with
K
denoting twice the number of orthogonal Cartesian directions in which
flow is encoded. The set contains independent Gaussian-distributed variables
with mean {
a
k
} and variance
{}
k
s
σ
2
. The deterministic signal component of the
PCMR pixel variable is given by
K
∑
1
A
=
a
2
,
(4.27)
k
k
=
The PDF of such a PCMR variable is given by
K
−
1
mm
m
2
+
A
2
m
A
2
p
()
m
=
exp
−
I
ε
()
m
.
(4.28)
m
K
σ
A
2
σ
σ
2
2
−
1
2
2
0 the PDF of the magnitude PCMR variable turns into a
gener-
alized Rayleigh PDF
:
When
A
→
2
2
m
K
−
1
m
2
p
()
m
=
exp
−
ε
(
m
)
.
(4.29)
m
2
σ
2
(
σ
)
Γ
(
K
/
2
)
K
Figure 4.2
shows the generalized Rician PDF for SNR
=
0 and SNR
=
3 and
for
K
=
2, 4, and 6 and
σ
=
1.
4.2.2.5
Moments of the Generalized Rician PDF
The general expression for the moments is written as
.
Γ
[(
K
+/
/
ν
)
2
]
ν
K
A
2
ν
[]( )
=
2
σ
22
ν
/
F
−,;−
(4.30)
E
m
11
Γ
(
K
2
)
2
2
2
σ
2
Again, the even moments turn out to be simple polynomials:
2
[]
=
KA
σ
2
+
2
.
(4.31)
E
m
4
[]
=
K
24
σ
+
2
K
σ
4
+
2
A K
2
σ
2
+
4
A
22
σ
+
A
4
.
(4.32)
E
m
For
A
=
0, we obtain the moments of the generalized Rayleigh PDF:
Γ
[(
K
+/
/
ν
)
2
]
ν
[]( )
=
2
σ
22
ν
/
.
(4.33)
E
m
Γ
(
K
2
)
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