Image Processing Reference
In-Depth Information
odd moments are much more complex. However, the confluent hypergeometric
function can be expressed in terms of modified Bessel functions, from which an
analytic expression of the odd moments can be derived. Hence, we have explicitly:
π
2
A
2
A
2
A
2
A
2
A
−
m
=
σ
e
1
+
I
+
I
,
(4.17)
E
2
4
σ
0
1
2
2
σ
4
σ
2
σ
4
σ
2
2
2
2
2
[]
=+σ
A
2
2
2
,
(4.18)
E
m
π
AA
I
2
4
A
2
2
A
−
[
3
=
σ
e
33
+
+
E
m
3
2
4
σ
0
2
σσ
2
2
4
4
σ
2
,
(4.19)
AA
I
A
2
4
2
+
2
+
σσ σ
1
2
4
2
4
2
4
[]
=+
A
4
8
σ
2
A
2
+ ,
8
σ
4
(4.20)
E
m
with
I
denoting the first-order modified Bessel function of the first kind.
1
4.2.2.3
Moments of the Rayleigh PDF
For completeness, we also mention the general expression for the moments of
the Rayleigh PDF, to which the Rician PDF tends at low SNR. For the Rayleigh
PDF, we have
,
ν
ν
[]( )
=
2
σ
22
ν
/
Γ
1
+
(4.21)
E
m
2
Explicitly, the first four moments are
π
σ
2
[
m
=
,
(4.22)
E
2
[]
= σ
2
2
,
(4.23)
E
m
π
σ
[
3
=
3
3
,
(4.24)
E
m
2
4
[]
= σ
8
4
.
(4.25)
E
m
4.2.2.4
Generalized Rician PDF
If magnitude data are computed from more than two Gaussian-distributed vari-
ables, the underlying PDF is the
. Such data are found
in phased array magnitude MR images, where use is made of multiple receiver
generalized Rician PDF
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